THE LEVELS OF MODULAR REPRESENTATIONS

Kevin Buzzard Trinity College, Cambridge.

A dissertation submitted for the degree of Doctor of Philosophy

at the University of Cambridge To my Father

Declaration.

Except where otherwise stated, this dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration.

Acknowledgements.

I would like to thank my supervisor, Dr Richard Taylor, for the endless amount of time and energy which he has devoted to me over the last three years. Without him this dissertation could not possibly exist.

I would also like to thank various other mathematicians for the help they have given me whilst I was learning the background material for this thesis. The learning process would have taken a lot longer were it not for the helpful comments of Carayol, Diamond, Ribet, Shepherd-Barron and others. Special thanks are due in particular to Gross and Edixhoven for their helpful comments on the papers [G] and [E] respectively.

Finally, thanks to Karsten, Tamzin, Steve, Lance and others for moral support and encouragement during the last three years.

Contents.

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1. The levels of modular representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2. Summary of this dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Chapter 2. Curves and Jacobians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 2.1. Curves over finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2. Abelian varieties over finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3. Dieudonné modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4. Curves over local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Chapter 3. Modular curves, Igusa curves and modular forms. . . . . . . . . . . . . . . . .26 3.1. Morphisms of moduli problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2. Modular curves and Igusa curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3. Modular forms and q-expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4. Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5. Differentials and regular differentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 3.6. Representations attached to cusp forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

Chapter 4. Adding the first prime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1. Statement of the main theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44 4.2. Maps between modular curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3. Beginning of proof of Theorem 4.1.3 for weights 3 ≤ k ≤ l. . . . . . . . . . . . . . . . . . . .50 4.4. Interlude: definite quaternion algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.5. 3 ≤ k ≤ l: End of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.6. The cases k = 2 and k = l + 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.7. Proof of Theorem 4.1.3 in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Chapter 5. Adding more primes to the level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.1. Indefinite quaternion algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2. False elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3. Shimura curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.4. New structures at l. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 5.5. Analysis of the new structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76 5.6. Results about these curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.7. The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Chapter 6. On modular mod 2 Galois representations. . . . . . . . . . . . . . . . . . . . . . . . .93 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2. Δ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94 6.3. Mod 2 representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96 6.4. The case m < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.5. The case m > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Chapter 7. On the number fields generated by coefficients of newforms …. 103 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Chapter 1. Introduction.

Summary. Here we give a brief account of some of the work done over the last ten years on characteristic 0 newforms and the conductors of the mod l representations arising from them. We discuss the main results of this dissertation, and finally give a brief summary of each chapter.

§1.1. The levels of modular representations.

Let l > 3 be a prime, and let N ≥ 1 and k ≥ 2 be integers. To any mod l cusp form f ∈ S_{k}(Γ_{1}(N); \overline{𝔽}_{l}) which is an eigenform for the Hecke operators, Deligne (following work of Eichler and Shimura in the case k = 2) has attached to f a mod l Galois representation

ρ_{f} : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{l})

which is continuous, odd, and semi-simple. We say that a mod l representation ρ is modular if it arises in this way. Several questions have arisen from this construction. In particular, a conjecture of Serre predicts that all continuous odd irreducible mod l representations are modular. If ρ is such a representation, a natural question to ask is for which levels N and weights k can one expect to find mod l eigenforms giving rise to ρ. More precisely, for which N and k does there exist a characteristic 0 newform F of level N and weight k such that the mod l representation ρ_{f} arising from the mod l reduction f of F is isomorphic to ρ.

Serre in [S] attaches, to any continuous odd irreducible mod l Galois representation ρ, positive integers N(ρ) ≥ 1 and k(ρ) ≥ 2, and conjectures that ρ arises from the mod l reduction of a characteristic 0 newform F of level N(ρ) and weight k(ρ). Serre defines N(ρ) to be the conductor of ρ, which is by definition prime to l, and his definition of k(ρ) depends only on the restriction of ρ to a decomposition group at l. Serre also conjectures that N(ρ) and k(ρ) are minimal, in the sense that if ρ arises from the mod l reduction of a newform of level N and weight k ≥ 2 then N ≥ N(ρ) (in fact N is a multiple of N(ρ)) and k ≥ k(ρ). These conjectures imply the Shimura-Taniyama Conjecture and also Fermat’s Last Theorem.

The question of whether every continuous odd irreducible mod l representation is modular is still open. However, if one assumes ρ to be modular, then much progress can be made on finding at which weights and levels there are newforms giving rise to ρ. In particular, the main theorem of [D3] (see also [R2]) summarises the work done by many people on minimising the level and weight and says that, for an odd prime l, any irreducible modular representation ρ is indeed modular of level N(ρ) and weight k(ρ), and moreover that N(ρ) and k(ρ) are minimal in the sense defined above.

Carayol in [C2] (see also [Li]) shows, by examining how the conductor of an l-adic representation can degenerate under reduction, that if N is some positive integer prime to l for which there is a newform F of level N whose mod l reduction gives rise to ρ, then N must be of the form

N = N(ρ) ∏_{p} p^{α(p)}

where N(ρ) is the conductor of ρ, and for each p with α(p) > 0, at least one of the following holds:

(1) p ∤ N(ρ), p(tr(ρ(Frob_{p})))^{2} = (1 + p)^{2}det(ρ(Frob_{p})) and α(p) = 1,

(2) p ≡ −1 mod l and one of the following holds:

(a) p ∤ N(ρ), tr(ρ(Frob_{p})) = 0 and α(p) = 2,

(b) p ∥ N(ρ), det(ρ) is unramified at p, and α(p) = 1,

(3) p ≡ 1 mod l and one of the following holds:

(a) p ∤ N(ρ) and α(p) = 2,

(b) p^{2} ∤ N(ρ) or the power of p dividing N(ρ) is equal to the power of p dividing the conductor of (det(ρ)), and α(p) = 1.

We say that a positive integer N prime to l is a possible level for ρ if it is of this form. Note that the restriction that N be prime to l is natural if one thinks of mod l modular forms of weight k as global sections of a certain sheaf ω^{⊗k} on the modular curve X_{1}(N)_{𝔽_{l}}.

Let (†) be the following statement.

Let ρ be an irreducible modular mod l representation. Let N be a possible level for ρ. Then ρ does indeed arise from the mod l reduction of a newform of level N.

Ribet has conjectured that (†) holds. In this thesis, we prove (†) in the case when ρ is not “D_{l}-scalar” (see below). In particular, our result applies in many cases where ρ does not arise from a form of weight at most l − 1.

Before we talk about our work on (†), we will summarise what was already known. The subtlest primes to deal with are those in case (1) above. Firstly, we introduce another definition.

Let N ≥ 1 be an integer, and l ≥ 5 be a prime not dividing N. Let p be a prime not dividing Nl. We say a form F ∈ S_{k}(Γ_{1}(Np); \overline{ℚ}_{l}) is p-new if it is in the kernel of the two trace maps from S_{k}(Γ_{1}(Np), \overline{ℚ}_{l}) to S_{k}(Γ_{1}(N), \overline{ℚ}_{l}). Let d be a squarefree integer coprime to Nl. We say that a form F ∈ S_{k}(Γ_{1}(Nd); \overline{ℚ}_{l}) is d-new if it is p-new for all primes p dividing d.

Ribet was the first to prove a result which can be thought of as adding a prime of the form (1) to the level. The main result of [R1] is that if ρ arises from a mod l eigenform of level N, weight 2 and trivial character, and if p ∤ Nl is in case (1) above, then ρ arises from the mod l reduction of a p-new eigenform of level Np, weight 2 and trivial character. The heart of the proof was to show (using arithmetical methods) that Gal(\overline{ℚ}/ℚ) acts on a certain kernel via its abelianisation Gal(\overline{ℚ}/ℚ)^{ab}, i.e. that this kernel is in some sense “Eisenstein”. This technique was refined in [D2] and more recently in [DT1], a paper which uses geometric methods to prove that certain kernels in a more general setting are Eisenstein, and proves (†) in the case where ρ arises from a form of weight k < l − 1. Later in [DT2] the case k = l − 1 was also proved.

The only time that [DT1] needed the assumption of low weight is in the proof of their Theorem B, which is the theorem used to add primes of the form (1) to the level. To prove (†) in full generality then, it would suffice to prove the following generalisation of Theorem B of [DT1].

Suppose ρ : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{l}) is an irreducible representation which arises from a mod l eigenform of weight k ≥ 2 on Γ_{1}(N), where N is an integer prime to l. Suppose d is a product of distinct primes, and is coprime to Nl. Suppose that for all p dividing d we have

p(tr(ρ(Frob_{p})))^{2} = (1 + p)^{2}det(ρ(Frob_{p})).

Then ρ arises from the mod l reduction of a d-new form on Γ_{1}(N) ∩ Γ_{0}(d).

Call this statement (∗). A natural way to attack (∗) would be to prove it for a representation ρ arising from a form of weight k satisfying 2 ≤ k ≤ l + 1, and then use the fact that every modular representation arises up to twist from a form with weight in this region and deduce the general result from this fact. Because of the results of [DT1] and [DT2] (where (∗) is proved for representations ρ arising from forms of weight k satisfying 2 ≤ k ≤ l − 1) we see that to prove (∗) it suffices to do the following:

(I) To prove (∗) in the case where ρ arises from an eigenform of weight l or l + 1.

(II) To reduce the general question to a question involving only representations which arise from forms of weight at most l + 1.

Here are a few remarks about these obstacles. Firstly, it seems that the methods of [DT1] and [DT2] will not generalise to solve (I) in their present form because they rely on work of Faltings which is only valid for weights k ≤ l − 1. In fact, for weights k > l − 1 there is not even a conjectural analogue of Faltings’ work. Moreover, it seems that some people are at present questioning the validity of the work of Faltings upon which [DT1] relies. The problem with (II) is that although some twist of ρ will arise from a form of weight at most l + 1, one has to make sure that primes which one has taken pains to add to the level do not come out again whilst one is untwisting back.

This thesis deals with (II) completely, and with (I) in the case when ρ is not D_{l}-scalar. We say a representation ρ : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{l}) is D_{l}-scalar if the image under ρ of a decomposition group at l is contained within the scalar matrices. We shall now give a brief overview of the proofs.

Firstly we concentrate on the case k = l of (I). We proceed by induction on the number of primes dividing d, i.e. we add primes one at a time, taking care not to let any drop out once they are added. We firstly discuss the case when d is prime. This has been dealt with by Diamond in [D2] but we give a new geometric proof. As in [R1] we reduce the problem to showing that the kernel of a certain map between Jacobians is Eisenstein, i.e. does not contain in its support any maximal ideals of the Hecke algebra which correspond to irreducible representations. Now by analysing the l-divisible groups attached to the Jacobians of curves over finite fields, we show that if a certain map between spaces of differentials is injective, then the l-torsion of the kernel of a certain map between their Jacobians is multiplicative. Note that we had to consider the full l-divisible groups and not just their l-torsion to establish this result. We apply the result to the irreducible components of the special fibre of a semi-stable model of X_{1}(Nl) over ℤ_{l}[ζ_{l}]. The injectivity of the maps between differentials is established using a q-expansion argument, and we deduce a multiplicativity result. This result implies that the kernel which we are analysing can only contain in its support maximal ideals corresponding to reducible or D_{l} -scalar representations. Because we are excluding D_{l}-scalar representations from our calculations, this is enough to prove the result when d is prime.

For general d, we proceed by induction on the number of primes dividing d. By using the Jacquet-Langlands correspondence to switch to a quaternion algebra, we see that the inductive step is equivalent to adding one prime to the level as before, but for modular forms over a quaternion algebra D/ℚ. If D is definite then this is done in [DT1]. If D is indefinite then we use exactly the same techniques as for adding the first prime. To justify this approach we have to prove that the bad reduction of certain Shimura curves is analogous to the bad reduction of modular curves in certain cases. Once we have done this, the techniques that we have used in the case d prime extend immediately. This deals with the case k = l. We remark that these techniques could also probably be used for weights less than l also, giving another proof of some of the results of [DT1]. We also remark that in [G], Gross also had problems dealing with the case when ρ is D_{l} -scalar, and it is also the one low weight case for which Edixhoven in [E] does not prove a mod l multiplicity 1 result, although we will not rely on any multiplicity 1 results in our work.

The case k = l +1 of (I) is treated totally differently. We again switch to a quaternion algebra, this time ramified at l, and then the problem turns into a problem about weight 2 forms on this quaternion algebra, which we then solve using methods analogous to those in [DT1].

Switching to modular forms over a quaternion algebra is also the key to our attack on problem (II). We use an algebraic approach which has its origins in [AS], and in particular a cohomological analogue of the operator θ of [K2] which twists systems of eigenvalues occurring in spaces of modular forms on quaternion algebras.

We present the argument as follows. Firstly, we give a detailed proof of

Theorem 1.1.1. Suppose ρ : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{l}) is irreducible and arises from a mod l eigenform f of level N prime to l and weight k ≥ 2. Suppose ρ is not D_{l}-scalar. Suppose that p is a prime not dividing Nl such that

p(tr(ρ(Frob_{p})))^{2} = (1 + p)^{2}det(ρ(Frob_{p})).

Then ρ arises from the mod l reduction g of a p-new eigenform G on Γ_{1}(N) ∩ Γ_{0}(p).

Our proof is as discussed above, and relies on results proved in [DR] about the bad reduction of modular curves. We then prove analogous results about the bad reduction of certain Shimura curves, and once we get round the problem of not having q-expansions in this setting, the techniques we have set up prove

Theorem 1.1.2. Suppose ρ : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{l}) is irreducible and arises from a mod l eigenform f of level M prime to l and weight k ≥ 2. Suppose ρ is not D_{l}-scalar. Suppose that d is a positive squarefree integer prime to Ml such that for all p dividing d we have

p(tr(ρ(Frob_{p})))^{2} = (1 + p)^{2}det(ρ(Frob_{p})).

Then ρ arises from the mod l reduction g of a p-new eigenform G on Γ_{1}(M) ∩ Γ_{0}(d). We also show how to deduce from [DT1] and Theorem 1.1.2 the following corollary. Corollary 1.1.3. Let ρ be an irreducible modular mod l representation. Let M prime

to l be a possible level for ρ. If ρ is not D_{l}-scalar then ρ arises from the mod l reduction of a newform of level M.

The bulk of the work in the proof of Theorem 1.1.1 given here is to prove the theorem when ρ arises from a form of weight at most l + 1, and under this restriction, the theorem is already proved in most cases in [D2]. However, the techniques of [D2] do not seem to extend to prove Theorem 1.1.2, whereas our methods are constructed to do so.

The preceding paragraphs refer to Chapters 2–5 of this dissertation. In the course of learning about mod l modular representations, we have proved some other results which are presented in Chapters 6 and 7, which we shall now briefly describe.

The main result of Chapter 6 was motivated by a question of Wiles about what could be said about reducible, indecomposable (and hence not semi-simple) representations ρ : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝔽_{l}). One can ask whether these representations are in some sense the reduction of an l-adic representation associated to a cusp form. We show that in the case l = 2 they always are, by constructing theta functions giving rise to these representations.

The results in Chapter 7 were motivated by a question of Ramakrishnan on whether anything could be said about the extension of ℚ generated by the eigenvalues of a characteristic 0 eigenform. We used a Galois-theoretic lemma and some computations to give examples where the eigenvalues generated extensions contained in no soluble extension of ℚ. A version [Bu] of this chapter has been accepted for publication in the Journal of Number Theory. Two comments, one in Chapter 7 and one in the appendix, are due to the referee of the paper, and attention is drawn to this on the occasion of each comment.

§1.2. Summary of this dissertation.

The following is a brief summary of each chapter.

Chapter 2 sets up some preliminary algebraic geometry. The main result is a theorem which gives us some information about the kernel of a certain map between two semi-abelian schemes over a local ring, under the assumption that a certain map between differentials is an injection. The information is that the l-torsion in the special fibre of this kernel is multiplicative.

In Chapter 3 we recall various facts about modular curves, all of which are known in some form or other. In particular, we study mod l modular forms of weight at most l + 1. We recall various facts relating mod l modular forms to differentials on Igusa curves, we recall the technique of lifting mod l forms of low weight to characteristic 0, and we look at the l-divisible groups attached to these liftings.

Theorem 1.1.1 is proved in Chapter 4. We apply the main result of Chapter 2 to a certain map between the Néron models of Jacobians of modular curves, and deduce from the geometry that a certain maximal ideal cannot be in the support of the kernel of this map. We deduce that the maximal ideal in question is in the support of a congruence module, and use this fact to prove the theorem for representations arising from forms of weight at most l. We use the theory of quaternion algebras to prove the theorem in weight l + 1, and also to deduce the general theorem from this low weight result. Quaternion algebras are used to assure that primes added to the level stay in the level.

Chapter 5 contains the results about the bad reduction of Shimura curves which we shall need, and a proof of Theorem 1.1.2. This theorem is deduced from the Jacquet-Langlands Theorem, results in [DT1], and a result analogous to Theorem 1.1.1 for modular forms over an indefinite quaternion algebra. Many of the techniques used are immediate generalisations of results used to prove Theorem 1.1.1. We also show how to deduce Corollary 1.1.3 from Theorem 1.1.2 and results in [DT1].

In Chapter 6 we consider reducible mod 2 representations. In particular, we show that a certain class of indecomposable reducible mod 2 representations arise as the “naïve” reduction of representations attached to characteristic 0 eigenforms, of level equal to the conductor of the representation, up to a power of 2.

Finally, in Chapter 7 we examine the eigenvalues of Hecke operators acting on eigenforms of level 1. By doing a calculation using MATHEMATICA, we prove a theorem showing that the extensions of ℚ generated by these eigenvalues may be contained in no soluble extension of ℚ.

Chapter 2. Curves and Jacobians.

Summary: This chapter is mostly arithmetic algebraic geometry. At the heart of the chapter is Proposition 2.3.4, which says that for two curves X and Y over a finite field of characteristic l, and two maps f and g from X to Y, if the induced map on differentials f^{∗} + g^{∗} : H^{0}(Y, Ω^{1})^{2} → H^{0}(X, Ω^{1}) is injective, then the l-torsion in the kernel K of the induced map of Jacobians f^{∗} + g^{∗} : Jac(Y)^{2} → Jac(X) is multiplicative. We use this to deduce a similar-looking result (Theorem 2.4.6) about certain semi-stable curves over the ring of integers of a local field. Briefly, this latter theorem gives us a geometric fact about a certain kernel which will enable us later to prove that in the arithmetic applications, the kernel is in many cases Eisenstein.

§2.1. Curves over finite fields.

Let l be a rational prime, and k a finite field of characteristic l. A curve over k is a 1-dimensional separated scheme C of finite type over k such that C\overline{k} = C ×_{Spec k} Spec \overline{k} is integral. Let X be a complete non-singular curve defined over k, and let us assume moreover that the set of k-rational points X(k) of X is non-empty. Fix a k-rational point P ∈ X(k). The curve X has a Jacobian J_{X}, also defined over k, and there is a canonical map

f^{P} : X → J_{X}

defined over k, sending P to 0 (see e.g. [Mil2], §2). For G/k a commutative group scheme, let O_{e} be the local ring of G at the identity, and let 𝔪_{e} be its maximal ideal. Then O_{e}/𝔪_{e} ≅ k, and 𝔪_{e}/𝔪_{e}^{2} is a k-vector space. We define the tangent space to G at the identity, Lie(G), to be its vector space dual (𝔪_{e}/𝔪_{e}^{2})^{∨}. For α : G → H a homomorphism of commutative group schemes over k, there is a homomorphism Lie(α) : Lie(G) → Lie(H) of k-vector spaces.

Let L_{X} = Lie(J_{X}). Then L_{X} is a finite-dimensional vector space over k. Let L_{X}^{∨} denote its dual. Then, as is well-known, there is a canonical isomorphism of k-vector spaces

\begin{matrix}h_{X} : H^{0}(J_{X}, Ω_{J_{X}}^{1} ) & ⟶^{≅}L_{X}^{∨} \\ ω & ↦ ω(e) \end{matrix}

sending a differential form on J_{X} to its value at the origin of J_{X}. See e.g. [Sha], Chapter III §5.2.

The map f^{P} induces a map on differentials

(f^{P})^{∗} : H^{0}(J_{X}, Ω_{J_{X}}^{1} )⟶^{≅}H^{0}(X, Ω_{X}^{1}),

which is independent of P and also an isomorphism of k-vector spaces by [Mil2] Proposition 2.2.

Composing the inverse of (f^{P})^{∗} with h_{X} gives us an isomorphism

j_{X} : H^{0}(X, Ω_{X}^{1})⟶^{≅}L_{X}^{∨},

also independent of P. Now let Y be another curve defined over k. If

g : X → Y

is a morphism defined over k, then Q = g(P) is a k-valued point of Y, and so we can define the canonical morphism

f^{Q} : Y → J_{Y}

as above, and we have a collection of maps

\begin{matrix}X & ⟶^{f^{P}} & J_{X} \\ ↓ & g \\ Y & ⟶^{f^{Q}} & J_{Y} \end{matrix} .

The composite f^{Q} ∘ g is a morphism from X to an abelian variety, sending P to 0, and hence ([Mil2], Proposition 6.1) factors through a unique homomorphism of abelian varieties Alb(g) : J_{X} → J_{Y}, i.e. the diagram

X ⟶^{f^{P}} J_{X}
↓^{g} ↓^{Alb(g)}
Y ⟶^{f^{Q}} J_{Y}

commutes. The map Alb(g) in turn induces

d_{g} = Lie(Alb(g)) : L_{X} → L_{Y} ,

the derivative of Alb(g) at 0, and d_{g} has a vector space dual

(d_{g})^{∨} : L_{Y}^{∨} → L_{X}^{∨}.

The map g also defines a map on global differentials

g^{∗} : H^{0}(Y, Ω_{Y}^{1} ) → H^{0}(X, Ω_{X}^{1}).

Lemma 2.1.1. The diagram

H^{0}(Y, Ω_{Y}^{1} ) ⟶^{j_{Y}} L_{Y}^{∨}
↓^{g^{∗}}  ↓^{(d_{g})^{∨}}
H^{0}(X, Ω_{X}^{1}) ⟶^{j_{X}} L_{X}^{∨}

commutes.

Proof. By the definition of j_{X} and j_{Y}, it suffices to show that the diagram

H^{0}(Y, Ω_{Y}^{1} ) ⟵^{(f^{Q})^{∗}} H^{0}(J_{Y} , Ω_{J_{Y}}^{1} ) ⟶^{h_{Y}} L_{Y}^{∨}
↓^{g^{∗}}  ↓^{Alb(g)^{∗}}  ↓^{(d_{g})^{∨}}
H^{0}(X, Ω_{X}^{1}) ⟵^{(f^{P})^{∗}} H^{0}(J_{X}, Ω_{J_{X}}^{1} ) ⟶^{h_{X}} L_{X}^{∨}

commutes. Here the superscript ^{∗} indicates an induced map on differentials. That the left-hand square commutes is clear, as

Alb(g) ∘ f^{P} = f^{Q} ∘ g

by definition. That the right-hand square commutes is clear from the definition of pulling back differentials—see for example [Sha] Chapter III §5.1.

□

§2.2. Abelian varieties over finite fields.

In this section we recall some general theorems about abelian varieties in characteristic l. If G is a commutative group scheme over k, then let G[l] denote the kernel of the multiplication by l homomorphism from G to G. If A is an abelian variety over k, then A[l] is a finite group scheme over k.

Lemma 2.2.1. The exact sequence

0 → A[l]⟶^{i}A⟶^{×l}A

of group schemes over k induces an exact sequence

0 → Lie(A[l]) ⟶^{Lie(i)}Lie(A)⟶^{×l}Lie(A)

of k-vector spaces. In particular, as k is of characteristic l, the multiplication by l map from Lie(A) to Lie(A) is the zero map, and hence Lie(A[l]) ≅ Lie(A).

Proof. It suffices to show that the derivative of the multiplication by l map from A to A is indeed the multiplication by l map from Lie(A) to Lie(A). But this is standard—see for instance [Mu] p43.

□

Now we briefly recall some facts about dual abelian varieties. The multiplication by l map

(l) : A → A

on an abelian variety over k is an isogeny with kernel A[l]. The dual of the multiplication by l map is the multiplication by l map

(l) : A^{∨} → A^{∨},

and this map has kernel A^{∨}[l] which is canonically isomorphic, via k_{l} say, to the Cartier dual A[l]^{∨} of A[l]. See [O] Theorem 1.1 for an explicit construction of k_{l}. Note however that later we shall follow [KM] 2.8 for our conventions about pairings on elliptic curves, and [KM] uses the opposite convention to [O]. To fix our ideas then, we shall define k_{l} to be minus the map ν_{l}^{−1} defined in [O] Theorem 1.1.

Now we recall various consequences of self-duality of the Jacobian. Firstly we need some definitions. A k-scheme is a scheme over Spec(k). A pointed k-scheme is a pair (S, s) where S is a connected k-scheme and s is a k-valued point of S. A divisoral correspondence between two pointed k-schemes (S, s) and (T, t) is an invertible sheaf L on S ×_{k} T such that L|_{S×{t}} and L|_{{s}×T} are both trivial. (X, P), (J_{X}, 0) and (J_{X}^{∨}, 0) are all pointed k-schemes. There is a canonical divisoral correspondence P_{X}, the Poincaré sheaf, between (J_{X}, 0) and (J_{X}^{∨}, 0), and also a canonical divisoral correspondence M^{P} between (X, P) and (J_{X}, 0). We define

τ_{P} : J_{X}^{∨} → J_{X}

to be the unique homomorphism of abelian varieties such that the sheaves (f^{P} × 1)^{∗}P_{X} and (1 × τ_{P})^{∗}M^{P} on X × J_{X}^{∨} are isomorphic. See [Mil2] Theorem 1.2 for the details.

The map τ_{P} is an isomorphism of abelian varieties by [Mil2] Lemma 6.9. Furthermore, τ_{P} is independent of P. For −τ_{P}^{−1} is the map ϕ_{L(Θ_{P})} : J_{X} → J_{X}^{∨} induced by a certain divisor Θ_{P} on J_{X}. Changing our choice of P ∈ X(k) replaces Θ_{P} by a translate and hence by the theorem of the square does not change ϕ_{L(Θ_{P})}.

The dual of τ_{P} is the map

τ_{P}^{∨} : J_{X}^{∨} → (J_{X}^{∨})^{∨} ≅ J_{X},

where the isomorphism ≅ is a canonical one.

Define α_{X} to be the composite of the maps below.

α_{X} : J_{X}[l]^{∨}⟶^{k_{l}}J_{X}^{∨}[l]⟶^{i}J_{X}^{∨} ⟶^{τ_{P}^{∨}}((J_{X})^{∨})^{∨}⟶^{≅}J_{X}.

The map g : X → Y induces by Picard functoriality a map

Pic(g) : J_{Y} → J_{X}

between Jacobians, and hence a map

Pic(g) : J_{Y} [l] → J_{X}[l].

Cartier duality then gives us a map

(Pic(g))^{∨} : J_{X}[l]^{∨} → J_{Y} [l]^{∨}.

Lemma 2.2.2. The diagram

J_{X}[l]^{∨} ⟶^{α_{X}} J_{X}
↓^{(Pic(g))^{∨}} ↓^{Alb(g)}
J_{Y} [l]^{∨} ⟶^{α_{Y}} J_{Y}

commutes.

Proof. We take α_{X} and α_{Y} apart. The map Pic(g) : J_{Y} → J_{X} induces by duality of abelian varieties a map

(Pic(g))^{∨} : J_{X}^{∨} → J_{Y}^{∨},

which restricts to a map

(Pic(g))^{∨} : J_{X}^{∨}[l] → J_{Y}^{∨}[l].

It suffices to show that

J_{X}[l]^{∨} ⟶^{k_{l}} J_{X}^{∨}[l] ⟶^{i} J_{X}^{∨} ⟶^{τ_{P}^{∨}} (J_{X}^{∨})^{∨} ⟶^{≅} J_{X}
↓^{(Pic(g))^{∨}} ↓^{(Pic(g))^{∨}} ↓^{(Pic(g))^{∨}} ↓^{(Alb(g)^{∨})^{∨}} ↓^{Alb(g)}
J_{Y} [l]^{∨} ⟶^{k_{l}} J_{Y}^{∨}[l] ⟶^{i} J_{Y}^{∨} ⟶^{τ_{Q}^{∨}} (J_{Y}^{∨})^{∨} ⟶^{≅} J_{Y}

commutes. We deal with each square separately.

The first square commutes by [O] Theorem 1.1. Note that the ∨ signs in the three (Pic(g))^{∨}s mean different things—the first is Cartier duality, the other two duality of abelian varieties.

The second square commutes by definition.

To prove that the third square commutes, we show that the diagram

J_{Y}^{∨} ⟶^{τ_{Q}} J_{Y}
↓^{(Alb(g))^{∨}} ↓^{Pic(g)}
J_{X}^{∨} ⟶^{τ_{P}} J_{X}

commutes, and the third square is the dual of this diagram. We shall prove that this latter diagram commutes by chasing sheaves round the following commutative diagram:

Y × J_{Y}  ⟵^{g×1}  X × J_{Y}  ⟶^{1×γ}  X × J_{X} ↑^{1×τ_{Q}}  ↑^{1×τ_{Q}}  ↑^{1×τ_{P}} Y × J_{Y}^{∨}  ⟵^{g×1}  X × J_{Y}^{∨} ⟶^{1×(Alb(g))^{∨}} X × J_{X}^{∨} ↓_{f^{Q}×1}  ↓_{f^{P} ×1}  ↓_{f^{P} ×1} J_{Y} × J_{Y}^{∨} ⟵^{Alb(g)×1} J_{X} × J_{Y}^{∨} ⟶^{1×(Alb(g))^{∨}} J_{X} × J_{X}^{∨},

where

γ = τ_{P} ∘ (Alb(g))^{∨} ∘ τ_{Q}^{−1}.

We claim that (1 × γ)^{∗}M^{P} ≅ (g × 1)^{∗}M^{Q} on X × J_{Y}. To show this, it suffices to prove that these sheaves become isomorphic when pulled back to X × J_{Y}^{∨}, as τ_{Q} is an isomorphism. But

(1 × τ_{Q})^{∗}(1 × γ)^{∗}M^{P}
=(1 × (Alb(g))^{∨})^{∗}(1 × τ_{P})^{∗}M^{P}
≅(1 × (Alb(g))^{∨})^{∗}(f^{P} × 1)^{∗}P_{X} (by definition of τ_{P})
=(f^{P} × 1)^{∗}(1 × (Alb(g))^{∨})^{∗}P_{X}
≅(f^{P} × 1)^{∗}(Alb(g) × 1)^{∗}P_{Y} (by definition of dual morphism (see [Mil1] §11))
=(g × 1)^{∗}(f^{Q} × 1)^{∗}P_{Y}
≅(g × 1)^{∗}(1 × τ_{Q})^{∗}M^{Q} (by definition of τ_{Q})
=(1 × τ_{Q})^{∗}(g × 1)^{∗}M^{Q}

and this establishes the claim. Now Pic(g) is the unique morphism from J_{Y} to J_{X} with this property, and hence γ = Pic(g), which establishes the commutativity of the third square.

That the fourth square commutes is a general fact about Abelian varieties—see for example [La1] Chapter 5, §2, Proposition 8.

□

Note that Lie(α_{X}) and Lie(α_{Y}) are isomorphisms of k-vector spaces, by Lemma 2.2.1, and hence as a corollary of this proposition we deduce that

Lie(J_{X}[l]^{∨}) ⟶^{≅} L_{X}
↓^{Lie((Pic(g))^{∨})} ↓^{d_{g}}
Lie(J_{Y} [l]^{∨}) ⟶^{≅} L_{Y}

commutes, where the superscript ^{∨} indicates Cartier duality. By putting together this diagram with the dual of the diagram in Lemma 2.1.1, we can summarise our results in the following proposition:

Proposition 2.2.3. If we define an isomorphism

η_{X} : Lie(J_{X}[l]^{∨}) → H^{0}(X, Ω_{X}^{1})^{∨}

as the composite

Lie(J_{X}[l]^{∨}) ⟶^{Lie(k_{l})}Lie(J_{X}^{∨}[l]) ⟶^{Lie(i)}Lie(J_{X}^{∨}) ⟶^{Lie(τ_{P}^{∨})}

⟶^{Lie(τ_{P}^{∨})}Lie((J_{X}^{∨})^{∨})⟶^{≅}Lie(J_{X}) = L_{X} ⟶^{j_{X}^{∨}}H^{0}(X, Ω_{X}^{1})^{∨},

and η_{Y} similarly, then η_{X} is independent of the choice of P, and the diagram

Lie(J_{X}[l]^{∨}) ⟶^{η_{X}} H^{0}(X, Ω_{X}^{1})^{∨}
↓^{Lie((Pic(g))^{∨})}  ↓^{(g^{∗})^{∨}}
Lie(J_{Y} [l]^{∨}) ⟶^{η_{Y}} H^{0}(Y, Ω_{Y}^{1} )^{∨}

commutes.

□

§2.3. Dieudonné modules.

We now shift our attention to the theory of Dieudonné modules. We refer the reader to [F] for details.

If G is a finite group scheme over the finite field k, we may associate to G its Dieudonné module M(G), a finitely-generated module over W (k), the Witt vectors of k, equipped with semi-linear endomorphisms F and V satisfying F V = V F = l. We may canonically identify the k-vector space Lie(G^{∨}) with the kernel _{V} M(G) of V on M(G), via the map c_{G} say. A definition of c_{G} is in [F] §3 Proposition 3.2.

If H is another finite group scheme over k, then a homomorphism α : G → H defined over k induces a homomorphism of W (k)-modules M(α) : M(H) → M(G) which commutes with the actions of F and V, and hence induces a map from _{V} M(H) to _{V} M(G). Also, α induces a map θ^{∨} : H^{∨} → G^{∨} by Cartier duality.

Lemma 2.3.1. The diagram

Lie(H^{∨}) ⟶^{c_{H}}  _{V} M(H)
↓^{Lie(α^{∨})} ↓^{M(θ)}
Lie(G^{∨}) ⟶^{c_{G}}  _{V} M(G)

commutes.

Proof. Unwind the definitions in [F].

□

Now we undertake a more detailed analysis of the Dieudonné modules associated to various groups connected with Abelian varieties.

If J is an abelian variety over k, of dimension d, then the inductive limit

G_{J} = \underrightarrow{lim}_{n} J[l^{n}]

is an l-divisible group of height 2d over k, and the Dieudonné module

M(G_{J}) = \underleftarrow{lim}_{n} M(G_{J}[l^{n}])

is a free W (k)-module of rank 2d.

The exact sequence

0 → G_{J}[l]⟶^{i}G_{J} ⟶^{×l}G_{J} → 0

induces an exact sequence

0 → M(G_{J})⟶^{×l}M(G_{J}) → M(G_{J}[l]) → 0.

Write M for M(G_{J}) and N for M(G_{J}[l]) = M(J[l]). Then N ≅ M/lM.

Lemma 2.3.2. Im(F) = ker(V) and Im(V) = ker(F) on N.

Proof. We will show that Im(F) = ker(V). The same argument is used to show Im(V) = ker(F).

Firstly, note that V F = l, which acts as 0 on N, and hence Im(F) ⊆ ker(V). Now say x̃ is an element of ker(V) on N. Then we can lift x̃ to an element x of M, and the fact that V x̃ = 0 implies that V x ∈ lM. So we can choose y in M such that V x = ly. Then l(x − F y) = F V (x − F y) = F V x − F ly = F (V x − ly) = 0, and because M is a free W (k)-module this implies that x − F y = 0, i.e. that x = F y. Hence if ỹ is the image of y in N, we see that F ỹ = x̃, and because x̃ was arbitrary, we deduce ker(V) ⊆ Im(F), which establishes the claim.

□

We now have enough information to prove the main result we need on Dieudonné modules. Firstly, some definitions.

Let A and B be abelian varieties over k. Say

α_{ab} : A → B

is a homomorphism. Set G = A[l] and H = B[l]. Then α_{ab} induces a map α : G → H, with kernel K say. K is a finite group scheme over k.

Proposition 2.3.3. If the map

α^{∨} : H^{∨} → G^{∨}

induces a surjection Lie(α^{∨}) : Lie(H^{∨}) → Lie(G^{∨}) of tangent spaces, then K is multiplicative, i.e. the Cartier dual of K is étale.

Proof. The exact sequence

0⟶K⟶^{i}G⟶^{α}H

induces an exact sequence

M(H) ⟶^{M(α)}M(G)⟶^{M(i)}M(K)⟶0.

M(α) induces a map from _{V} M(H)to _{V} M(G), and the fact that Lie(α^{∨}) is surjective, together with Lemma 2.3.1, shows that this map between kernels is a surjection.

We now show that V is a bijection on M(K). Say z ∈ M(K). Then because M(i) is surjective, there is y ∈ M(G) such that M(i)y = z. Now F y ∈ _{V} M(G) by Lemma 2.3.2, and so there is x ∈ _{V} M(H) such that M(α)x = F y. The kernel of V on M(H) is equal to the image of F, by Lemma 2.3.2, and hence we can write x = F w for some w ∈ M(H). Then

F (y − M(α)w)
=F y − M(α)F w
=F y − M(α)x
=F y − F y
=0

and hence y − M(α)w ∈ ker(F) in M(G). But ker(F) = Im(V) on M(G), again by Lemma 2.3.2, and hence y − M(α)w = V t for some t ∈ M(G). Now applying M(i) we see that M(i)y − M(i)M(α)w = M(i)V t, i.e. that z = V M(i)t, as M(i)M(α) = 0. Since z was arbitrary, this implies that V is surjective on M(K), and because M(K) is a finite object, this implies that V is bijective. But this implies that K is multiplicative—see [De] Chapter 3, theorem on p69.

□

Putting this together with the work on curves earlier in this chapter, we can deduce the main result of this section. We firstly establish some notation. X and Y are curves over k, and p_{1} and p_{2} are morphisms from X to Y. The maps p_{i}, i = 1, 2, induce homomorphisms (p_{i})^{∗} : H^{0}(Y, Ω_{Y}^{1} ) → H^{0}(X, Ω_{X}^{1}), and also homomorphisms of abelian varieties Pic(p_{i}) : J_{Y} → J_{X}. Define

\begin{matrix}γ : H^{0}(Y, Ω_{Y}^{1} )^{2} & → H^{0}(X, Ω_{X}^{1}) \\ (a, b) & ↦ (p_{1})^{∗}a + (p_{2})^{∗}b \end{matrix}

and

α_{ab} : J_{Y}^{2} → J_{X}
(s, t) ↦ Pic(p_{1})s + Pic(p_{2})t.

Then α_{ab} restricts to a map α : J_{Y}^{2} [l] → J_{X}[l]. Say the kernel of α is the group scheme L. L is then a finite group scheme over k.

Proposition 2.3.4. If γ is injective, then L is multiplicative.

Proof. We have done all the work. By replacing k by a finite extension if necessary, we may assume that X and Y have k-rational points. From Proposition 2.2.3 we see that

Lie(J_{X}[l]^{∨}) ⟶^{η_{X}} H^{0}(X, Ω_{X}^{1})^{∨}
↓^{Lie((Pic(p_{i}))^{∨})}  ↓^{(p_{i}^{∗})^{∨}}
Lie(J_{Y} [l]^{∨}) ⟶^{η_{Y}} H^{0}(Y, Ω_{Y}^{1} )^{∨}

commutes, and it follows immediately that the map

Lie(J_{X}[l]^{∨}) ⟶^{η_{X}} H^{0}(X, Ω_{X}^{1})^{∨}
↓^{Lie(α^{∨})}  ↓^{γ^{∨}}
Lie(J_{Y}^{2} [l]^{∨}) ⟶^{η_{Y}^{2}} (H^{0}(Y, Ω_{Y}^{1} )^{2})^{∨}

commutes. By assumption, γ is injective and hence γ^{∨} is surjective, which implies that Lie(α^{∨}) is surjective. The result now immediately follows from Proposition 2.3.3.

□

§2.4. Curves over local rings.

In this section, we consider the geometry of curves over Spec R, where R will ultimately be the ring of integers of a local field. A typical curve we shall consider will have an irreducible generic fibre, but a more complex special fibre—typically the union of two irreducible curves, crossing each other transversally at a finite set of points. The motivating examples are certain semi-stable models of modular curves X_{1}(Nl) over Spec(ℤ_{l}[ζ_{l}]), where N and l are coprime. A detailed analysis of the geometric properties of the curves we shall consider is made in [BoLüRa], and we shall below extract what we need.

Let R be a discrete valuation ring, with field of fractions K, and residue field k of characteristic l > 0. Set S = Spec(R). We recall the following definition.

Definition 2.4.1. A scheme X/S is a semi-stable curve if:

• The structure morphism X → S is proper and flat,

• The geometric fibres X\overline{s} of this map are reduced, connected, 1-dimensional, and only have ordinary double points as singularities.

For the purposes of this dissertation the following definition will also be very useful.

Definition 2.4.2. We say a scheme X/S is amenable if:

• X/S is semi-stable,

• X is normal,

• The generic fibre X_{K} of X is geometrically irreducible,

• The special fibre X_{k} of X has just two irreducible components X_{1} and X_{2}.

Note that the normalisation X̃_{k} of X_{k} is simply X_{1} ∐ X_{2}. Let X be an amenable curve, let J_{X_{K}} be the Jacobian of X_{K} and let J_{X} be its Néron

model over S. Let J_{X}^{0} be the identity component of this group scheme, by which we mean the open subscheme consisting of the connected components of the identity in each fibre. Let \overline{J}_{X}^{0} be the special fibre of J_{X}^{0}. We shall need the following theorem.

Theorem 2.4.3.

(i) There is a canonical exact sequence of group schemes over k

1⟶T_{X}⟶Pic_{X_{k}/k}^{0}⟶^{π_{X}^{∗}} ∏_{i=1}^{2} Pic_{X_{i}/k}^{0}⟶1,

where

π_{X} : X̃_{k} = ∐_{i=1}^{2} X_{i} → X_{k}

is the canonical map, π_{X}^{∗} is Pic(π_{X}), and T_{X} is a torus.

(ii) The canonical morphism Pic_{X/S}^{0}⟶J_{X}^{0} is an isomorphism.

Proof. (i) is [BoLüRa] Chapter 9, Section 2, Example 8 applied to X_{k}. (ii) is [BoLüRa] Chapter 9, Section 7, Corollary 2.

□

We now wish to analyse certain morphisms between two amenable curves. So say X and Y are two amenable curves over S, and θ : X → Y is a morphism between them, defined over S. Suppose moreover that there are maps θ_{i} : X_{i} → Y_{i} for 1 ≤ i ≤ 2 of the components of the special fibres of these curves, such that the diagram

X_{1} ∐ X_{2} ⟶^{π_{X}} X_{k}
↓^{θ_{1} ∐ θ_{2}} ↓^{θ_{k}}
Y_{1} ∐ Y_{2} ⟶^{π_{Y}} Y_{k}

commutes, where θ_{k} is the pullback of θ to the special fibre. Then θ_{k} induces by Picard functoriality a map

θ_{k}^{∗} = Pic(θ_{k}) : Pic_{Y_{k}/k} → Pic_{X_{k}/k}

which defines by restriction a map Pic_{Y_{k}/k}^{0} → Pic_{X_{k}/k}^{0}. Similarly, θ_{i} induces

β_{θ,i} = Pic(θ_{i}) : Pic_{Y_{i}/k}^{0} → Pic_{X_{i}/k}^{0}

and so by taking Pic^{0} of the commutative diagram above, and using Theorem 2.4.3 (i), we see that we can write down the following commutative diagram:

1 ⟶ T_{Y} ⟶ Pic_{Y_{k}/k}^{0} ⟶^{π_{Y}^{∗}} ∏_{i=1}^{2} Pic_{Y_{i}/k}^{0} ⟶ 1 ↓^{θ_{k}^{∗}}  ↓^{∏ β_{θ,i}} 1 ⟶ T_{X} ⟶ Pic_{X_{k}/k}^{0} ⟶^{π_{X}^{∗}} ∏_{i=1}^{2} Pic_{X_{i}/k}^{0} ⟶ 1 ,

in which the rows are exact sequences.

Next note that we can fill in a map T_{Y} → T_{X} because the image of T_{Y} in Pic_{X_{k}/k}^{0} is killed by π_{X}^{∗}, by exactness and commutativity. Finally we would like to replace Pic_{X_{k}/k}^{0}

with the special fibre \overline{J}_{X}^{0} of J_{X}^{0}, so we need to check the following:

Lemma 2.4.4. The diagrams

Pic_{Y/S}^{0} ⟶^{≅} J_{Y}^{0}
↓  ↓
Pic_{X/S}^{0} ⟶^{≅} J_{X}^{0}

and

Pic_{Y_{k}/k}^{0} ⟶^{≅} \overline{J}_{Y}^{0}
↓^{θ_{k}^{∗}}  ↓
Pic_{X_{k}/k}^{0} ⟶^{≅} \overline{J}_{X}^{0}

commute, where the vertical maps are morphisms induced by Picard functoriality and the horizontal maps are the canonical morphisms.

Proof. The generic fibre of the first diagram commutes, and hence the first diagram commutes by the Néron property. The second diagram is the special fibre of the first, and hence must also commute.

In particular, we may replace Pic_{X_{k}/k}^{0} with \overline{J}_{X}^{0}, and similarly for Y, and deduce

Proposition 2.4.5. If X and Y are amenable curves over S, and θ is a morphism between them such that θ maps X_{i} to Y_{i}, then there is a commutative diagram

1 ⟶ T_{Y} ⟶ \overline{J}_{Y}^{0} ⟶ ∏_{i=1}^{2} Pic_{Y_{i}/k}^{0} ⟶ 1
↓^{t_{θ}}  ↓^{α_{θ}}  ↓^{∏_{i} β_{θ,i}}
1 ⟶ T_{X} ⟶ \overline{J}_{X}^{0} ⟶ ∏_{i=1}^{2} Pic_{X_{i}/k}^{0} ⟶ 1 ,

(note that we have christened the unnamed vertical maps) where the rows are exact and the middle and right hand vertical maps are the ones induced by Picard functoriality.

Proof. This is merely a summary of what we have established so far.

□

Now say ψ is another map X → Y inducing maps ψ_{i} : X_{i} → Y_{i} on the special fibres. By the above lemma there is a commutative diagram

1 ⟶ T_{Y}^{2} ⟶ (\overline{J}_{Y}^{0} )^{2} ⟶ ∏_{i=1}^{2}(Pic_{Y_{i}/k}^{0})^{2} ⟶ 1
↓^{t_{θ}×t_{ψ}}  ↓^{α_{θ}×α_{ψ}}  ↓^{∏_{i} β_{θ,i}×β_{ψ,i}}
1 ⟶ T_{X}^{2} ⟶ (\overline{J}_{X}^{0})^{2} ⟶ ∏_{i=1}^{2}(Pic_{X_{i}/k}^{0})^{2} ⟶ 1 ,

where the vertical maps are the product of the maps induced by θ and ψ. Finally, the multiplication maps m induce a commutative diagram

1 ⟶ T_{X}^{2} ⟶ (\overline{J}_{X}^{0})^{2} ⟶ ∏_{i=1}^{2}(Pic_{X_{i}/k}^{0})^{2} ⟶ 1
↓^{m}  ↓^{m}  ↓^{m}
1 ⟶ T_{X} ⟶ \overline{J}_{X}^{0} ⟶ ∏_{i=1}^{2} Pic_{X_{i}/k}^{0} ⟶ 1 ,

and hence by putting these two diagrams together we get a commutative diagram

1 ⟶ T_{Y}^{2} ⟶ (\overline{J}_{Y}^{0} )^{2} ⟶ ∏_{i=1}^{2}(Pic_{Y_{i}/k}^{0})^{2} ⟶ 1
↓^{t_{θ,ψ}}  ↓^{α_{θ,ψ}}  ↓^{∏_{i} β_{θ,ψ,i}}
1 ⟶ T_{X} ⟶ \overline{J}_{X}^{0} ⟶ ∏_{i=1}^{2} Pic_{X_{i}/k}^{0} ⟶ 1 ,

where t_{θ,ψ} = m ∘ (t_{θ} × t_{ψ}) and so on.

Now we can deduce the main theorem of this chapter.

Theorem 2.4.6. Let R be a discrete valuation ring, and let X and Y be amenable (see Definition 2.4.2) curves over R. Let θ and ψ be two morphisms from X to Y which induce morphisms θ_{i}, ψ_{i} : X_{i} → Y_{i} on the components of the special fibres of X and Y. Let J_{X}^{0} be the identity component of the Néron model of Jac(X_{K}), and define J_{Y}^{0} similarly. Let K be the kernel of the map

Pic(θ) + Pic(ψ) = α_{θ,ψ} : (J_{Y}^{0} )^{2} → J_{X}^{0}.

If the induced maps on differentials

\begin{matrix}θ_{i}^{∗} + ψ_{i}^{∗} : H^{0}(Y_{i}, Ω_{Y_{i}}^{1})^{2} & → H^{0}(X_{i}, Ω_{X_{i}}^{1}) \\ (a, b) & ↦ θ_{i}^{∗}a + ψ_{i}^{∗}b \end{matrix}

are injective for i = 1, 2, then the special fibre of K[l] is multiplicative.

Proof. We will use the notation above. By the snake lemma, there is an exact sequence of commutative group schemes

0 → ker t_{θ,ψ} → ker α_{θ,ψ} → ∏_{i=1}^{2} ker β_{θ,ψ,i} ,

and hence an exact sequence of finite group schemes

0 → (ker t_{θ,ψ})[l] → (ker α_{θ,ψ})[l] → ∏_{i=1}^{2}(ker β_{θ,ψ,i})[l],

over k. By Proposition 2.3.4 applied to the maps θ_{i}, ψ_{i} we have that (ker (β_{θ,ψ,i})[l]) is multiplicative. Moreover, (ker t_{θ,ψ})[l] is multiplicative, as it is a subgroup of a torus.

These facts force (ker α_{θ,ψ})[l] to be multiplicative, as can be seen for example by a simple calculation of the Dieudonné modules of the terms in the exact sequence.

□

Finally, we mention a slightly more refined version of this theorem, the proof of which needs no new ideas, and which we shall need in Chapter 5.

Say X is an amenable curve over R as before, but now let X be equipped with an action of the finite group (ℤ/lℤ)^{×}, defined over R. We shall write the action as maps

⟨d⟩_{l} : X → X

for d ∈ (ℤ/lℤ)^{×}. Assume that this action preserves the components X_{1} and X_{2} of the special fibres of X. Then for 1 ≤ i ≤ 2 the 𝔽_{l}-vector space H^{0}(X_{i}, Ω^{1}) becomes equipped with an action of (ℤ/lℤ)^{×} and this action splits it up into the direct sum of eigenspaces. More precisely, we have

H^{0}(X_{i}, Ω^{1}) = ⨁_{j=1}^{l−1} H^{0}(X_{i}, Ω^{1})(j)

where

H^{0}(X_{i}, Ω^{1})(j) = {x ∈ H^{0}(X_{i}, Ω^{1}) | ⟨d⟩_{l}^{∗}x = d^{j}x}.

Now say that the curve Y is also equipped with such an action, and the maps θ and ψ : X → Y preserve this action. From the maps

θ_{i}^{∗} + ψ_{i}^{∗} : H^{0}(Y_{i}, Ω_{Y_{i}}^{1})^{2} → H^{0}(X_{i}, Ω_{X_{i}}^{1})

we get maps

θ_{i}^{∗} + ψ_{i}^{∗} : H^{0}(Y_{i}, Ω_{Y_{i}}^{1})(j)^{2} → H^{0}(X_{i}, Ω_{X_{i}}^{1})(j)

for 1 ≤ j ≤ l − 1. The special fibre K[l]_{k} of K[l] also splits up as a sum of eigenspaces, namely

K[l]_{k} = ⨁_{j=1}^{l−1} K[l]_{k}(j).

By following the proof of Theorem 2.4.6 we deduce Theorem 2.4.7. If for some j the induced maps on differentials

\begin{matrix}θ_{i}^{∗} + ψ_{i}^{∗} : H^{0}(Y_{i}, Ω_{Y_{i}}^{1})(j)^{2} & → H^{0}(X_{i}, Ω_{X_{i}}^{1})(j) \\ (a, b) & ↦ θ_{i}^{∗}a + ψ_{i}^{∗}b \end{matrix}

are injective for i = 1, 2, then the group scheme K[l]_{k}(j) is multiplicative.

□

Chapter 3. Modular curves, Igusa curves and modular forms.

Summary: In this chapter, we recall some of the aspects of the theory of modular curves and modular forms. In particular, we discuss how to relate spaces of differentials on certain models of modular curves and Igusa curves to various spaces of cusp forms. Many results here are either in [G] in some form, or based on results in [G]. Many have their origins in [DR] and [KM].

§3.1. Morphisms of moduli problems.

We define various moduli problems and morphisms between them. We will follow the conventions of [KM]. Many results in this chapter are in [G] in some form, but [G] uses slightly different conventions, and so some of the results we obtain are slightly different to those found in [G]. The main difference is that Gross works with embeddings of μ_{N} to define a Γ_{1}(N)-structure, whereas we follow [KM] and work with points of exact order N (see below). As a consequence, some formulae we obtain will differ slightly from results found in [G]. Also, we will need results about curves classifying elliptic curves with Γ_{0}(p)- structures as well as Γ_{1}(N)-structures, and so in some cases the results in [G] are slightly generalised here.

Let N ≥ 5 be a positive integer, and let l ≥ 3 and p be distinct primes neither of which divide N. Let R = ℤ_{l}[ζ_{l}], and fix once and for all ζ_{l} ∈ R a primitive l-th root of unity. Let K be the field of fractions of R. We will keep this notation throughout this chapter. We sometimes use π = 1 − ζ_{l} for a uniformiser in R. Because we will later want to talk about q-expansions, and are already using p and l, we frequently use the letter r to denote a prime.

We now describe well-known semi-stable models of modular curves. Much of what is here is in [KM] but most of it in this case dates back to [DR]. We shall stick to the notation of [KM]. Firstly, we recall the following fundamental definitions of various structures on elliptic curves over schemes. For details see [KM]. Let E/S be an elliptic curve, where S is a scheme over Spec(R).

A Γ_{1}(N)-structure on E/S/R is a homomorphism of groups ϕ : ℤ/Nℤ → E[N](S) which is a ℤ/Nℤ-structure on E/S in the language of [KM]. The related moduli problem on (Ell/R) is called [Γ_{1}(N)] ⊗ R. There is an action ⟨ ⟩_{N} of (ℤ/Nℤ)^{×} on these structures, defined thus. If d ∈ (ℤ/Nℤ)^{×}, and (E, ϕ) is an elliptic curve with a Γ_{1}(N)-structure, then we define

⟨d⟩_{N}(E, ϕ) = (E, dϕ)

which is also an elliptic curve with a Γ_{1}(N)-structure.

A canonical balanced Γ_{1}(l)-structure on E/S/R is a collection of data

(π, π^{t}, P, P′),

where • π : E → E′ is a cyclic isogeny of degree l,

• π^{t} : E′ → E is the dual isogeny, • P ∈ E(S) generates the kernel of π, in the sense of [KM] (3.2.4),

• P′ ∈ E′(S) generates the kernel of π^{t}, such that the canonical pairing ⟨ , ⟩_{π} on ker(π) × ker(π^{t}) satisfies ⟨P, P′⟩_{π} = ζ_{l}. To fix our ideas, we will use the pairing given in [KM] 2.8.1. This is the inverse of that used in [O]. The related moduli problem on (Ell/R) is technically called ([bal.Γ_{1}(l)] ⊗ ℤ_{l})^{can}. There is an action ⟨ ⟩_{l} of (ℤ/lℤ)^{×} on this problem. If d ∈ (ℤ/lℤ)^{×} then define

⟨d⟩_{l}(E, π, π^{t}, P, P′) = (E, π, π^{t}, dP, d^{−1}P′).

Note also that if (π, π^{t}, P, P′) is a canonical balanced Γ_{1}(l)-structure on E/S then (π^{t}, π, P′, −P) is a canonical balanced Γ_{1}(l)-structure on E′/S, as ⟨P, P′⟩_{π}⟨P′, P ⟩_{π^{t}} = 1.

A Γ_{0}(p)-structure on E/S is a cyclic p-isogeny ψ : E → E_{1}. The related moduli problem is called [Γ_{0}(p)] ⊗ R.

Let n ∈ ℤ satisfy np ≡ 1 mod l.

Construction 3.1.1. Let E/S be an elliptic curve with a Γ_{1}(N)-structure ϕ, a canonical balanced Γ_{1}(l)-structure (π, π^{t}, P, P′), and a Γ_{0}(p)-structure ψ : E → E_{1}. Then ψ induces a Γ_{1}(N)-structure ϕ_{1} and a canonical balanced Γ_{1}(l)-structure (π_{1}, π_{1}^{t}, P_{1}, P′_{1}) on E_{1} in the following way:

Define ϕ_{1} = ψ ∘ ϕ : ℤ/Nℤ → E_{1}(S). Then ϕ_{1} is a Γ_{1}(N)-structure on E_{1}.

The composite π ∘ ψ^{t} : E_{1} → E′ is cyclic of degree lp by [KM] Proposition 6.7.10 and hence has a standard factorisation as

E_{1}⟶^{π_{1}}E′_{1}⟶^{ρ}E′,

where π_{1} has degree l and ρ has degree p. Set P_{1} = ψ(P) and P′_{1} = nρ^{t}(P′) where n ∈ ℤ is as above. Then (π_{1}, π_{1}^{t}, P_{1}, P′_{1}) is a canonical balanced Γ_{1}(l)-structure on E_{1}, using [KM] (2.8.3) and (2.8.4.1). Moreover, the construction

(E, ϕ) ↦ (E_{1}, ϕ_{1})

commutes with the action of (ℤ/Nℤ)^{×} and the construction

(E, π, π^{t}, P, P′) ↦ (E_{1}, π_{1}, π_{1}^{t}, P_{1}, P′_{1})

commutes with the action of (ℤ/lℤ)^{×}.

Construction 3.1.2. Say E/S is an elliptic curve with a Γ_{1}(N)-structure ϕ, a canonical balanced Γ_{1}(l)-structure (π, π^{t}, P, P′) where π : E → E′, and a Γ_{0}(p)-structure ψ : E → E_{1}. Then the structure at l induces a Γ_{1}(N)-structure ϕ′ and a Γ_{0}(p)-structure ψ′ on E′ as follows. The composite

ϕ′ = π ∘ ϕ : ℤ/Nℤ → E′(S)

is a Γ_{1}(N)-structure, and the composite ψ∘π^{t} : E′ → E_{1} has degree lp and hence factorises as π_{l} ∘ ψ′ with ψ′ of degree p and π_{l} of degree l. Then ψ′ is a Γ_{0}(p)-structure on E′. Hence if (E/S, ϕ, ψ, π, π^{t}, P, P′) is an elliptic curve with a Γ_{1}(N), Γ_{0}(p) and canonical balanced Γ_{1}(l)-structure, then so is (E′/S, ϕ′, ψ′, π^{t}, π, P′, −P).

We turn now to the case where S is a scheme over 𝔽_{l}.

Construction 3.1.3. Say E/S/𝔽_{l} is an elliptic curve with a Γ_{1}(N)-structure ϕ and a Γ_{0}(p)-structure ψ. Say F : E → E^{(l)} is the Frobenius map. Then

ϕ^{(l)} = F ∘ ϕ : ℤ/Nℤ → E^{(l)}(S)

is a Γ_{1}(N)-structure on E^{(l)}, and ψ : E → E_{1} induces

ψ^{(l)} : E^{(l)} → E_{1}^{(l)},

which can be checked to be a Γ_{0}(p)-structure on E^{(l)}. Moreover,

(E, ϕ) ↦ (E^{(l)}, ϕ^{(l)})

commutes with the action of (ℤ/Nℤ)^{×}.

The reduction of the moduli problem ([bal.Γ_{1}(l)] ⊗ ℤ_{l})^{can} on (Ell/R) to (Ell/𝔽_{l}) is called [bal.Γ_{1}(l); det = 1]. A (bal.Γ_{1}(l); det = 1)-structure on an elliptic curve E/S/𝔽_{l} is a collection (π, π^{t}, P, P′) with

π : E → E′ a cyclic l-isogeny with kernel generated by P ∈ E(S), π^{t} : E′ → E the dual isogeny with kernel generated by P′ ∈ E′(S), such that ⟨P, P′⟩_{π} = 1. The action of (ℤ/lℤ)^{×} on ([bal.Γ_{1}(l)] ⊗ ℤ_{l})^{can} induces an

action of (ℤ/lℤ)^{×} on [bal.Γ_{1}(l); det = 1].

If S is a scheme over 𝔽_{l} then there is one other structure on E/S that we shall need. An Igusa structure at level l, or just an Igusa structure, is a point P ∈ E^{(l)}(S) generating the kernel of the Verschiebung map V : E^{(l)} → E. The corresponding moduli problem over 𝔽_{l} is called [Ig(l)]. There is an action ⟨ ⟩_{l} of (ℤ/lℤ)^{×} on this problem:

⟨d⟩_{l}(E, P) = (E, dP).

Construction 3.1.4. If E/S/𝔽_{l} has an Igusa structure P ∈ E^{(l)} then

(F, V, 0, P)

is a (bal.Γ_{1}(l); det = 1)-structure on E, and

(V, F, P, 0)

is a (bal.Γ_{1}(l); det = 1)-structure on E^{(l)}. Moreover,

(E, P) ↦ (E^{(l)}, V, F, P, 0)

commutes with the action of (ℤ/lℤ)^{×}, and the construction

(E, P) ↦ (E, F, V, 0, P)

inverts the action, that is

⟨d⟩_{l}(E, P) ↦ ⟨d⟩_{l}^{−1}(E, F, V, 0, P).

§3.2. Modular curves and Igusa curves.

We will now briefly describe many things which are known about the moduli problems [Γ_{1}(N)] ⊗ R and ([Γ_{1}(N)] ⊗ R, ([bal.Γ_{1}(l)] ⊗ ℤ_{l})^{can}). Everything we say about these problems will have analogues for the problems ([Γ_{1}(N)], [Γ_{0}(p)]) ⊗ R and (([Γ_{1}(N)], [Γ_{0}(p)]) ⊗ R, ([bal.Γ_{1}(l)] ⊗ ℤ_{l})^{can}), and we shall sometimes allude to these analogues, particularly when establishing notation.

[Γ_{1}(N)]⊗R is finite and étale over (Ell/R). The problems [Γ_{1}(N)]⊗R and ([Γ_{1}(N)]⊗ R, ([bal.Γ_{1}(l)] ⊗ ℤ_{l})^{can}) are representable, and are represented by elliptic curves over schemes

Y_{1}(N)_{R} = M([Γ_{1}(N)] ⊗ R)

and

Y_{1}(Nl)_{R} = M([Γ_{1}(N)] ⊗ R, ([bal.Γ_{1}(l)] ⊗ ℤ_{l})^{can})

respectively. Note that the N and l appear to be on an equal footing in the expression Y_{1}(Nl)_{R}, but in this case the structure at l is a canonical balanced one, and the structure at N is not. In the case where we are also interested in a [Γ_{0}(p)]-structure, we shall write

Y_{1}(N; p)_{R} = M([Γ_{1}(N)], [Γ_{0}(p)])

and so on.

If A is any R-algebra, and Y_{R} is any scheme over R, we shall write Y_{A} for the pullback of Y_{R} to A. We shall frequently omit the suffix if it is R. So, for example, Y_{1}(N) will be taken to mean Y_{1}(N)_{R}.

The schemes Y_{1}(N) and Y_{1}(Nl) are affine over Spec(R). Using techniques from [KM] Chapter 8 we may form the compactifications

X_{1}(N)_{R} = \overline{M}([Γ_{1}(N)] ⊗ R)

and

X_{1}(Nl)_{R} = \overline{M}([Γ_{1}(N)] ⊗ R, ([bal.Γ_{1}(l)] ⊗ ℤ_{l})^{can}).

The special fibre X_{1}(N)_{𝔽_{l}} of X_{1}(N) is a smooth curve, because l does not divide N. The special fibre of X_{1}(Nl) is more subtle. Using techniques from [KM] Chapter 13, it can be shown that X_{1}(Nl) is an amenable curve as defined in Definition 2.4.2. In fact, its special fibre is the union of two irreducible curves C_{1} and C_{2} over 𝔽_{l}, which cross transversally at finitely many points, corresponding to supersingular elliptic curves in characteristic l. The curves C_{1} and C_{2} are both isomorphic to the curve

I_{1}(N)_{𝔽_{l}} = \overline{M}([Γ_{1}(N)] ⊗ 𝔽_{l}, [Ig(l)]).

The analogous result for X_{1}(Nl; p) is that it is an amenable curve, and that both components of its special fibre are isomorphic to

I_{1}(N; p)_{𝔽_{l}} = \overline{M}(([Γ_{1}(N)], [Γ_{0}(p)]) ⊗ 𝔽_{l}, [Ig(l)]).

Because the Igusa curves I_{1}(N) and I_{1}(N; p) only exist in characteristic l, we will omit the suffix if it is 𝔽_{l}. So I_{1}(N) = I_{1}(N)_{𝔽_{l}}.

An important remark is that the compactified curve X_{1}(N) is also the solution to a moduli problem, that is, the problem of classifying generalised elliptic curves with a Γ_{1}(N)-structure. Because of this fact, the universal elliptic curve E_{Y_{1}(N)} over Y_{1}(N) is in fact the pullback of a universal generalised elliptic curve E_{X_{1}(N)} over X_{1}(N).

Note also that in many cases to define a morphism between compactified structures of this form, for example a morphism from X_{1}(N; p) to X_{1}(N), we shall give a recipe for a morphism between non-compactified structures, i.e. a morphism between moduli problems, with the (often implicit) assertion that it extends uniquely to a morphism between compactified structures. This assertion can either be verified by examining the construction of the compactifications in [KM] Chapter 8, or by thinking of the compactified structures as the solutions to a moduli problem for generalised elliptic curves as in [DR].

We can define a morphism

j_{1} : M([Γ_{1}(N)] ⊗ 𝔽_{l}, [Ig(l)]) → Y_{1}(Nl)_{𝔽_{l}}

by the morphism of moduli problems

(E/S/𝔽_{l}, ϕ, P ∈ E^{(l)}) ↦ (E/S/𝔽_{l}, ϕ, (F, V, 0, P)),

and this morphism extends uniquely to give a morphism

j_{1} : I_{1}(N) → X_{1}(Nl)_{𝔽_{l}}.

The exotic morphism

j_{2} : M([Γ_{1}(N)] ⊗ 𝔽_{l}, [Ig(l)]) → Y_{1}(Nl)_{𝔽_{l}}

given by

(E/S/𝔽_{l}, ϕ, P ∈ E^{(l)}) ↦ (E^{(l)}/S/𝔽_{l}, ϕ^{(l)}, (V, F, P, 0)),

where ϕ^{(l)} is as in Construction 3.1.3, extends uniquely to a map between compactified structures

j_{2} : I_{1}(N) → X_{1}(Nl)_{𝔽_{l}}.

These maps j_{i}, 1 ≤ i ≤ 2 identify the two components of the special fibre of X_{1}(Nl) with I_{1}(N). Say C_{i} is the image of j_{i}.

For any integer d prime to N the automorphism ⟨d⟩_{N} of [Γ_{1}(N)]⊗R sending (E, ϕ) to (E, dϕ) induces an action of (ℤ/Nℤ)^{×} on Y_{1}(Nl), which extends to an action on X_{1}(Nl). This action induces an action of (ℤ/Nℤ)^{×} on the special fibre C_{1} ∪ C_{2}, which in turn induces actions on C_{1} and on C_{2}. The action of (ℤ/Nℤ)^{×} on [Γ_{1}(N)] ⊗ R also induces an action on I_{1}(N), and the maps j_{i}, 1 ≤ i ≤ 2 commute with this action.

For any integer d prime to l the automorphism ⟨d⟩_{l} of [bal.Γ_{1}(l) ⊗ ℤ_{l}]^{can} sending (E, π, π^{t}, P, P′) to (E, π, π^{t}, dP, d^{−1}P′) induces an action of (ℤ/lℤ)^{×} on X_{1}(Nl). Again this induces actions of (ℤ/lℤ)^{×} on the special fibre X_{1}(Nl)_{𝔽_{l}} and on its components C_{1} and C_{2}. The action of (ℤ/lℤ)^{×} on [Ig(l)] induces an action on I_{1}(N). This time, the action of (ℤ/lℤ)^{×} commutes with j_{2}, but the action on I_{1}(N) coincides with the inverse of the action on C_{1} when identified via j_{1}. When we talk of the action of (ℤ/lℤ)^{×} on C_{1} we shall always think of this as the action induced by the action on X_{1}(Nl), and not as the action which is natural when C_{1} is identified with I_{1}(N).

§3.3. Modular forms and q-expansions.

We now define various invertible sheaves on the curves described above. Let E/X_{1}(N) be the universal generalised elliptic curve over X_{1}(N), with p : E → X_{1}(N). We define the sheaf ω_{X_{1}(N)} to be the dual of the sheaf of Lie algebras Lie(E^{reg}) on X_{1}(N). Then ω_{X_{1}(N)} is invertible. Let ω_{I_{1}(N)} on I_{1}(N) be the pullback of ω_{X_{1}(N)} via the natural maps I_{1}(N) → X_{1}(N)_{𝔽_{l}} → X_{1}(N).

Define the sheaf Ω_{X_{1}(N)} on X_{1}(N) to be the sheaf of differentials Ω_{X_{1}(N)/R}, and define the sheaf Ω_{I_{1}(N)} on I_{1}(N) to be the sheaf of differentials Ω_{I_{1}(N)/𝔽_{l}}.

We now define spaces of modular and cusp forms of level N. For k ≥ 2 define the space of weight k modular forms for X_{1}(N) to be

M_{k}(Γ_{1}(N); R) = H^{0}(X_{1}(N), ω_{X_{1}(N)}^{⊗k}).

Define the space of weight k cusp forms for X_{1}(N) to be

S_{k}(Γ_{1}(N); R) = H^{0}(X_{1}(N), ω_{X_{1}(N)}^{⊗(k−2)} ⊗ Ω_{X_{1}(N)}).

There is an isomorphism (the Kodaira-Spencer isomorphism)

ω_{X_{1}(N)}^{⊗2} ≅ Ω_{X_{1}(N)}(cusps),

where (cusps) is the divisor consisting of the cusps of X_{1}(N). This isomorphism gives us a canonical embedding

S_{k}(Γ_{1}(N); R) ↪ M_{k}(Γ_{1}(N); R).

Similar constructions work for X_{1}(N; p) giving us the spaces M_{k}(Γ_{1}(N) ∩ Γ_{0}(p); R) and S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); R). If A is any R-algebra then the universal generalised elliptic curve over X_{1}(N)_{A} gives by a similar construction the spaces M_{k}(Γ_{1}(N); A) and S_{k}(Γ_{1}(N); A).

Note that because k ≥ 2 we have that M_{k}(Γ_{1}(N); A) = M_{k}(Γ_{1}(N); R) ⊗ A, and similarly for spaces S_{k} of cusp forms.

If we choose an embedding R ↪ ℂ then ℂ becomes an R-algebra and our definitions coincide with the usual ones, for example S_{k}(Γ_{1}(N); ℂ) coincides with the usual definition of cusp forms of weight k and level N with coefficients in ℂ.

A useful tool in understanding maps between spaces of modular forms is that of qexpansions. In fact we shall define q-expansions, not only for cusp forms, but also for differentials on various curves. We again make use of the fact that our compactified curves classify generalised elliptic curves with extra structure.

We wish to look at q-expansions of several kinds of objects. To define these qexpansions, we need to look at a specific generalised elliptic curve, the Tate curve. Say A is any commutative ring. Set S = Spec(A[[q]]). There is a generalised elliptic curve, the curve Tate(q)/S, which can loosely be thought of as 𝔾_{m}/q^{ℤ}. We can put various structures on Tate(q)/S, if A has various properties, as detailed below.

If there is a primitive N-th root of unity ζ_{N} in A then we can define ϕ : ℤ/Nℤ → Tate(q)(S) by 1 ↦ ζ_{N}. Then ϕ is a Γ_{1}(N)-structure on Tate(q).

If A contains a primitive l-th root of unity then we can define the following canonical balanced Γ_{1}(l)-structure on Tate(q). Set P = ζ_{l} ∈ Tate(q)(S). Then we define π such that P generates the kernel of π thus.

π : Tate(q) → Tate(q)/⟨ζ_{l}⟩
= 𝔾_{m}/⟨ζ_{l}, q⟩
≅ 𝔾_{m}/q^{lℤ}
= Tate(q^{l}).

Set P′ = q ∈ Tate(q^{l})(S) generating the kernel of π^{t}. It can be checked that ⟨P, P′⟩_{π} = ζ_{l} and hence that (π, π^{t}, P, P′) is a canonical balanced Γ_{1}(l)-structure on Tate(q)/S.

By replacing l by p in the definition of π above we get a map ψ : Tate(q) → Tate(q^{p}). This is a Γ_{0}(p)-structure on Tate(q).

Finally, if A is an 𝔽_{l}-algebra, we can define an Igusa structure on Tate(q) thus. We have V : Tate(q^{l}) → Tate(q) defined by the natural surjection. We set

P_{Ig} = q ∈ Tate(q^{l}) = Tate(q)^{(l)}.

Now if A = R[ζ_{N}] and S = Spec(A[[q]]) then (Tate(q)/S, ϕ) is a generalised elliptic curve with a [Γ_{1}(N)]-structure and hence defines a morphism

τ_{∞} : Spec(R[ζ_{N}][[q]]) → X_{1}(N).

Similarly the generalised elliptic curve (Tate(q), ϕ, π, π^{t}, P, P′) defines a morphism

τ_{∞} : Spec(R[ζ_{N}][[q]]) → X_{1}(Nl).

If A = 𝔽_{l}[ζ_{N}] and S = Spec(A[[q]]) then (Tate(q), ϕ, P_{Ig}) defines a morphism

τ_{∞} : Spec(𝔽_{l}[ζ_{N}][[q]]) → I_{1}(N).

In all three cases, the image of the point q = 0 is by definition the cusp ∞, and q is a uniformiser in a neighbourhood of this cusp.

The first class of objects for which we will define q-expansions is modular forms with coefficients in R. Given

f ∈ M_{k}(Γ_{1}(N); R) = H^{0}(X_{1}(N), ω_{X_{1}(N)}^{⊗k}),

we can form the pullback

τ_{∞}^{∗}f ∈ H^{0}(S, ω_{Tate(q)/S}^{⊗k}) = H^{0}(Tate(q), Ω_{Tate(q)/S}^{⊗k})

and this pullback will be of the form s.ω_{can}^{⊗k} for some s ∈ R[ζ_{N}][[q]], where ω_{can} is the canonical differential dt/t on Tate(q). We define the q-expansion of f to be s, and write

q-exp(f) = s.

A similar construction works if f ∈ M_{k}(Γ_{1}(N); B) for B any R-algebra. Because the space of cusp forms can be identified with a subspace of the space of modular forms, we also obtain a notion of q-expansion for spaces of cusp forms.

Next we define the q-expansion of an element of H^{0}(X_{1}(Nl), Ω_{X_{1}(Nl)/R}), where the sheaf Ω_{X_{1}(Nl)/R} is the dualising sheaf on X_{1}(Nl). We have

τ_{∞} : Spec(R[ζ_{N}][[q]]) → X_{1}(Nl).

Pulling back differentials gives us a notion of q-expansion. If S = Spec(R[ζ_{N}][[q]]) then the regular differential

ν ∈ H^{0}(X_{1}(Nl), Ω_{X_{1}(Nl)/R})

pulls back via τ_{∞}^{∗} to

τ_{∞}^{∗}ν ∈ H^{0}(S, Ω_{S/R[ζ_{N}]})

which can be written as

τ_{∞}^{∗}ν = s(dq/q)

where s ∈ R[ζ_{N}][[q]]. Here s/q is the expansion of the differential ν at ∞. We define s to be the q-expansion of ν.

An analogous construction gives the notion of a q-expansion on H^{0}(X_{1}(N), Ω_{X_{1}(N)}).

Next we define q-expansions of elements of H^{0}(I_{1}(N), ω_{I_{1}(N)}^{⊗k}). If we define S = Spec(𝔽_{l}[ζ_{N}][[q]]), then for any

ν ∈ H^{0}(I_{1}(N), ω_{I_{1}(N)}^{⊗k})

we can form the pullback of ν via the map

τ_{∞} : S → I_{1}(N)

to give us

τ_{∞}^{∗}ν ∈ H^{0}(S, ω_{Tate(q)/S}^{⊗k}) = H^{0}(Tate(q), Ω_{Tate(q)/S}^{⊗k}).

Hence τ_{∞}^{∗}ν = s(dt/t)^{⊗k} for some s ∈ 𝔽_{l}[ζ_{N}][[q]]. We define the q-expansion of ν to be s. Finally, we define q-expansions of elements of H^{0}(I_{1}(N), Ω_{I_{1}(N)}). If

ν ∈ H^{0}(I_{1}(N), Ω_{I_{1}(N)})

then we can pull ν back to an element

τ_{∞}^{∗}ν ∈ H^{0}(S, Ω_{S/𝔽_{l}[ζ_{N}]})

which can be written as

τ_{∞}^{∗}ν = s(dq/q).

Here s is an element of 𝔽_{l}[ζ_{N}][[q]]. We define the q-expansion of ν to be s. This is just the expansion of the differential ν in a neighbourhood of infinity.

The Γ_{0}(p)-structure defined above on the Tate curve enables us to use similar techniques to define q-expansions on the following objects:

M_{k}(Γ_{1}(N) ∩ Γ_{0}(p); R),
H^{0}(X_{1}(Nl; p), Ω_{X_{1}(Nl;p)/R}),
H^{0}(X_{1}(N; p), Ω_{X_{1}(N;p)/R}),
H^{0}(I_{1}(N; p), ω_{I_{1}(N;p)}^{⊗k}),
and H^{0}(I_{1}(N; p), Ω_{I_{1}(N;p)/𝔽_{l}}).

§3.4. Hecke operators.

We now discuss the theory of Hecke operators. These operators can be viewed as correspondences on curves or as endomorphisms of spaces of modular forms, cusp forms, or differentials. We briefly recall their definitions.

Firstly we have the diamond operators, which have already been defined as endomorphisms ⟨d⟩_{N} of X_{1}(N). These endomorphisms induce endomorphisms ⟨d⟩_{N} of H^{0}(X_{1}(N), ω_{X_{1}(N)}^{⊗k}), and also by pullback endomorphisms ⟨d⟩_{N}^{∗} of H^{0}(X_{1}(N), Ω_{X_{1}(N)}).

Next we define T_{r}, for any prime r ≠ l not dividing N. Consider the curve

X_{1}(N; r)_{R} = \overline{M}(([Γ_{1}(N)], [Γ_{0}(r)]) ⊗ R).

We define two maps

1_{r}, η_{r} : X_{1}(N; r) → X_{1}(N)

as follows. The map 1_{r} forgets the structure at r, and the map η_{r} quotients out by this structure, using Construction 3.1.1 (with r replacing p). These two maps give rise to a correspondence T_{r} on X_{1}(N), which on non-cuspidal points can be thought of as the map

(E, α) ↦ ∑_{σ}(σE, σα)

where σ runs through the cyclic r-isogenies with domain E and the level structure α is pushed forward to a structure σα on the image of σ as in Construction 3.1.1. This correspondence induces an endomorphism (1_{r})_{∗}(η_{r})^{∗} on the space H^{0}(X_{1}(N), Ω_{X_{1}(N)}) of regular differentials. Here (1_{r})_{∗} is the trace map.

We also define, for a prime r not dividing N, an action of T_{r} on H^{0}(X_{1}(N), ω_{X_{1}(N)}^{⊗k}) analogously to [K1] (1.11.0.2) (note that Katz has a full level N structure). If r = l and we are considering spaces of forms mod l, then we shall frequently refer to the operator T_{l} as U_{l}.

The effect of these Hecke operators on q-expansions is well-known. We recall that if f ∈ H^{0}(X_{1}(N), ω_{X_{1}(N)}^{⊗k}), with

q-exp(f) = ∑ a_{n}q^{n}
and q-exp(f|⟨r⟩_{N}) = ∑ b_{n}q^{n},

then we have

q-exp(f|T_{r}) = ∑ a_{nr}q^{n} + r^{k−1} ∑ b_{n}q^{nr}.

Similarly, if ω ∈ H^{0}(X_{1}(N), Ω_{X_{1}(N)}) such that

q-exp(ω) = ∑ a_{n}q^{n}
and q-exp(⟨r⟩_{N}^{∗}ω) = ∑ b_{n}q^{n},

then we have

q-exp(T_{r}^{∗}ω) = ∑ a_{nr}q^{n} + r ∑ b_{n}q^{nr}.

For any prime r dividing N there is also a standard correspondence U_{r} on X_{1}(N), which also induces endomorphisms of modular forms and differentials in essentially the same way as T_{r}. On q-expansions it is well-known that if

q-exp(f) = ∑ a_{n}q^{n}

then

q-exp(f|U_{r}) = ∑ a_{nr}q^{n}.

Note that the inclusion

H^{0}(X_{1}(N), Ω_{X_{1}(N)}) ↪ H^{0}(X_{1}(N), ω_{X_{1}(N)}^{⊗2})

given by the Kodaira-Spencer isomorphism preserves q-expansions and also the actions of T_{r}, U_{r} and ⟨d⟩_{N}.

We define the Hecke operator S_{r}, for r prime to N, as follows. S_{r} acts on spaces of modular forms, and is equal to r^{k−2}⟨r⟩_{N} on forms of weight k and level N. S_{r} also acts on spaces of differentials as ⟨r⟩_{N}.

Note that the endomorphisms T_{r}, U_{r}, ⟨d⟩_{N} and S_{r} all commute when considered either as endomorphisms of modular forms or of differentials. Note also that we will write endomorphisms of spaces of modular forms and cusp forms on the right, and endomorphisms of spaces of differentials on the left. This will hopefully cause no confusion when we identify H^{0}(X_{1}(N), Ω_{X_{1}(N)}) with a subspace of H^{0}(X_{1}(N), ω_{X_{1}(N)}^{⊗2}) because all these operators commute.

There are natural correspondences T_{r} for r prime to Nl, and U_{r} for r dividing N, on the curve I_{1}(N). These correspondences are defined in essentially exactly the same way as the ones on X_{1}(N), and induce maps T_{r}^{∗} and U_{r}^{∗} on H^{0}(I_{1}(N), Ω_{X_{1}(N)}).

Now we turn our attention to the curve X_{1}(Nl)_{R}. The generic fibre X_{1}(Nl)_{K} of this curve has a model X_{1}(Nl)_{ℚ} over ℚ, which classifies generalised elliptic curves with a Γ_{1}(N)- structure and a Γ_{1}(l)-structure over ℚ-schemes. We can define as above correspondences ⟨d⟩_{N}, ⟨d⟩_{l}, T_{r} for r prime to Nl, and U_{r} for r dividing Nl, on X_{1}(Nl)_{ℚ}, and these correspondences induce commuting endomorphisms of H^{0}(X_{1}(Nl)_{ℚ}, Ω_{X_{1}(Nl)_{ℚ}/ℚ}), and hence of H^{0}(X_{1}(Nl)_{K}, Ω_{X_{1}(Nl)_{K}/K}). By the discussion in [G] §8 we have a lattice

L = H^{0}(X_{1}(Nl)_{R}, Ω_{X_{1}(Nl)/R})

in H^{0}(X_{1}(Nl)_{K}, Ω_{X_{1}(Nl)_{K}/K}), where the sheaf Ω_{X_{1}(Nl)/R} is the dualising sheaf. By [G] Proposition 8.7, the endomorphisms T_{r}^{∗}, U_{r}^{∗}, ⟨d⟩_{l}^{∗} and ⟨d⟩_{N}^{∗} extend to endomorphisms of L.

We define the automorphism w = w_{ζ_{l}} of X_{1}(Nl) as follows. It is induced by Construction 3.1.2, and is

w : (E/S, ϕ, π, π^{t}, P, P′) ↦ (E′/S, ϕ′, π^{t}, π, P′, −P)

using the notation of Construction 3.1.2, on non-cuspidal points. A similar construction gives us a map w : X_{1}(Nl; p) → X_{1}(Nl; p). Note that w interchanges the two components of the special fibre of these curves. The map w gives us an endomorphism w^{∗} of L. Note however that w^{∗} does not commute with the actions of T_{r} and U_{r} on L in general, and in fact

w^{∗}T_{r}^{∗} = ⟨r⟩_{l}^{∗}T_{r}^{∗}w^{∗}

for r not dividing Nl, and

w^{∗}U_{r}^{∗} = ⟨r⟩_{l}^{∗}U_{r}^{∗}w^{∗}

for r dividing N.

§3.5. Differentials and regular differentials.

We now examine I_{1}(N) more closely, with its action of (ℤ/lℤ)^{×}. This action extends to an action of (ℤ/lℤ)^{×} on the space of differentials H^{0}(I_{1}(N), Ω_{I_{1}(N)}). This action decomposes the finite dimensional 𝔽_{l}-vector space H^{0}(I_{1}(N), Ω_{I_{1}(N)}) into the direct sum of eigenspaces

H^{0}(I_{1}(N), Ω_{I_{1}(N)}) = ⨁_{j=1}^{j=l−1} H^{0}(I_{1}(N), Ω_{I_{1}(N)})(j)

where, for 1 ≤ j ≤ l − 1,

H^{0}(I_{1}(N), Ω_{I_{1}(N)})(j) = {x ∈ H^{0}(I_{1}(N), Ω_{I_{1}(N)}) : ⟨d⟩_{l}^{∗}x = d^{j}x}.

There is a Hasse invariant

A ∈ M_{k}(Γ_{1}(N); 𝔽_{l})

which has a canonical (l − 1)-th root

a = A^{1/(l−1)} ∈ H^{0}(I_{1}(N), ω_{I_{1}(N)}),

which satisfies

q-exp(a) = 1,

and

a|⟨d⟩_{l} = da,

where here the action of ⟨d⟩_{l} is induced by the action on I_{1}(N).

The explicit construction in [KM] Theorem 12.8.8 gives us an isomorphism

S_{l+1−j}(Γ_{1}(N); 𝔽_{l})⟶^{≅}H^{0}(I_{1}(N), Ω_{I_{1}(N)})(j)  (†)

for 1 ≤ j ≤ l − 1. If π : I_{1}(N)_{𝔽_{l}} → X_{1}(N)_{𝔽_{l}} is the projection, then this isomorphism is given by f ↦ (π^{∗}f)/a^{⊗(l−1−j)}. Moreover, this isomorphism preserves q-expansions (as a has q-expansion equal to 1) and also the action of (ℤ/Nℤ)^{×}. However, it twists the actions of T_{r} for r prime to Nl, and of U_{r} for r dividing N. One way of seeing how the Hecke operators behave with respect to this isomorphism is to calculate explicitly their behaviour on q-expansions on both sides. However, we shall sketch another approach.

We may interpret an element of S_{k}(Γ_{1}(N); 𝔽_{l}) as a rule which, to an elliptic curve E/S/𝔽_{l} equipped with a Γ_{1}(N)-structure, attaches a section of ω_{E/S}^{⊗k} = π_{∗}Ω_{E/S} where π : E → S. This rule must satisfy certain axioms (base change, conditions at cusps and so on). An element of H^{0}(I_{1}(N), ω^{⊗k}) may be interpreted similarly. It can be checked that if (E, ϕ, P) is an elliptic curve with a Γ_{1}(N)-structure ϕ and an Igusa structure P, and ψ : E → E_{1} is an isogeny of degree r, then with the definition of the element a of H^{0}(I_{1}(N), ω_{I_{1}(N)}) given in [KM], we have

ψ^{∗}a(E_{1}, ψ ∘ ϕ, ψ^{(l)}(P)) = ra(E, ϕ, P),

where ψ^{∗} is the natural map from ω_{E_{1}/S} to ω_{E/S}. Now a calculation analogous to [G] Proposition 3.15 shows us that the Hecke operator T_{r} on S_{l+1−j}(Γ_{1}(N); 𝔽_{l}) gets identified via the isomorphism (†) above with the operator ⟨r⟩_{l}^{−1}T_{r} = r^{l−1−j}T_{r} on H^{0}(I_{1}(N), Ω_{I_{1}(N)})(j). Similarly, the operator on cusp forms U_{r} for r dividing N becomes identified with the operator r^{l−1−j}U_{r} on differentials.

Recall L = H^{0}(X_{1}(Nl), Ω_{X_{1}(Nl)/R}), where Ω_{X_{1}(Nl)/R} is the sheaf of regular differentials on X_{1}(Nl). We have that L is a free R-module of finite rank, such that

L ⊗ K = H^{0}(X_{1}(Nl)_{K}, Ω_{X_{1}(Nl)_{K}/K})

for K = ℚ_{l}(ζ_{l}), and

L ⊗ 𝔽_{l} = H^{0}(X_{1}(Nl)_{𝔽_{l}}, Ω_{X_{1}(Nl)_{𝔽_{l}} /𝔽_{l}})) = H^{0}(C_{1} ∪ C_{2}, Ω_{C_{1}∪C_{2}/𝔽_{l}}),

where, as before, the special fibre of X_{1}(Nl)_{R} is C_{1} ∪ C_{2}, and again Ω_{C_{1}∪C_{2}/𝔽_{l}} is the dualising sheaf. The space H^{0}(C_{1} ∪ C_{2}, Ω_{C_{1}∪C_{2}/𝔽_{l}}) can be identified with the space of meromorphic differentials (ν_{1}, ν_{2}) on C_{1} ∐ C_{2} which are regular outside C_{1} ∩ C_{2}, which have at worst simple poles on C_{1} ∩ C_{2}, and for which Res_{s}(ν_{1})+Res_{s}(ν_{2}) = 0 for all s ∈ C_{1} ∩ C_{2}. We have the notion of q-expansion for ν_{1} and ν_{2} because we can define q-exp(ν_{i}) = ∑ a_{n}q^{n} where (∑ a_{n}q^{n})(dq/q) is the expansion of ν_{i} in a neighbourhood of ∞, when C_{i} is identified with I_{1}(N). We can see the relationship between these qexpansions and the q-expansion of ω = (ν_{1}, ν_{2}) ∈ L as follows. If ω ∈ L is such that q-exp(ω) = ∑ A_{n}q^{n} and q-exp(w^{∗}ω) = ∑ B_{n}q^{n} then by the commutative diagrams

Spec(𝔽_{l}[ζ_{N}][[q]]) ⟶^{τ_{∞}} I_{1}(N)
↓ ↓^{j_{1}}
Spec(R[ζ_{N}][[q]]) ⟶^{τ_{∞}} X_{1}(Nl)

and

Spec(𝔽_{l}[ζ_{N}][[q]]) ⟶^{τ_{∞}} I_{1}(N)
↓ ↓^{j_{2}}
Spec(R[ζ_{N}][[q]]) ⟶^{w∘τ_{∞}} X_{1}(Nl)

we have that q-exp(ν_{1}) = ∑ a_{n}q^{n} and q-exp(ν_{2}) = ∑ b_{n}q^{n} with a_{n} the reduction of A_{n} modulo the maximal ideal of R[ζ_{N}] and similarly for B_{n} and b_{n}. There is an action of (ℤ/lℤ)^{×} on L induced by the action on X_{1}(Nl), and this induces an action on L/πL = \overline{L}, where π = 1 − ζ_{l}. The action breaks \overline{L} up into a direct sum

\overline{L} = ⨁_{j=1}^{j=l−1} \overline{L}(j).

Now say ω = (ν_{1}, ν_{2}) ∈ \overline{L}(j) for some j with 1 ≤ j ≤ l − 1. Because (ℤ/lℤ)^{×} acts trivially on the supersingular points in characteristic l, we see that if j ≠ l − 1 then both ν_{1} and ν_{2} are regular differentials on C_{1} and C_{2}. Hence if j ≠ l − 1, we have that any ω in \overline{L}(j) gives rise to

ν_{1}(ω) ∈ H^{0}(C_{1}, Ω_{C_{1}/𝔽_{l}})(j)

and

ν_{2}(ω) ∈ H^{0}(C_{2}, Ω_{C_{2}/𝔽_{l}})(j).

Conversely, any pair of ν_{i}, 1 ≤ i ≤ 2 of this form give rise to an ω ∈ \overline{L}(j). Hence we have

\overline{L}(j) ≅ H^{0}(C_{1}, Ω_{C_{1}/𝔽_{l}})(j) ⊕ H^{0}(C_{2}, Ω_{C_{2}/𝔽_{l}})(j)

for 1 ≤ j ≤ l − 2. Checking the isomorphisms j_{i} : I_{1}(N) → C_{i}, we recall that j_{1} inverts the action of (ℤ/lℤ)^{×} and j_{2} preserves it. So we can think of ν_{1}(ω) as an element of H^{0}(I_{1}(N), Ω_{I_{1}(N)})(l−1−j) and ν_{2}(ω) as an element of H^{0}(I_{1}(N), Ω_{I_{1}(N)})(j). Furthermore, we have identified these spaces with spaces of cusp forms, and so ν_{1}(ω) becomes identified with f_{1}(ω) ∈ S_{j+2}(Γ_{1}(N); 𝔽_{l}) and ν_{2}(ω) with f_{2}(ω) ∈ S_{l+1−j}(Γ_{1}(N); 𝔽_{l}). We conclude that if 1 ≤ j ≤ l − 2 then

\begin{matrix}\overline{L}(j) & ≅ S_{j+2}(Γ_{1}(N); 𝔽_{l}) ⊕ S_{l+1−j}(Γ_{1}(N); 𝔽_{l}) \\ ω & ↦ (f_{1}(ω), f_{2}(ω)). \end{matrix}

We see from [G] Proposition 8.13 that if ω ↦ (f_{1}, f_{2}) under this isomorphism, then

⟨d⟩_{N}^{∗}ω ↦ (f_{1}|⟨d⟩_{N}, f_{2}⟨d⟩_{N}),
T_{r}^{∗}ω ↦ (f_{1}|T_{r}, r^{j}f_{2}|T_{r}) for r prime to Nl,
U_{r}^{∗}ω ↦ (f_{1}|U_{r}, r^{j}f_{2}|U_{r}) for r dividing N,
and U_{l}^{∗}ω ↦ (f_{1}|U_{l}, 0).

We remark that the formula given in [G] Proposition 8.13 holds in our setting, despite the fact that our definition of X_{1}(N) is slightly different to that in [G]. In general, it is the formulae involving Igusa curves which tend to differ by a twist from formulae in [G].

We define for k ≥ 2 and 1 ≤ j ≤ l − 1 the space

S_{k}(Γ_{1}(N); 𝔽_{l})[j]

to be the space S_{k}(Γ_{1}(N); 𝔽_{l}) with the action of the Hecke operators twisted such that ⟨d⟩_{N} acts as usual, but T_{r} now acts as r^{j}T_{r} used to, and U_{r} now acts as r^{j}U_{r} used to. We deduce

Proposition 3.5.1. If 1 ≤ j ≤ l − 2, then we have an isomorphism

\overline{L}(j) ≅ S_{j+2}(Γ_{1}(N); 𝔽_{l}) ⊕ S_{l+1−j}(Γ_{1}(N); 𝔽_{l})[j]

which respects the action of ⟨d⟩_{N}, T_{r} for r prime to Nl and U_{r} for r dividing Nl.

□

We have yet to deal with \overline{L}(l−1). This can be identified with the submodule of H^{0}(C_{1}∪ C_{2}, Ω_{C_{1}∪C_{2}/𝔽_{l}}) where (ℤ/lℤ)^{×} acts trivially. For ω = (ν_{1}, ν_{2}) ∈ H^{0}(C_{1} ∪ C_{2}, Ω_{C_{1}∪C_{2}/𝔽_{l}}) we see that ν_{1} is a meromorphic differential on C_{1} with poles of order at most 1 at the supersingular points. There is a unique element f of S_{l+1}(Γ_{1}(N); 𝔽_{l}) with the same qexpansion as ν_{1}, and the corresponding map

\begin{matrix}\overline{L}(l − 1) & → S_{l+1}(Γ_{1}(N); 𝔽_{l}) \\ ω & ↦ f \end{matrix}

is a surjection. Any differential in the kernel of this map must be of the form (0, ν_{2}) on C_{1} ∪ C_{2} and, because 0 is regular at the supersingular points, we deduce that ν_{2} must be regular everywhere. Hence ν_{2} corresponds to an element of S_{2}(Γ_{1}(N); 𝔽_{l}). Hence we get an exact sequence

0 → S_{2}(Γ_{1}(N); 𝔽_{l}) → \overline{L}(l − 1) → S_{l+1}(Γ_{1}(N); 𝔽_{l}) → 0.

Both the surjection and the injection can be checked to be compatible with the action of T_{r} for r prime to Nl and of U_{r} for r dividing N. The surjection commutes with the action of U_{l}.

We have correspondences ⟨d⟩_{N}, ⟨d⟩_{l}, T_{r} for r prime to Nl and U_{r} for r dividing Nl, all correspondences of X_{1}(Nl)_{ℚ}. By Albanese functoriality these correspondences induce endomorphisms of Jac(X_{1}(Nl)_{ℚ}) = J_{1}(Nl)_{ℚ}. Let H ⊆ End(J_{1}(Nl)_{ℚ}) be the subring generated by these endomorphisms. We have that H is a finite free ℤ-algebra. Moreover, H acts on Jac(X_{1}(Nl)_{K}) and hence, by the Néron property, on its Néron model J(X_{1}(Nl)_{R}), on the identity component J^{0}(X_{1}(Nl)_{R}) of J(X_{1}(Nl)_{R}), and hence on the cotangent space of J^{0}(X_{1}(Nl)_{R}), namely L = H^{0}(X_{1}(Nl)_{R}, Ω_{X_{1}(Nl)/R}). Note that in this way we recover the actions of T_{r}, U_{r}, ⟨d⟩_{N} and ⟨d⟩_{l} already defined. Hence H acts on L/πL and (L/πL) ⊗ \overline{𝔽}_{l}.

Using the facts established above about the structure of \overline{L}(j) for 1 ≤ j ≤ l − 2, we can deduce that if f is an eigenform in S_{k}(Γ_{1}(N); \overline{𝔽}_{l}) and 3 ≤ k ≤ l then there is an element ω_{f} = (ν_{f}, 0) of \overline{L}(k − 2) which is an eigenvector for H. We define the maximal ideal 𝔪 of H attached to f to be the kernel of the action of H on ω_{f}. We also see, as in [G] Proposition 9.3, that there is a lifting F ∈ S_{2}(Γ_{1}(Nl); \overline{ℚ}_{l}) of f, i.e. an eigenform F such that the reduction of its q-expansion is equal to the q-expansion of f.

§3.6. Representations attached to cusp forms.

Let f ∈ S_{k}(Γ_{1}(N); \overline{𝔽}_{l}) be a mod l eigenform of level N prime to l, weight k satisfying 3 ≤ k ≤ l, and character χ. We have a lifting F of f as above, and a maximal ideal 𝔪 of H attached to f. We will now give some of the details of how to attach an l-adic Galois representation to F, and hence how to attach a mod l Galois representation to f.

By construction, F is an eigenform, and so there is a newform F_{1}, of level Ml for some M dividing N, such that the eigenvalues of T_{r} acting on F_{1} and F are equal for all r not dividing Nl. Note that F_{1} has level divisible by l because the character of F at l is non-trivial.

Because F_{1} is a newform, it is an eigenvector for H_{1}, where H_{1} ⊆ End(Jac(X_{1}(Ml)_{ℚ})) is defined analogously to H above. Hence F_{1} induces a map χ_{F_{1}} : H_{1} → O_{\overline{ℚ}_{l}}, sending an operator to its eigenvalue. Let T be the l-adic Tate module of the Jacobian Jac(X_{1}(Ml)_{ℚ}). Then T is a finite free ℤ_{l}-module with an action of Gal(\overline{ℚ}/ℚ), and a commuting action of H_{1}. The Hecke and Galois actions commute because the Hecke operators we are considering are defined over ℚ. Let V = T ⊗_{ℤ_{l}} \overline{ℚ}_{l}, and let W be the subspace of this vector space where H_{1} acts via the character χ_{F_{1}} associated to F_{1}. In this case, W is known to be 2-dimensional, and the l-adic representation

ρ_{F} : Gal(\overline{ℚ}/ℚ) → Aut(W)

is the l-adic representation attached to F. This representation depends upon which lifting F of f has been chosen.

A standard trick enables us to find a 2-dimensional lattice Λ ⊆ W stabilised by Gal(\overline{ℚ}/ℚ) and the reduction of this lattice is a representation to GL_{2}(\overline{𝔽}_{l}). The semisimplification of this representation is a representation

(ρ̃_{F})^{ss} : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{l}).

This representation is determined by its trace at the Frobenius elements at q for all primes q not dividing Nl. These traces are just the a_{q} occuring in the q-expansion of f, and so we see that (ρ̃_{F})^{ss} depends only on f and not on the lifting F. We will refer to (ρ̃_{F})^{ss} as the mod l representation attached to f, and call it ρ_{f}.

We will restrict attention to the case where ρ_{f} is irreducible. Reducible representations may arise, and this phenomenon occurs when f is congruent to a characteristic 0 Eisenstein series.

By choosing an embedding \overline{ℚ} into \overline{ℚ}_{l} we may identify Gal(\overline{ℚ}_{l}/ℚ_{l}) with a decomposition group D_{l} ⊆ Gal(\overline{ℚ}/ℚ). We shall now describe the restriction

ρ_{f}|_{D_{l}} : D_{l} → GL_{2}(\overline{𝔽}_{l})

of ρ_{f} to the decomposition group D_{l}. First we need some notation. For any a ∈ \overline{𝔽}_{l}^{×} we define

λ(a) : Gal(\overline{ℚ}_{l}/ℚ_{l}) → \overline{𝔽}_{l}^{×}

to be the unramified character sending the Frobenius element to a. We define

χ : Gal(\overline{ℚ}_{l}/ℚ_{l}) → 𝔽_{l}^{×}

to be the mod l reduction of the cyclotomic character. Define ϵ : (ℤ/Nℤ)^{×} → \overline{𝔽}_{l}^{×} such that ⟨d⟩_{N}f = ϵ(d)f. Finally, we say that f = Σ_{n≥1}a_{n}q^{n} ∈ S_{k}(Γ_{1}(N); \overline{𝔽}_{l}) is ordinary if a_{l} ≠ 0.

Theorem 3.6.1. (Deligne, Fontaine) If f is ordinary and 2 ≤ k ≤ l + 1 then

ρ_{f}|_{D_{l}} ≅ ( \begin{matrix}χ^{k−1}λ(ϵ(l)/a_{l}) & ∗ \\ 0 & λ(a_{l}) \end{matrix} ) .

If f is not ordinary then ρ_{f}|_{D_{l}} is irreducible.

Proof. References are [G] §12 for the case where f is ordinary (proved by Deligne) and [E], Chapter 6 for the case where f is not ordinary (proved by Fontaine).

□

If f is ordinary, we say that ρ_{f} is D_{l}-scalar if ρ_{f}|_{D_{l}} is isomorphic to two copies of a character of Gal(\overline{ℚ}_{l}/ℚ_{l}). If f ∈ S_{k}(Γ_{1}(N); \overline{𝔽}_{l}) with character ϵ and 2 ≤ k ≤ l + 1, and ρ_{f} is D_{l}-scalar, then we see from Theorem 3.6.1 that k = l and a_{l}^{2} = ϵ(l). We also see that ρ_{f} must be unramified at l.

We will now study the group scheme J^{0}(X_{1}(Nl)_{R})[𝔪], the kernel of the action of the maximal ideal 𝔪 on J^{0}(X_{1}(Nl)_{R}). The analysis of this object splits into two cases, which are covered in Section 12 of [G] and Chapter 6 of [E], and we shall briefly recount the methods used and results.

Let J_{n} = J^{0}(X_{1}(Nl)_{R})[l^{n}]. Then J_{n} is a quasi-finite, flat group scheme over R. Let J_{∞} = \underrightarrow{lim}_{n} J_{n}. Then J_{∞} has an action of H ⊗ ℤ_{l} = H_{l}. Now H is a free ℤ-module of finite

rank, and hence the completion H_{𝔪} of H with respect to the maximal ideal 𝔪 is a direct summand of H_{l}, i.e.

H_{l} = H_{𝔪} × H′_{𝔪}

for some subring H′_{𝔪} of H_{l}. The element 1 of H_{l} decomposes as

1 = ε + ε′,

with ε ∈ H_{𝔪} and ε′ ∈ H′_{𝔪}. Using ε we can construct a more well-behaved object than J_{n}, namely the group G_{n} = εJ_{n} = J_{n}[ε′].

Lemma 3.6.2. The group G_{n} is finite and flat over R.

Proof. This phenomenon occurs because the non-finiteness is concentrated where ⟨d⟩_{l} acts trivially on J^{0}(X_{1}(Nl)_{R}). More specifically, if J_{n}^{f} is the finite part of J_{n}, then ⟨d⟩_{l} acts trivially on J_{n}/J_{n}^{f} (see [E] Section 6.8) and 𝔪 arises from an element of \overline{L}(j) where 1 ≤ j ≤ l − 2 and so ⟨d⟩_{l} acts non-trivially on \overline{L}(j). Hence G_{n} is finite. That G_{n} is flat can be deduced from the fact that J_{n} is flat, and J_{n} = G_{n} ×_{R} J_{n}[ε].

□

Hence the group G = εJ = J[ε′] = \underrightarrow{lim}_{n} G_{n} is an l-divisible group over R.

We remark that Theorem 2.4.3 (i) applied to X_{1}(Nl)_{R} gives us an exact sequence

1 → T → \overline{J} ^{0}(X_{1}(Nl)_{R}) → ∏_{i=1}^{2} Jac(C_{i}) → 1

where T is a torus, \overline{J} ^{0}(X_{1}(Nl)_{R}) is the special fibre of J^{0}(X_{1}(Nl)_{R}), and C_{1} ∪ C_{2} is the special fibre of X_{1}(Nl)_{R}. It can be checked that ⟨d⟩_{l} acts trivially on T and hence the special fibre of G can be regarded as a subgroup of the l-divisible group attached to ∏_{i=1}^{2} Jac(C_{i}). We refer to [E] Section 6.6 for the formulae giving the action of H on ∏_{i=1}^{2} Jac(C_{i}).

We now recall what is known about the l-divisible group G. The theory splits into two distinct cases, depending on whether our original f = Σa_{n}q^{n} ∈ S_{k}(Γ_{1}(N); \overline{𝔽}_{l}) is ordinary or not.

Theorem 3.6.3. G is an l-divisible group over R, with endomorphisms by the ℤ_{l}- algebra H_{𝔪}. The special fibre \overline{G} of G is canonically isomorphic to G_{1} ⊕ G_{2}, where G_{i} = ε(\underrightarrow{lim} (Jac(C_{i})[l^{n}])). (Note that even though H_{𝔪} does not act on Jac(C_{2}), it does act on the subgroup of Jac(C_{2}) where ∑_{d∈(ℤ/lℤ)^{×}}⟨d ⟩_{l} = 0, by the formulae in [E] Section 6.6). The l-divisible groups G_{i} over 𝔽_{l} factor as

G_{i} = G_{i}^{e} ⊕ G_{i}^{ll} ⊕ G_{i}^{m},

a direct sum into étale, local-local and multiplicative parts.

(i) If a_{l} ≠ 0 then G_{1} = G_{1}^{m}, G_{2} = G_{2}^{e}, and the decomposition G = G_{1}^{m} ⊕ G_{2}^{e} induces an exact sequence

0 → G^{0} → G → G^{e} → 0

of l-divisible groups over ℚ_{l}, with good reduction over R. There is an induced exact sequence of 𝔽 = H_{𝔪}/𝔪-vector-space schemes

0 → G^{0}[𝔪] → G[𝔪] → G^{e}[𝔪] → 0

over ℚ_{l}, with finite flat extensions to R. Both G^{0}[𝔪] and G^{e}[𝔪] are non-trivial group schemes. The group Gal(\overline{ℚ}_{l}/ℚ_{l}) acts on G^{0}[𝔪]_{ℚ_{l}}(\overline{ℚ}_{l}) via a character.

(ii) If a_{l} = 0 then G_{i} = G_{i}^{ll} and G_{i}^{ll} is a local-local 𝔽-vector space scheme. The group scheme G[𝔪] over R is finite flat and non-trivial.

The generic fibre G[𝔪]_{K} has a model G[𝔪]_{ℚ} over ℚ. The corresponding Galois representation G[𝔪]_{ℚ}(\overline{ℚ}) is isomorphic to ⊕_{i=1}^{d}ρ_{f} for some integer d ≥ 1.

Proof. See [G] §12 for the ordinary case, and [E] Chapter 6 for the non-ordinary case. Note that Gross only considers forms f which lift to characteristic 0 newforms. However he does not use this fact in the proof of (i).

The last sentence of the theorem is proved in [BoLeRi].

□

Edixhoven would like us to point out that there is a minor error in the proof of Theorem 2.6 contained in Chapter 6 of [E], and the following correction. Ten lines up on p583, replace “We conclude that Ḡ_{E} = Ḡ_{E}^{l}” with “We conclude that Ḡ_{E} ≠ 0”, and on the fifth line of p584, replace “we know that Ḡ[p^{n}] = Ḡ[p^{n}]^{l}.” with “we know that Ḡ[p^{n}]^{l} ≠ 0.”. Because local-local group schemes extend uniquely over the D′ occurring in this proof, we can conclude by the end of the first paragraph on p584 that Ḡ is indeed local-local.

Chapter 4. Adding the first prime.

Summary: In this chapter, we prove the first of the main theorems of this dissertation, namely Theorem 1.1.1 of the introduction. Note that there is no bound on the weight of the eigenform giving rise to ρ. We reduce the theorem to a problem about forms of low weight using group cohomological techniques, and then prove the theorem in this form by showing that a certain kernel cannot contain the maximal ideal corresponding to our representation in its support. This latter ingredient, an analogue of a lemma of Ihara in [R1], is provided by the geometry developed in the previous two chapters. Note that we can remove the condition that ρ is not D_{l}-scalar of Theorem 1.1.1 in many cases if we use the main result of [D2]. However, the arithmetic technique used in [D2] does not appear to generalise to the Shimura curve case, and so we cannot remove the condition from Theorem 1.1.2.

§4.1. Statement of the main theorem.

Let l ≥ 5 be a prime, and N ≥ 5 be an integer prime to l. Let p be a prime not dividing Nl. If we fix an isomorphism \overline{ℚ}_{l} ≅ ℂ then we may examine the ℂ-valued points of the curves X = X_{1}(Nl; p)_{R} and Y = X_{1}(Nl)_{R}. We have a classical description of these Riemann surfaces as quotients of the upper half plane, compactified by adding cusps. In fact, the Riemann surface Y (ℂ) is compact and is isomorphic to H^{∗}/Γ_{Y}, where H^{∗} is the union of the upper half plane and the cusps ℙ_{ℚ}^{1}, and Γ_{Y} = Γ_{1}(Nl).

Definition 4.1.1. Define 𝕋 to be the polynomial ring over ℤ generated by the independent variables T_{r} and S_{r} for r prime. If M is a positive integer then define 𝕋^{(M)} to be the subring of 𝕋 generated by T_{r} and S_{r} for all primes r not dividing M.

Let k ≥ 2 be an integer. There are two degeneracy maps 1 and η, mapping the space of forms S_{k}(Γ_{1}(Nl); ℂ) to S_{k}(Γ_{1}(Nl; p); ℂ). These maps are induced by the 2 degeneracy maps between the modular curves over R. We define

S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); ℂ)^{p-old},

the p-old subspace of S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); ℂ), to be the (direct) sum of the images of these maps. We define

S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); ℂ)^{p-new},

the p-new subspace of S_{k}(Γ_{1}(N)∩Γ_{0}(p); ℂ), to be the orthogonal complement of this space under the Petersson inner product. Because we have fixed an isomorphism of \overline{ℚ}_{l} with ℂ, we have an isomorphism

S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); ℂ) ≅ S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{ℚ}_{l}),

and using this isomorphism we can define the p-old and p-new subspaces of the space of cusp forms S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{ℚ}_{l}). Let λ be the maximal ideal of the integers of \overline{ℚ}_{l}. We define the space S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{𝔽}_{l})^{p-new} to be the subspace of elements of S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{𝔽}_{l}) whose q-expansions are equal to the mod λ reduction of the q-expansion of some form in S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{ℚ}_{l})^{p-new}.

Warning. It is sometimes misleading to think of the eigenforms in S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{𝔽}_{l})^{p-new} as “p-newforms”. For the system of eigenvalues arising from an eigenform in S_{k}(Γ_{1}(N)∩Γ_{0}(p); \overline{𝔽}_{l})^{p-new} may occur at a level which is not a multiple of p. This is because there may exist a congruence between an eigenform in S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{ℚ}_{l})^{p-old} and an eigenform in S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{ℚ}_{l})^{p-new}. If this occurs, then an element of S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{𝔽}_{l})^{p-new} will have a p-old system of eigenvalues.

If d is a product of distinct primes and is prime to Nl, then we define S_{k}(Γ_{1}(N) ∩ Γ_{0}(d); \overline{𝔽}_{l})^{d-new} to be the forms which are mod λ reductions of characteristic 0 forms on Γ_{1}(N) ∩ Γ_{0}(d) which are new at p for all p dividing d.

Theorem 4.1.2. Let k ≥ 2 and N ≥ 5 be integers, and let l be a prime not dividing N. Say f ∈ S_{k}(Γ_{1}(N); \overline{𝔽}_{l}) is an eigenform for 𝕋^{(Nl)}, and say ρ_{f} : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{l}) is the representation attached to f. Assume that ρ_{f} is irreducible and not D_{l}-scalar. Let p be a prime not dividing Nl such that

p(trρ_{f}(Frob_{p}))^{2} = (1 + p)^{2}detρ_{f}(Frob_{p}) ∈ \overline{𝔽}_{l}.

Then there exists k′ ≥ 2 and an eigenform g ∈ S_{k′}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{𝔽}_{l})^{p-new} such that ρ_{f} ≅ ρ_{g}.

This is the main result of this chapter. Note that it implies Theorem 1.1.1. In fact, the only difference between Theorem 4.1.2 and Theorem 1.1.1 is the condition N ≥ 5, which is easily removed from the conditions of Theorem 4.1.2 using [DT1] Lemma 11 and the remarks following it. Briefly, the argument is that if N ≤ 4, then it is possible to find a prime q such that Nq ≥ 5 and that q cannot divide the conductor of any lift of ρ. Hence we may consider the form of level N as being a form of level Nq, apply the theorem, and then drop the q.

We will now exhibit an equivalent form of Theorem 4.1.2.

If 𝔞 is a maximal ideal of 𝕋^{(Nl)} in the support of S_{k}(Γ_{1}(N); ℂ) such that l ∈ 𝔞, then 𝔞 is the kernel of the homomorphism θ_{f} : 𝕋^{(Nl)} → \overline{𝔽}_{l} defined by some mod l eigenform f ∈ S_{k}(Γ_{1}(N); \overline{𝔽}_{l}). Then f induces a semisimple representation ρ_{f} : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{l}), which is determined uniquely by 𝔞 and even by 𝔞′ = 𝔞 ∩ 𝕋^{(Nlp)}. We shall refer to this representation as ρ_{𝔞}. We say 𝔞 ⊆ 𝕋^{(Nl)} of the above form is Eisenstein if there exists an integer d > 0 such that for all but finitely many primes r ≡ 1 (mod d) we have that T_{r} − 2 ∈ 𝔞 and S_{r} − 1 ∈ 𝔞. By [DT1] Proposition 2 we see that 𝔞 is Eisenstein iff ρ_{𝔞} is reducible. We see that Theorem 4.1.2 is equivalent to the following theorem.

Theorem 4.1.3. Let k ≥ 2 and N ≥ 5 be integers, and let l be a prime not dividing N. Let 𝔞 ⊆ 𝕋^{(Nl)} be a maximal ideal in the support of S_{k}(Γ_{1}(N); \overline{𝔽}_{l}). Assume that 𝔞 is not Eisenstein, and that the corresponding Galois representation

ρ_{𝔞} : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝕋^{(Nl)}/𝔞})

attached to 𝔞 is not D_{l}-scalar. Let p be a prime not dividing Nl such that

T_{p}^{2} − (1 + p)^{2}S_{p} ∈ 𝔞.

Then 𝔞′ = 𝔞 ∩ 𝕋^{(Nlp)} is in the support of S_{k′}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{𝔽}_{l})^{p-new} for some k′ ≥ 2.

Remark. Our methods sometimes show that k′ can be taken to equal k. However, there are still some obstacles to get over before we can state that k′ may always be taken to equal k using these methods. We shall refer to this fact in more detail at the end of the chapter.

We will prove Theorem 4.1.3 in this chapter. We shall break it up into cases depending on the value of k. Firstly, we shall prove the theorem for weight k with 3 ≤ k ≤ l, and then for weights 2 and l + 1. Finally we shall use these results to prove the theorem for general k ≥ 2. Note that for weights 2 ≤ k ≤ l + 1 the result is already known, by work of Diamond. However, the method of proof that we shall give will generalise to situations where Diamond’s proof does not.

We begin with a preliminary section which is an application of the results of Chapter 2.

§4.2. Maps between modular curves.

We have

X = X_{1}(Nl; p)_{R} = \overline{M}(([Γ_{1}(N)], [Γ_{0}(p)]) ⊗ R, ([bal.Γ_{1}(l)] ⊗ ℤ_{l})^{can})

and

Y = X_{1}(Nl)_{R} = \overline{M}([Γ_{1}(N)] ⊗ R, ([bal.Γ_{1}(l)] ⊗ ℤ_{l})^{can}).

There are two canonical maps from X to Y, which can be easily written down from what we already know. The map

1 : X → Y

is induced by the morphism of moduli problems which forgets the [Γ_{0}(p)]-structure, and the map

η : X → Y

is induced by the following construction on non-compactified structures. If E/S/R is an elliptic curve equipped with a Γ_{1}(N)-structure ϕ and a canonical balanced Γ_{1}(l)-structure (π, π^{t}, P, P′), and ψ : E → E_{1} is a cyclic p-isogeny, then

η : (E/S/R, ϕ, ψ, (π, π^{t}, P, P′)) ↦ (E_{1}/S/R, ϕ_{1}, (π_{1}, π_{1}^{t}, P_{1}, P′_{1}))

is defined by Construction 3.1.1. It is the two maps 1 and η to which we would like to apply Theorem 2.4.6.

Let the special fibre of Y be Y_{1} ∪ Y_{2}, where

Y_{1} ≅ Y_{2} ≅ I_{1}(N) = \overline{M}([Γ_{1}(N)] ⊗ 𝔽_{l}, [Ig(l)]),

via the isomorphisms

j_{Y,1} : \overline{M}([Γ_{1}(N)] ⊗ 𝔽_{l}, [Ig(l)]) → Y_{1},

and

j_{Y,2} : \overline{M}([Γ_{1}(N)] ⊗ 𝔽_{l}, [Ig(l)]) → Y_{2}

defined in §3.2. Similarly, let the special fibre of X be X_{1} ∪ X_{2}, with

X_{1} ≅ X_{2} ≅ I_{1}(N; p) = \overline{M}(([Γ_{1}(N)], [Γ_{0}(p)]) ⊗ 𝔽_{l}, [Ig(l)]),

with isomorphisms j_{X,1} and j_{X,2} defined analogously. Note that the isomorphisms j_{X,1} and j_{Y,1} invert the action of (ℤ/lℤ)^{×} because we have defined the actions of (ℤ/lℤ)^{×} on X_{1} and Y_{1} as those induced from the ones on X and Y.

Define the following maps from I_{1}(N; p) to I_{1}(N).

1_{1} = 1_{2} : I_{1}(N; p) → I_{1}(N)

is just forgetting the [Γ_{0}(p)]-structure.

η_{1} : I_{1}(N; p) → I_{1}(N)

sends (E/S/𝔽_{l}, ϕ, P, ψ) to (E_{1}/S/𝔽_{l}, ψ ∘ ϕ, nψ^{(l)}(P)), where n is as always an integer such that np ≡ 1 mod l, and

ψ : E → E_{1}

induces

ψ^{(l)} : E^{(l)} → E_{1}^{(l)}.

Finally,

η_{2} : I_{1}(N; p) → I_{1}(N)

sends (E/S/𝔽_{l}, ϕ, P, ψ) to (E_{1}/S/𝔽_{l}, ψ∘ϕ, ψ^{(l)}(P)). By the isomorphisms j_{X_{i}} and j_{Y_{i}}, these maps 1_{i}, η_{i} induce morphisms with the same names between X_{i} and Y_{i}, so for example the diagram

I_{1}(N; p) ⟶^{j_{X,1}} X_{1}
↓^{η_{1}} ↓^{η_{1}}
I_{1}(N) ⟶^{j_{Y,1}} Y_{1}

commutes.

Lemma 4.2.1. The maps 1 and η : X → Y commute with the actions of (ℤ/Nℤ)^{×} and (ℤ/lℤ)^{×}, and induce the maps 1_{i}, η_{i} : X_{i} → Y_{i}, 1 ≤ i ≤ 2 on the special fibres.

Proof. This is just a case of checking that the four relevant diagrams commute, for example

\begin{matrix}I_{1}(N; p) & ⟶^{j_{X,1}} & X_{1} & ↪ & X_{1}(Nl; p) \\ ↓^{η_{1}} & & ↓^{η_{1}} & & ↓^{η} \\ I_{1}(N) & ⟶^{j_{Y,1}} & Y_{1} & ↪ & X_{1}(Nl) \end{matrix} .

It suffices to check these diagrams commute on non-cuspidal points, where it becomes an exercise in elliptic curve chasing.

□

The maps 1_{i} and η_{i} : I_{1}(N; p) → I_{1}(N) induce by pullback maps on differentials 1_{i}^{∗} and η_{i}^{∗} : H^{0}(I_{1}(N), Ω^{1}) → H^{0}(I_{1}(N; p), Ω^{1}). By the results in the previous chapter there is a notion of q-expansion on these differentials. We now work out the effect that the maps 1_{i}^{∗} and η_{i}^{∗} have on these q-expansions.

Lemma 4.2.2. Say ν ∈ H^{0}(I_{1}(N), Ω_{I_{1}(N)}), with

q-exp(ν) = ∑ a_{n}q^{n}
q-exp(⟨n⟩_{l}^{∗}⟨p⟩_{N}^{∗}ν) = ∑ b_{n}q^{n}
and q-exp(⟨p⟩_{N}^{∗}ν) = ∑ c_{n}q^{n}.

Then we have the following formulae:

q-exp(1_{i}^{∗}ν) = ∑ a_{n}q^{n}, 1 ≤ i ≤ 2,
q-exp(η_{1}^{∗}ν) = p ∑ b_{n}q^{pn}
and q-exp(η_{2}^{∗}ν) = p ∑ c_{n}q^{pn}.

Proof. Again this is just a case of writing down the correct diagram. For example, to prove the middle formula we observe that

Spec(𝔽_{l}[ζ_{N}][[q]]) ⟶^{τ_{∞}} I_{1}(N; p) ⟶^{η_{1}} I_{1}(N)
↓^{r↦q^{p}} ↓^{=}
Spec(𝔽_{l}[ζ_{N}][[r]]) ⟶^{τ_{∞}} I_{1}(N) ⟶^{⟨p⟩_{N} ⟨n⟩_{l}} I_{1}(N)

commutes, and we can immediately deduce that if ν ∈ H^{0}(I_{1}(N), Ω_{I_{1}(N)}) then the pullback

τ_{∞}^{∗}(⟨n⟩_{l}^{∗}⟨p⟩_{N}^{∗}ν) = (∑ b_{n}r^{n})(dr/r)

is related to the pullback

τ_{∞}^{∗}(η_{1}^{∗}ν) = (∑ d_{n}q^{n})(dq/q)

by the formula

\begin{aligned}(∑ d_{n}q^{n})(dq/q) & = (∑ b_{n}(q^{p})^{n})(d(q^{p})/q^{p}) \\ & = p(∑ b_{n}q^{pn})(dq/q). \end{aligned}

□

We will now verify that the criteria of Theorem 2.4.6 are satisfied in this case.

Lemma 4.2.3. The maps

1_{i}^{∗} + η_{i}^{∗} : H^{0}(Y_{i}, Ω_{Y_{i}/𝔽_{l}})^{2} → H^{0}(X_{i}, Ω_{X_{i}/𝔽_{l}}^{1})

are injective for 1 ≤ i ≤ 2.

Note. In this case we are able to deduce this easily from the q-expansion principle and ideas dating back to [AL]. However, this idea will not generalise to the case of Shimura curves, as there is no notion of q-expansion. This lemma is the only step of the argument which is substantially different in the Shimura curve case. See Chapter 5 for details of how the analogue of this lemma is proved.

Proof. 1_{i}^{∗} and η_{i}^{∗} commute with the action of (ℤ/lℤ)^{×} and hence induce maps

α_{i,j} = 1_{i}^{∗} + η_{i}^{∗} : H^{0}(Y_{i}, Ω_{Y_{i}/𝔽_{l}}^{1})(j)^{2} → H^{0}(X_{i}, Ω_{X_{i}/𝔽_{l}}^{1})(j)

for 1 ≤ j ≤ l − 1. It suffices to show that these maps are all injective. We have seen that these spaces are isomorphic to spaces of cusp forms. In fact, we can identify α_{1,j} with a map

α_{1,j} : S_{k_{1}}(Γ_{1}(N); 𝔽_{l})^{2} → S_{k_{1}}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l}),

where k_{1} = j + 2 if 1 ≤ j ≤ l − 2, and k_{1} = 2 if j = l − 1. Similarly, α_{2,j} can be identified with a map

α_{2,j} : S_{k_{2}}(Γ_{1}(N); 𝔽_{l})^{2} → S_{k_{2}}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l}),

where k_{2} = l + 1 − j.

We can evaluate the maps α_{i,j} explicitly on q-expansions. This is because we know that ⟨d⟩_{l} acts as d^{j} on H^{0}(I_{1}(N), Ω^{1})(j), that the identification of this with the space of cusp forms S_{l+1−j}(Γ_{1}(N); 𝔽_{l}) preserves the notion of q-expansion, and the fact that we have evaluated the effect of 1_{i}^{∗} and η_{i}^{∗} on q-expansions in Lemma 4.2.2. We find that if f, g ∈ S_{k_{1}}(Γ_{1}(N); 𝔽_{l}), and the q-expansions of f, g, and g|⟨p⟩_{N} are the following:

q-exp(f) = ∑_{n≥1} a_{n}q^{n},

q-exp(g) = ∑_{n≥1} b_{n}q^{n},

q-exp(g|⟨p⟩_{N}) = ∑_{n≥1} c_{n}q^{n},

then

q-exp(α_{1,j}(f, g)) = ∑_{n≥1} a_{n}q^{n} + p^{k_{1}−1} ∑_{n≥1} c_{n}q^{np}.

Now say (f, g) ∈ ker(α_{1,j}). We shall construct for all t ≥ 0 an element g_{t} of S_{k_{1}}(Γ_{1}(N); 𝔽_{l}) with q-expansion

q-exp(g_{t}) = ∑_{n≥1} b_{n}q^{np^{t}}.

Set g_{0} = g and g_{1} = −p^{1−k_{1}}f|⟨p⟩_{N}^{−1}. Say t ≥ 1 and we have constructed g_{t−1} and g_{t}. The q-expansion of g_{t}|T_{p} is

∑_{n≥1} b_{n}q^{np^{t−1}} + p^{k_{1}−1} ∑_{n≥1} c_{n}q^{np^{t+1}},

and hence we may set

g_{t+1} = p^{1−k_{1}}(g_{t}|T_{p} − g_{t−1})|⟨p⟩_{N}^{−1}.

Then g_{t+1} has the required q-expansion. Now if the q-expansion of g is non-zero then this is a contradiction, because we see that there can be no linear relation between the g_{t}, t ≥ 0, and yet S_{k}(Γ_{1}(N); 𝔽_{l}) is a finite-dimensional space. Hence g = 0 and because (f, g) ∈ ker(α_{1,j}) we see that the q-expansion of f is 0, and hence f = 0. So α_{1,j} is injective for all j.

One can also check that if f, g ∈ S_{k_{2}}(Γ_{1}(N); 𝔽_{l}) then

q-exp(α_{2,j}(f, g)) = ∑_{n≥1} a_{n}q^{n} + p ∑_{n≥1} c_{n}q^{np},

and so a similar proof will work for α_{2,j}, establishing the injectivity result we need.

□

Hence we are in a position to apply Theorem 2.4.6. Recall that J_{X}^{0} is the identity component of the Néron model of Jac(X_{K}), and J_{Y}^{0} is defined using Y similarly. Then 1 and η induce a map

1^{∗} + η^{∗} : (J_{Y}^{0} )^{2} → J_{X}^{0}

with kernel K say. Theorem 2.4.6 then implies

Proposition 4.2.4. The special fibre of K[l] is multiplicative.

□

§4.3. Beginning of proof of Theorem 4.1.3 for weights 3 ≤ k ≤ l.

The strategy here is to prove the theorem by deducing it from a proposition on weight 2 forms at level Nl. This proposition will be deduced from a result saying that a certain kernel cannot contain a certain non-Eisenstein maximal ideal in its support. This latter result will be proved, not using group-theoretic techniques as in [R1], [D1] or [D2], but more along the lines of [DT1], using the geometric results that we have established in §3.6 and §4.2. We start by establishing this result.

Say f is a cusp form satisfying the conditions of Theorem 4.1.2. Say moreover 3 ≤ k ≤ l. In these cases we have defined a maximal ideal 𝔪 of a Hecke algebra H ⊆ End(Jac(X_{1}(Nl)_{ℚ})) and also a characteristic zero form F ∈ S_{2}(Γ_{1}(Nl); \overline{ℚ}_{l}) such that the reduction of the q-expansion of F is the q-expansion of f. Recall J_{X_{1}(Nl)} is the Néron model over R of the Jacobian of X_{1}(Nl)_{K}, and J_{X_{1}(Nl)}^{0} is the identity component of this model.

We will ultimately apply the lemma below to a group scheme closely related to the group scheme K defined in §4.2.

Lemma 4.3.1. Say G_{R} is a closed finite subgroup scheme of (J_{X_{1}(Nl)}^{0})^{2} with the following properties:

(i) The generic fibre G_{K} has a model G_{ℚ} over ℚ. (ii) The l-torsion in the special fibre G_{𝔽_{l}} of G is multiplicative. (iii) G_{ℚ} is preserved by the action of H.

Then the finite group scheme (Jac(X_{1}(Nl)_{ℚ}))^{2}[𝔪] over ℚ has trivial intersection with G_{ℚ}, where 𝔪 is the maximal ideal defined above, coming from an eigenform f satisfying the criteria of Theorem 4.1.2.

Proof. In the notation of §3.6, we have to show that

(G[𝔪]_{ℚ})^{2} ∩ G_{ℚ} = 0

as subgroup schemes of (Jac(X_{1}(Nl)_{ℚ}))^{2} over ℚ.

Assume for a contradiction that G[𝔪]_{ℚ}^{2} ∩ G_{ℚ} ≠ 0. Because the Galois representation attached to G[𝔪]_{ℚ} is isomorphic to ⊕_{i=1}^{d}ρ_{f} by [BoLeRi] Theorems 1 and 2, we have G[𝔪]_{ℚ}^{2} ≅ ⊕_{i=1}^{2d}ρ_{f}, and there is a subgroup scheme 0 ≠ X_{ℚ} ⊆ G[𝔪]_{ℚ}^{2} ∩ G_{ℚ} whose associated Galois representation is ρ_{f}, and which is a two-dimensional H/𝔪-vector space scheme.

The group scheme G[𝔪]_{K}^{2} extends (by taking its closure in G[l]_{R}) to a finite flat group scheme G[𝔪]_{R}^{2} over R. Now X_{K} ⊆ G[𝔪]_{K}^{2} and because G[𝔪]_{R}^{2} ∩ G_{R} is closed in G[𝔪]_{R}^{2}, it must contain the scheme-theoretic closure X_{R} of X in G[𝔪]_{R}^{2}. Now X_{R} is a finite flat group scheme over R. Moreover, the special fibre X_{𝔽_{l}} of X_{R} is multiplicative, because it is contained in G_{𝔽_{l}}[l]. If f is not ordinary, then this is immediately a contradiction, as X_{𝔽_{l}} is non-trivial, but G[𝔪]_{𝔽_{l}} is local-local. If f is ordinary then X_{𝔽_{l}} ⊆ G^{0}[𝔪]_{R}^{2} and hence X_{R} ⊆ G^{0}[𝔪]_{R}^{2} where G^{0}[𝔪]_{R} is defined as in Theorem 3.6.3. Hence X_{K} ⊆ G^{0}[𝔪]_{K}^{2} and X_{ℚ_{l}} ⊆ G^{0}[𝔪]_{ℚ_{l}}^{2}. This means that the Galois representation attached to X_{ℚ_{l}} is diagonal, and hence that ρ_{f} is D_{l}-scalar, contrary to our assumptions.

□

We now show how to use this lemma to deduce the weight 2 result alluded to at the beginning of this section. Say 3 ≤ k ≤ l, and 𝔞 and p are as in the conditions of Theorem 4.1.3. By Proposition 3.5.1 we see that 𝔞 ∈ Supp_{𝕋^{(Nl)}}(\overline{L}(j)), where j = k − 2, L = H^{0}(X_{1}(Nl), Ω_{X_{1}(Nl)/R}), and the bar represents reduction modulo π = 1 − ζ_{l}. Set

L′ = H^{0}(X_{1}(Nl; p), Ω_{X_{1}(Nl;p)/R})

where here again the sheaf Ω_{X_{1}(Nl;p)/R} is the dualising sheaf, and note that

L′ ⊗_{R} \overline{ℚ}_{l} ≅ H^{0}(X_{1}(Nl; p)_{\overline{ℚ}_{l}}, Ω_{X_{1}(Nl;p)_{\overline{ℚ}_{l}} /\overline{ℚ}_{l}}).

We have defined the p-new subspace S_{2}(Γ_{1}(Nl) ∩ Γ_{0}(p); \overline{ℚ}_{l})^{p-new} of S_{2}(Γ_{1}(Nl) ∩ Γ_{0}(p); \overline{ℚ}_{l}), and because S_{2}(Γ_{1}(Nl) ∩ Γ_{0}(p); \overline{ℚ}_{l}) ≅ (L′ ⊗_{R} \overline{ℚ}_{l}), the p-new subspace of S_{2}(Γ_{1}(Nl) ∩ Γ_{0}(p); \overline{ℚ}_{l}) is identified with a subspace (L′ ⊗_{R} \overline{ℚ}_{l})^{p-new} of L′ ⊗_{R} \overline{ℚ}_{l}. We define (L′)^{p-new} = L′ ∩ (L′ ⊗ \overline{ℚ}_{l})^{p-new} and note that (L′)^{p-new} ⊗_{R} \overline{ℚ}_{l} = (L′ ⊗ \overline{ℚ}_{l})^{p-new}. We have an action of 𝕋^{(Nl)} on L, recalling that S_{r} acts as ⟨r⟩_{Nl} for primes r not dividing Nl, and we also have an action of 𝕋^{(Nlp)} on L′ which induces an action on (L′)^{p-new}.

Proposition 4.3.2. If 𝔞 ⊆ 𝕋^{(Nl)} is the maximal ideal above, in the support of \overline{L}(j), where j = k − 2, and if T_{p}^{2} − (1 + p)^{2}S_{p} ∈ 𝔞, then 𝔞′ = 𝔞 ∩ 𝕋^{(Nlp)} is in the support of (L′)^{p-new}/π(L′)^{p-new}.

Proof. Firstly we recall the canonical isomorphism

S_{2}(Γ_{1}(Nl); ℂ) ≅ H^{1}(X_{1}(Nl)_{ℂ}, ℝ)

of real vector spaces, where the H^{1} on the right hand side is Betti cohomology of the compact Riemann surface X_{1}(Nl)_{ℂ}. Moreover, the correspondences T_{r} for prime r not dividing Nl, and U_{r} for prime r dividing Nl, induce endomorphisms of H^{1}(X_{1}(Nl)_{ℂ}, ℤ) and hence of H^{1}(X_{1}(Nl)_{ℂ}, ℝ), which commute with this isomorphism. For the rest of this proof we shall drop the suffix ℂ on modular curves, and so for example X_{1}(Nl) will represent the compact Riemann surface X_{1}(Nl)_{ℂ}, and if we want to talk about the model of X_{1}(Nl) over R we will use the term X_{1}(Nl)_{R}, that is we will mention R explicitly.

We reduce the proposition to a question involving cohomology groups, and then to the question of proving that 𝔞′ is in the support of a certain cohomology congruence module.

There is a map

i = 1^{∗} + η^{∗} : H^{1}(X_{1}(Nl), ℚ_{l})^{2} → H^{1}(X_{1}(Nl; p), ℚ_{l})

given by the 2 degeneracy maps of complex manifolds 1, η : X_{1}(Nl; p) → X_{1}(Nl). We recall the cup product pairings

∪ : H^{1}(X_{1}(Nl), ℚ_{l}) × H^{1}(X_{1}(Nl), ℚ_{l}) → H^{2}(X_{1}(Nl), ℚ_{l}) ≅ ℚ_{l}

and

∪ : H^{1}(X_{1}(Nl; p), ℚ_{l}) × H^{1}(X_{1}(Nl; p), ℚ_{l}) → H^{2}(X_{1}(Nl; p), ℚ_{l}) ≅ ℚ_{l}.

With respect to these pairings the Hecke operators behave thus. Both pairings satisfy

T_{r}a ∪ b = a ∪ ⟨r⟩_{Nl}^{−1}T_{r}b

and

⟨r⟩_{Nl}a ∪ b = a ∪ ⟨r⟩_{Nl}^{−1}b,

for prime r not dividing Nlp. With respect to these pairings one can form the dual homomorphism

i^{†} : H^{1}(X_{1}(Nl; p), ℚ_{l}) → H^{1}(X_{1}(Nl), ℚ_{l})^{2}.

The composite i^{†}i is represented by the 2 × 2 matrix

( \begin{matrix}p + 1 & T_{p} \\ ⟨p⟩_{Nl}^{−1}T_{p} & p + 1 \end{matrix} ) .

Now 𝔞 ∈ Supp(L) and hence 𝔞 ∈ Supp H^{1}(X_{1}(Nl), ℤ_{l})^{2}. Then 𝔞 must also be in the support of

H^{1}(X_{1}(Nl), ℤ_{l})^{2}/ ( \begin{matrix}p + 1 & T_{p} \\ ⟨p⟩_{Nl}^{−1}T_{p} & p + 1 \end{matrix} ) H^{1}(X_{1}(Nl), ℤ_{l})^{2},

because of our condition on p. Also i^{†}i : H^{1}(X_{1}(Nl), ℚ_{l})^{2} → H^{1}(X_{1}(Nl), ℚ_{l})^{2} is an isomorphism, because the eigenvalues of T_{p} on H^{1}(X_{1}(Nl), ℂ) cannot be as large as p + 1 in absolute magnitude. Hence i^{†}i has an inverse, (i^{†}i)^{−1}, on H^{1}(X_{1}(Nl), ℚ_{l})^{2}.

Define

H^{1}(X_{1}(Nl; p), ℚ_{l})_{p-old} = iH^{1}(X_{1}(Nl), ℚ_{l})^{2},
H^{1}(X_{1}(Nl; p), ℤ_{l})_{p-old} = H^{1}(X_{1}(Nl; p), ℤ_{l}) ∩ H^{1}(X_{1}(Nl; p), ℚ_{l})_{p-old}
and H^{1}(X_{1}(Nl; p), ℤ_{l})^{p-old} = i(i^{†}i)^{−1}i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l}).

Then we see that

\begin{aligned}i^{†}iH^{1}(X_{1}(Nl), ℤ_{l})^{2} & ⊆ i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})_{p-old} \\ & ⊆ i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})^{p-old} = i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l}) \\ & ⊆ H^{1}(X_{1}(Nl), ℤ_{l})^{2}. \end{aligned}  (∗)

Now it can be easily shown that i and i^{†} are 𝕋^{(Nlp)}-module homomorphisms, and so if we set 𝔞′ = 𝔞 ∩ 𝕋^{(Nlp)} we see that 𝔞′ is a maximal ideal of 𝕋^{(Nlp)} and 𝔞′ must also be in the support of H^{1}(X_{1}(Nl), ℤ_{l})^{2}/i^{†}iH^{1}(X_{1}(Nl), ℤ_{l})^{2}, considered as a 𝕋^{(Nlp)}-module.

We claim that in fact

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})^{p-old})/(i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})_{p-old})

and to prove this it suffices by (∗) to show that

(i) 𝔞′ ∉ Supp_{𝕋^{(Nlp)}}(i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})_{p-old}/i^{†}iH^{1}(X_{1}(Nl), ℤ_{l})^{2}) and

(ii) 𝔞′ ∉ Supp_{𝕋^{(Nlp)}}(H^{1}(X_{1}(Nl), ℤ_{l})^{2}/i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})).

Firstly we deal with (i). By an easy application of the snake lemma,

\begin{matrix}H^{1}(X_{1}(Nl; p))_{p-old}/H^{1}(X_{1}(Nl), ℤ_{l})^{2} \\ ≅ & ker(\overline{i} : H^{1}(X_{1}(Nl), ℚ_{l}/ℤ_{l})^{2} → H^{1}(X_{1}(Nl; p), ℚ_{l}/ℤ_{l})). \end{matrix}

Now

H^{1}(X_{1}(Nl), ℝ/ℤ) ≅ Jac(X_{1}(Nl))(ℂ)

as modules for 𝕋^{(Nl)}, and similarly H^{1}(X_{1}(Nl; p), ℝ/ℤ) ≅ Jac(X_{1}(Nl; p))(ℂ) as modules for 𝕋^{(Nlp)}. Hence

ker(1^{∗} + η^{∗} : H^{1}(X_{1}(Nl), ℝ/ℤ)^{2} → H^{1}(X_{1}(Nl; p), ℝ/ℤ))
≅ker(1^{∗} + η^{∗} : Jac(X_{1}(Nl))^{2} → Jac(X_{1}(Nl; p)))(ℂ)
≅K(ℂ),

where K is defined as in §4.2. Because K(ℂ) is a finite group, K(ℂ) = K(\overline{ℚ}). Moreover, replacing ℝ/ℤ with ℚ_{l}/ℤ_{l} just replaces all Jacobians with their l^{∞}-torsion subgroups. We conclude that for some sufficiently large integer n we have

K[l^{n}](\overline{ℚ}) ≅ ker(\overline{i} : H^{1}(X_{1}(Nl), ℚ_{l}/ℤ_{l})^{2} → H^{1}(X_{1}(Nl; p), ℚ_{l}/ℤ_{l})).

Now assume for a contradiction that 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(K[l^{n}](\overline{ℚ})). Let \overline{𝕋}^{(Nlp)} be the image of 𝕋^{(Nlp)} in H, given by sending T_{r} to T_{r} and S_{r} to ⟨r⟩_{Nl}. Then \overline{𝕋}^{(Nlp)} is a finite free ℤ-algebra. Now K[l^{n}](\overline{ℚ}) is a module for \overline{𝕋}^{(Nlp)}/l^{n}\overline{𝕋}^{(Nlp)}, and 𝔞′ can be regarded as a maximal ideal of this finite ring. The localisation of this finite ring at 𝔞′ must be a direct summand of the ring, so we have an idempotent ε_{𝔞}′ ∉ 𝔞′ of \overline{𝕋}^{(Nlp)}/l^{n}\overline{𝕋}^{(Nlp)}, and the fact that 𝔞′ ∈ Supp(K[l^{n}](\overline{ℚ})) implies that ε_{𝔞}′K[l^{n}](\overline{ℚ}) ≠ 0. Hence ε_{𝔞}′K[l](\overline{ℚ}) ≠ 0. Let J = ε_{𝔞}′K[l]. Then J satisfies conditions (i) and (ii) of Lemma 4.3.1. Set G = HJ , the smallest group scheme containing J which has an action of H. We see that G satisfies the conditions of Lemma 4.3.1 and that 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(G(\overline{ℚ})).

Now G(\overline{ℚ}) is a non-zero H/lH-module, and hence there exists 𝔪′ a maximal ideal of H/lH such that 𝔪′ ∈ Supp_{H/lH}(G(\overline{ℚ})). Because 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(G(\overline{ℚ})) we see that the mod l representation arising from 𝔪′ is isomorphic to ρ_{f}, where we recall f is the eigenform giving rise to 𝔞′. Hence, by Lemma 4.3.1, we deduce that G[𝔪′] = 0 and by induction on r we see G[𝔪^{′r}] = 0 for all positive integers r. Hence 𝔪′ ∉ Supp_{H/lH}G(\overline{ℚ}), a contradiction. We conclude that 𝔞′ is not in Supp_{𝕋^{(Nlp)}}(K[l^{n}](\overline{ℚ})). Hence

𝔞′ ∉ Supp_{𝕋^{(Nlp)}}(H^{1}(X_{1}(Nl; p))_{p-old}/iH^{1}(X_{1}(Nl); ℤ_{l})^{2})

and this implies (i) above.

There is a natural duality between

A = H^{1}(X_{1}(Nl; p), ℤ_{l})_{p-old}/iH^{1}(X_{1}(Nl), ℤ_{l})^{2}

and

B = H^{1}(X_{1}(Nl), ℤ_{l})^{2}/i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})

thus: identify A with ker(\overline{i}) as above. Note that ker(\overline{i}) ⊆ H^{1}(X_{1}(Nl), ℚ_{l}/ℤ_{l})^{2} can be thought of as a subgroup of H^{1}(X_{1}(Nl), ℚ_{l})^{2}/H^{1}(X_{1}(Nl), ℤ_{l})^{2}. The pairing defined earlier on H^{1}(X_{1}(Nl), ℚ_{l})^{2} thus gives a map

{, } : A × B → ℚ_{l}/ℤ_{l},

such that

{T_{r}a, b} = {a, ⟨r⟩_{Nl}^{−1}T_{r}b}

and

{⟨r⟩_{Nl}a, b} = {a, ⟨r⟩_{Nl}^{−1}b}.

To prove (ii) we introduce the operator w_{ζ_{Nl}} as in [E] Section 6.4. We can define w_{ζ_{Nl}} on X_{1}(Nl)_{\overline{ℚ}_{l}} as in [E] 6.4.3 and on X_{1}(Nl; p)_{\overline{ℚ}_{l}} similarly. It is not difficult to show that

w_{ζ_{Nl}}^{−1}1^{∗}w_{ζ_{Nl}} = 1^{∗}

and that

w_{ζ_{Nl}}^{−1}η^{∗}w_{ζ_{Nl}} = ⟨p⟩_{Nl}^{−1}η^{∗}.

One way to verify these formulae is to check them on non-cuspidal points over ℂ, where it becomes an exercise in pairings on elliptic curves. Hence

w_{ζ_{Nl}}^{−1}Kw_{ζ_{Nl}} = (1, ⟨p⟩_{Nl}^{−1})K = K^{∗}

satisfies conditions (i) and (ii) of Lemma 4.3.1. Also

w_{ζ_{Nl}}^{−1}T_{r}w_{ζ_{Nl}} = ⟨r⟩_{Nl}^{−1}T_{r},

and

w_{ζ_{Nl}}^{−1}⟨r⟩_{Nl}w_{ζ_{Nl}} = ⟨r⟩_{Nl}^{−1},

hence we see

𝔞′ ∈ Supp(B) ⟺ 𝔞′ ∈ Supp(K^{∗}(\overline{ℚ})[l^{n}]).

By the same argument as we used to prove (i), we deduce that 𝔞′ ∉ Supp(K^{∗}(\overline{ℚ})[l^{n}]) and hence 𝔞′ ∉ Supp(B). This proves (ii). Hence our claim holds and we have

𝔞′ ∈ Supp(i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})^{p-old}/i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})_{p-old}).

We deduce that

𝔞′ ∈ Supp(i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})^{p-old}/i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})_{p-old})
= Supp(i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})/i^{†}H^{1}(X_{1}(Nl; p), ℤ_{l})_{p-old})
⊆ Supp(H^{1}(X_{1}(Nl; p), ℤ_{l})/H^{1}(X_{1}(Nl; p), ℤ_{l})_{p-old})
⊆ Supp(H^{1}(X_{1}(Nl; p), ℚ_{l})/H^{1}(X_{1}(Nl; p), ℚ_{l})_{p-old})
= Supp(H^{1}(X_{1}(Nl; p), ℚ_{l})^{p-new}).

Hence

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}H^{1}(X_{1}(Nl; p), ℚ_{l})^{p-new},

and so we deduce that

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(H^{0}(X_{1}(Nl; p)_{K}, Ω_{X_{1}(Nl;p)_{K}/K})^{p-new}),

then that

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}((L′)^{p-new}),

and finally that

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}((L′)^{p-new}/π(L′)^{p-new}).
□

To prove Theorem 4.1.3 for 3 ≤ k ≤ l, it suffices to prove that 𝔞′ is in the support of S_{k′}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new} for some k′. To control p-newness, we introduce the concept of modular forms on a quaternion algebra, which will be the topic of our next section.

§4.4. Interlude: definite quaternion algebras.

The purpose of this section is to describe briefly some of the theory of modular forms over a quaternion algebra. The reason we are interested in these objects is the Jacquet-Langlands Theorem, which relates the systems of eigenvalues found in these spaces to certain systems of eigenvalues found in spaces of classical modular forms. For more details of these applications of definite quaternion algebras and the Jacquet-Langlands Theorem we refer the reader to [DT1] §2 and §5.

We shall begin with some definitions. For g ≥ 0 an integer, we define the space V_{g} to be the free ℤ_{l}-module of rank g + 1 consisting of homogeneous polynomials of total degree g in two variables X and Y. We can regard V_{g} as being contained in the ring ℤ_{l}[X, Y]. We

equip the ring ℤ_{l}[X, Y] with a left action of GL_{2}(ℤ_{l}) by defining, for ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℤ_{l}),

( \begin{matrix}a & b \\ c & d \end{matrix} ) X = aX + cY

and

( \begin{matrix}a & b \\ c & d \end{matrix} ) Y = bX + dY,

and extending to an automorphism of the ring. This action preserves the spaces V_{g} and hence an action of GL_{2}(ℤ_{l}) is induced on these spaces. We see that for p a polynomial of degree g, we have

(( \begin{matrix}a & b \\ c & d \end{matrix} ) p) (X, Y) = p(aX + cY, bX + dY).

Note that V_{g} and Symm^{g}(ℤ_{l}^{2}), the g-th symmetric power of ℤ_{l}^{2} with its usual action of GL_{2}(ℤ_{l}), are isomorphic as left GL_{2}(ℤ_{l})-modules.

If μ is an integer, and M is a left ℤ_{l}[GL_{2}(ℤ_{l})]-module, then then we define the twisted module M[μ] to be the space M equipped with the following action of GL_{2}(ℤ_{l}). If g ∈ GL_{2}(ℤ_{l}) and m ∈ M such that gm = n under the old action, then gm with respect to the new action is defined to be (det(g))^{μ}n. If V is a left ℤ_{l}[GL_{2}(ℤ_{l})]-module then we define \overline{V} to be V/lV with its induced action of GL_{2}(ℤ_{l}). Note that when V = V_{g}[μ] for some integers g and μ, then \overline{V} = \overline{V}_{g}[μ] can be identified with the space of polynomials in two variables of total degree g over 𝔽_{l}, with the action of GL_{2}(ℤ_{l}) given by

(( \begin{matrix}a & b \\ c & d \end{matrix} ) f) (X, Y) = (ad − bc)^{μ}f(aX + cY, bX + dY).

Note in particular that this action factors through GL_{2}(𝔽_{l}), and hence \overline{V}_{g}[μ] may be thought of as a left GL_{2}(𝔽_{l})-module.

For g a non-negative integer, define I_{g} to be the space of homogeneous functions τ : 𝔽_{l}^{2} → 𝔽_{l} of degree g which vanish at the origin, that is the functions τ such that τ(0, 0) = 0 and τ(λx, λy) = λ^{g}τ(x, y) for λ ∈ 𝔽_{l}^{×}. Define an action of GL_{2}(𝔽_{l}) on I_{g} by

(( \begin{matrix}a & b \\ c & d \end{matrix} ) τ) (x, y) = τ(ax + cy, bx + dy).

We recall from [AS] Lemma 3.2 the following exact sequence of left GL_{2}(𝔽_{l})-modules. If 0 < g < l then

0⟶\overline{V}_{g}⟶^{α_{g}}I_{g}⟶^{β_{g}}\overline{V}_{l−1−g}[g]⟶0

is exact, where α_{g} is the evaluation map, and β_{g} is defined as in [AS] Section 3. Note that in [AS] the action of GL_{2}(𝔽_{l}) is on the right, but this makes no difference.

Let D/ℚ be a definite quaternion algebra over ℚ with discriminant d prime to l. Then d is the product of an odd number of distinct primes. Let 𝔸_{f} denote the finite adeles over ℚ and define D_{f} = D ⊗ 𝔸_{f}. For primes p not dividing d, fix an isomorphism D_{p} = D ⊗_{ℚ} ℚ_{p} ≅ M_{2}(ℚ_{p}). Define O_{D,p} ⊆ D_{p} corresponding to M_{2}(ℤ_{p}) under this isomorphism. For primes p dividing d, let O_{D,p} be the unique maximal order of D ⊗_{ℚ} ℚ_{p}.

If T is a positive integer prime to d, then we can define the principal congruence subgroup U_{T} of D_{f}^{×} as being the subgroup of ∏_{p} O_{D,p}^{×} consisting of elements congruent to

( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) modulo T. Note that this makes sense because T is prime to d, so at all primes p

dividing T we have identified O_{D,p} with a matrix group. Similarly we define the subgroup

U_{1}(T) to be the elements of ∏_{p} O_{D,p}^{×} congruent to ( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} ) modulo T, and U_{0}(T) to be the

elements congruent to ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) modulo T. Note that these are all open compact subgroups

of D_{f}^{×}.

If g is an element of D_{f}^{×}, and p is a prime, we write g_{p} to be the component of g at p. If U is a subset of D_{f}^{×}, we write U_{p} to be the set of all g_{p} as g runs through U, so U_{p} is the projection of U onto D_{p}^{×}. Recall that for p not dividing d, we have identified D_{p} with M_{2}(ℚ_{p}) and hence D_{p}^{×} is identified with GL_{2}(ℚ_{p}).

Say U ⊆ D_{f}^{×} is an open compact subgroup, M is a ℤ_{l}-module, and k ≥ 2. Assume moreover that U_{l}, the image of the projection of U to GL_{2}(ℚ_{l}), is contained in GL_{2}(ℤ_{l}). Then U acts on M^{2} via the projection U_{l}, and hence on the space Symm^{k−2}(M^{2}) ≅ Symm^{k−2}(ℤ_{l}^{2}) ⊗ M ≅ V_{k−2} ⊗ M. Define

L_{k}(U; M) = {f : D^{×}\D_{f}^{×} → V_{k−2} ⊗ M such that
f(gu) = u_{l}^{−1}f(g) for all u ∈ U}.

Define

L_{2}(U; M)^{triv} = {f ∈ L_{2}(U; M) such that f factors through det : D_{f}^{×} → 𝔸_{f}^{×}}

and set L_{k}(U; M)^{triv} = 0 if k > 2.

If U is a compact open subgroup of D_{f}^{×}, and A is a ℤ_{l}[U_{l}]-module, then define

B(U; A) = {f : D^{×}\D_{f}^{×} → A such that
f(gu) = u_{l}^{−1}f(g) for all u ∈ U, g ∈ D_{f}^{×}}.

So for example if M is a ℤ_{l}-module then Symm^{g}(M^{2}) inherits the structure of a left ℤ_{l}[GL_{2}(ℤ_{l})]-module and if U ⊆ D_{f}^{×} is a compact open subgroup such that U_{l} ⊆ GL_{2}(ℤ_{l}), we see that B(U; Symm^{g}(M^{2})) ≅ L_{g+2}(U; M). In particular, B(U, V_{g}) ≅ L_{g+2}(U; ℤ_{l}) and B(U, \overline{V}_{g}) ≅ L_{g+2}(U; 𝔽_{l}).

From now on we shall assume that U contains the principal congruence subgroup U_{T} for some integer T prime to d.

Say g ∈ D_{f}^{×} with g_{l} = 1. We define [UgU] : B(U, M) → B(U, M) thus. Let UgU = ∐ g_{i}U. Note that g_{i} ∈ UgU and hence (g_{i})_{l} ∈ U_{l}. We define

([UgU]f)(h) = ∑_{i}(g_{i})_{l}f(hg_{i}).

If r is a prime not dividing T ld, then define ϖ_{r} ∈ 𝔸_{f}^{×} by (ϖ_{r})_{v} = 1 if v ≠ r, and (ϖ_{r})_{r} = r.

Define η_{r} ∈ D_{f}^{×} by (η_{r})_{v} = 1 if v ≠ r, and (η_{r})_{r} = ( \begin{matrix}1 & 0 \\ 0 & r \end{matrix} ) . Then define T_{r} = [Uη_{r}U] and

S_{r} = [Uϖ_{r}U]. This makes B(U, M) into a 𝕋^{(T ld)}-module.

Theorem 4.4.1. There is an isomorphism

(L_{k}(U_{1}(N); ℚ_{l})/L_{k}(U_{1}(N); ℚ_{l})^{triv}) ⊗_{ℚ_{l}} ℂ ≅ S_{k}(Γ_{1}(N) ∩ Γ_{0}(d); ℂ)^{d-new}

of 𝕋^{(Nld)}-modules.

Proof. This can be read off from the Jacquet-Langlands Theorem [JL]. See [DT1] for details.

□

The submodule L_{k}(U_{1}(N); ℤ_{l}) of L_{k}(U_{1}(N); ℚ_{l}) is preserved by the action of 𝕋^{(Nld)}, and hence there is an induced action of 𝕋^{(Nld)} on L_{k}(U_{1}(N); ℤ_{l}) and on L_{k}(U_{1}(N); 𝔽_{l}). Similarly, the subspace L_{2}(U_{1}(N); ℤ_{l})^{triv} of L_{2}(U_{1}(N); ℚ_{l})^{triv}, and hence L_{2}(U_{1}(N); 𝔽_{l})^{triv}, inherit actions of 𝕋^{(Nld)}. We note that any maximal ideal of 𝕋^{(Nld)} in the support of L_{2}(U_{1}(N); ℚ_{l})^{triv} is Eisenstein, and hence we deduce

Corollary 4.4.2. The non-Eisenstein maximal ideals in

Supp_{𝕋^{(Nld)}}(L_{2}(U_{1}(N); 𝔽_{l}))

are precisely the non-Eisenstein maximal ideals in

Supp_{𝕋^{(Nld)}}(S_{2}(Γ_{1}(N) ∩ Γ_{0}(d); 𝔽_{l})^{d-new}).

If k > 2, then the maximal ideals in

Supp_{𝕋^{(Nld)}}(L_{k}(U_{1}(N); 𝔽_{l}))

are precisely the maximal ideals in

Supp_{𝕋^{(Nld)}}(S_{k}(Γ_{1}(N) ∩ Γ_{0}(d); 𝔽_{l})^{d-new}).

□

It sometimes helps to have the following description of B(U_{1}(N); M). If we decompose D_{f}^{×} as D_{f}^{×} = ∐_{i=1}^{r} D^{×}g_{i}U_{1}(N), then the map f ↦ (f(g_{i}))_{i=1}^{r} defines an isomorphism of ℤ_{l}-modules B(U_{1}(N); M) ≅ M^{r}. This follows as in §2 of [DT1], and uses the fact that N ≥ 5.

Let j ∈ ℤ be an integer. Let M be a left ℤ_{l}[GL_{2}[ℤ_{l}]]-module, and let U be a compact open subgroup of ∏_{r} O_{D,r}^{×} containing U_{T} for some integer T prime to d. Then B(U, M) inherits an action of ℤ_{l} and of 𝕋^{(T ld)}. We define the twisted module B(U, M)[j] as follows. As a ℤ_{l}-module, B(U, M)[j] = B(U, M), but we define a new action of 𝕋^{(T ld)} thus. Let T_{r} act as r^{j}T_{r} used to, and let S_{r} act as r^{2j}S_{r} used to. Note that r is prime to l, so r^{j} is a unit in ℤ_{l}.

Lemma 4.4.3. There is an isomorphism of 𝕋^{(T ld)}-modules

ι : B(U, M[j]) → B(U, M)[j].

Proof. Firstly we define a map τ_{l} : D_{f}^{×} → ℤ_{l}^{×} thus. There is a map det : D_{f}^{×} → 𝔸_{f}^{×}, and a norm map || || : 𝔸_{f}^{×} → ℚ^{×}. We identify ℚ^{×} with a subset of ℚ_{l}^{×}, and define τ_{l} : D_{f}^{×} → ℚ_{l}^{×} by τ_{l}(g) = (|| det g||(det g_{l})). It is easy to check that in fact τ_{l}(g) ∈ ℤ_{l}^{×}. Moreover, τ_{l}(gh) = τ_{l}(g)τ_{l}(h), τ_{l}(d) = 1 if d ∈ D^{×}, and τ_{l}(u) = det(u_{l}) if u ∈ U.

If f ∈ B(U, M[j]) then define ιf : D_{f}^{×} → M by
(ιf)(g) = τ_{l}(g)^{j}f(g).

If G ∈ D_{f}^{×}, u ∈ U, then

(ιf)(gu) = τ_{l}(gu)^{j}f(gu)
= τ_{l}(g)^{j}τ_{l}(u)^{j}((det u_{l}^{−1})^{j}u_{l}^{−1}f(g))
= τ_{l}(g)^{j}(det u_{l})^{j}(det u_{l}^{−1})^{j}u_{l}^{−1}f(g)
= u_{l}^{−1}ιf(g).

Hence ιf ∈ B(U, M). It is now a simple exercise to check that ι is the required isomorphism of 𝕋^{(T ld)}-modules. As an example, we shall check that the actions of T_{r} match up, for primes r not dividing T ld.

Set Uη_{r}U = ∐_{i=1}^{t} g_{i}U. Then for f ∈ B(U, M[j]) we have

(T_{r}f)(g) = ∑_{i}(g_{i})_{l}(det(g_{i})_{l})^{j}f(gg_{i})

and hence

(ιT_{r}f)(g) = τ_{l}(g)^{j} ∑_{i}(g_{i})_{l}(det(g_{i})_{l})^{j}f(gg_{i}).

On the other hand, ιf ∈ B(U, M)[j] satisfies

(T_{r}(ιf))(g) = r^{j} ∑_{i}(g_{i})_{l}(ιf)(gg_{i}))

= r^{j} ∑_{i}(g_{i})_{l}τ_{l}(gg_{i})^{j}f(gg_{i})

= r^{j}τ_{l}(g)^{j} ∑_{i}(g_{i})_{l}τ_{l}(g_{i})^{j}f(gg_{i})

and so to finish the proof we observe that for any g_{i} ∈ Uη_{r}U we have r^{j}τ_{l}(g_{i})^{j} = det((g_{i})_{l})^{j} in ℤ_{l}^{×}.

□

We now define the notion of twisting a maximal ideal of residue characteristic l by a character. Let 𝔞 ⊆ 𝕋^{(Nl)} be a maximal ideal with residue characteristic l, contained in the support of some space of cusp forms of weight k. Let χ : Gal(\overline{ℚ}/ℚ) → 𝔽_{l}^{×} be the mod l reduction of the cyclotomic character. Then we define the ideal 𝔞 ⊗ χ^{i} to be the kernel of the composite

\begin{matrix}α : 𝕋^{(Nl)} & → 𝕋^{(Nl)} → (𝕋^{(Nl)}/𝔞) \\ T_{r} & ↦ r^{i}T_{r} \\ S_{r} & ↦ r^{2i}S_{r}. \end{matrix}

Note that (𝕋^{(Nl)}/𝔞) is finite, so the kernel 𝔞 ⊗ χ^{i} is also a maximal ideal of 𝕋^{(Nl)}. The motivation behind this construction is clear. If 𝔞 is the maximal ideal attached to a mod l eigenform f𝔞 which gives rise to a mod l representation ρ𝔞, then 𝔞 ⊗ χ^{i} is attached to the form θ^{i}f, where θ is the operator defined by Katz in [K2], and the representation attached to θ^{i}f is ρ_{𝔞} ⊗ χ^{i}.

If 𝔞′ ⊆ 𝕋^{(Nld)} is a maximal ideal then we define 𝔞′ ⊗ χ^{i} similarly.

Finally in this section we record two propositions which will serve as the analogues of the operators θ and (multiplication by) A (the Hasse invariant) on spaces of mod l cusp forms.

In the following two propositions, d is the product of an odd number of distinct primes, and N ≥ 5 is an integer prime to d. Proposition 4.4.4 is the analogue of the θ operator.

Proposition 4.4.4. If k ≥ 2 and g ∈ S_{k}(Γ_{1}(N) ∩ Γ_{0}(d); \overline{𝔽}_{l})^{d-new} is an eigenvector for 𝕋^{(Nld)}, then there is an eigenform h ∈ S_{k+l+1}(Γ_{1}(N) ∩ Γ_{0}(d); \overline{𝔽}_{l})^{d-new} such that the representation ρ_{h} is isomorphic to ρ_{g} ⊗ χ.

Proof. We switch to the definite quaternion algebra D/ℚ with discriminant d. We have a maximal ideal 𝔞′ ⊆ 𝕋^{(Nld)} attached to g such that 𝔞′ ∈ Supp_{𝕋^{(Nld)}}(S_{k}(Γ_{1}(N) ∩ Γ_{0}(d); \overline{𝔽}_{l})^{d-new}). We must show that 𝔞′ ⊗ χ ∈ Supp_{𝕋^{(Nld)}}(S_{k+l+1}(Γ_{1}(N) ∩ Γ_{0}(d); \overline{𝔽}_{l})^{d-new}). By Corollary 4.4.2 it suffices to find an injection

θ : L_{k}(U_{1}(N); 𝔽_{l}) ↪ L_{k+l+1}(U_{1}(N); 𝔽_{l})

such that T_{r}θ = rθT_{r} and S_{r}θ = r^{2}θS_{r} for all primes r not dividing Nld, or equivalently a 𝕋^{(Nld)}-equivariant injection B(U_{1}(N); \overline{V}_{k−2}) ↪ B(U_{1}(N); \overline{V}_{k+l−1})[−1]. By Lemma 4.4.3 this is equivalent to finding a 𝕋^{(Nld)}-equivariant injection

θ : B(U_{1}(N); \overline{V}_{k−2}) ↪ B(U_{1}(N); \overline{V}_{k+l−1}[−1]).

We define θ : B(U_{1}(N); \overline{V}_{k−2}) → B(U_{1}(N); \overline{V}_{k+l−1}[−1]) by

(θf)(g) = (XY^{l} − Y X^{l})f(g).

It is an easy exercise to check that this is the map we require.

□

Proposition 4.4.5 is the analogue of multiplication by the Hasse invariant.

Proposition 4.4.5. If g ∈ S_{k}(Γ_{1}(N) ∩ Γ_{0}(d); \overline{𝔽}_{l})^{d-new} is an eigenform for 𝕋^{(Nld)}, 3 ≤ k ≤ l+1, and n is a positive integer, then there is an eigenform h ∈ S_{k+n(l−1)}(Γ_{1}(N)∩ Γ_{0}(d); \overline{𝔽}_{l})^{d-new} such that the representations ρ_{h} and ρ_{g} are isomorphic.

Proof. As in the proof of Proposition 4.4.4 we use the Jacquet-Langlands Theorem to reduce the problem to finding a Hecke equivariant injection from B(U_{1}(N); \overline{V}_{k−2}) to a subquotient of B(U_{1}(N); \overline{V}_{k−2+n(l−1)}). We recall from [AS] Lemma 3.2 that the evaluation homomorphism α_{g} : \overline{V}_{g} → I_{g} is injective if g < l and surjective if g ≥ l. The surjection α_{k−2+n(l−1)} : \overline{V}_{k−2+n(l−1)} → I_{k−2} identifies I_{k−2} with a subquotient of \overline{V}_{k−2+n(l−1)}, and α_{k−2} injects V_{k−2} into this subquotient. These maps induce an injection from B(U_{1}(N); \overline{V}_{k−2}) to B(U_{1}(N); I_{k−2}), a subquotient of B(U_{1}(N); \overline{V}_{k−2+n(l−1)}), and it is easily checked that this injection is Hecke equivariant and hence gives us the result we require.

□

§4.5. 3 ≤ k ≤ l: End of proof.

We now prove the analogue of Proposition 3.5.1 for p-new forms, namely Proposition 4.5.1. If 3 ≤ k ≤ l + 1 then any non-Eisenstein 𝔞′ ⊆ 𝕋^{(Nlp)} such that

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(((L′)^{p-new}/π(L′)^{p-new})(k − 2))

satisfies either

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new})

or

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{l+3−k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}[k − 2]).

Proof. Let D be the quaternion algebra over ℚ with discriminant p. Define Γ_{1}(l) ⊆

GL_{2}(ℤ_{l}) to be the subgroup of matrices congruent to ( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} ) mod l, and define Γ_{0}(l) ⊆

GL_{2}(ℤ_{l}) to be the subgroup of matrices congruent to ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) mod l. Note that Γ_{1}(l) is

the projection of U_{1}(l) onto GL_{2}(ℤ_{l}). Give 𝔽_{l} the trivial action of Γ_{1}(l).

The space B(U_{1}(N) ∩ U_{1}(l); 𝔽_{l}) of functions
{f : D^{×}\D_{f}^{×} → 𝔽_{l} such that
f(gu) = f(g) for all u ∈ U_{1}(N) ∩ U_{1}(l)}

has an action of 𝕋^{(Nlp)}, and by the Jacquet-Langlands Theorem we see that 𝔞′ is in its support. Moreover, if

B(U_{1}(N) ∩ U_{1}(l); 𝔽_{l})(k − 2)

is the submodule of B(U_{1}(N) ∩ U_{1}(l); 𝔽_{l}) consisting of functions f such that S_{r}f = r^{k−2}f for all primes r not dividing Nlp such that r ≡ 1 mod N, then the fact that 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(((L′)^{p-new}/π(L′)^{p-new})(k − 2)) tells us that 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(B(U_{1}(N) ∩ U_{1}(l); 𝔽_{l})(k − 2)).

Now let 𝔽_{l}{k − 2} be the space 𝔽_{l} with an action of Γ_{0}(l) given by

( \begin{matrix}a & b \\ 0 & d \end{matrix} ) x = d^{k−2}x.

Set A = B(U_{1}(N) ∩ U_{0}(l); 𝔽_{l}{k − 2}). So

A = {f : D^{×}\D_{f}^{×} → 𝔽_{l}{k − 2} such that
f(gu) = u_{l}^{−1}f(g) for all u ∈ U_{1}(N) ∩ U_{0}(l)}.

An easy calculation shows that A = B(U_{1}(N) ∩ U_{1}(l); 𝔽_{l})(k − 2). Moreover, the natural actions of 𝕋^{(Nlp)} on both sides agree, and hence 𝔞′ ∈ Supp(A).

For f ∈ B(U_{1}(N); I_{k−2}), define δ : D^{×}\D_{f}^{×} → 𝔽_{l} by δ(g) = (f(g))(0, 1). It is easily checked that f can be reconstructed from δ, and moreover that the map

\begin{matrix}B(U_{1}(N); I_{k−2}) & → A \\ f & ↦ δ \end{matrix}

is an isomorphism of 𝕋^{(Nlp)}-modules. Hence 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(B(U_{1}(N); I_{k−2})). The exact sequence

0⟶\overline{V}_{k−2} ⟶^{α_{k−2}}I_{k−2} ⟶^{β_{k−2}}\overline{V}_{l+1−k}[k − 2]⟶0

of modules for GL_{2}(𝔽_{l}) induces an exact sequence

0 → B(U_{1}(N); \overline{V}_{k−2}) → B(U_{1}(N); I_{k−2}) → B(U_{1}(N); \overline{V}_{l+1−k}[k − 2]) → 0

of 𝕋^{(Nlp)}-modules. One way to see that the latter sequence is exact is to recall that for N ≥ 5 we have an isomorphism

\begin{matrix}B(U_{1}(N); M) & ≅ M^{r} \\ f & ↦ (f(g_{i})) \end{matrix}

where D_{f}^{×} = ∐_{i=1}^{r} D^{×}g_{i}U_{1}(N).

Because 𝔞′ is in the support of the middle module, we deduce that either

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(B(U_{1}(N); \overline{V}_{k−2}))

or

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(B(U_{1}(N); \overline{V}_{l+1−k}[k − 2]))
= Supp_{𝕋^{(Nlp)}}(B(U_{1}(N); \overline{V}_{l+1−k})[k − 2])

by Lemma 4.4.3. By the results of §4.4 again we observe that this translates via the Jacquet-Langlands Theorem to

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new})

or

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{l+3−k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}[k − 2]),

the result we require.

□

We can now prove the main result of this section.

Theorem 4.5.2. Say 𝔞 is a maximal ideal of 𝕋^{(Nl)} in the support of S_{k}(Γ_{1}(N); \overline{𝔽}_{l}). Assume that the mod l representation attached to 𝔞 is irreducible and not D_{l}-scalar. Say T_{p}^{2} − (1 + p)^{2}S_{p} ∈ 𝔞. Write 𝔞′ = 𝔞 ∩ 𝕋^{(Nlp)}. Then either

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new})

or

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{l+3−k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}[k − 2]).

Proof. We have

𝔞 ∈ Supp_{𝕋^{(Nl)}}(S_{k}(Γ_{1}(N); 𝔽_{l})).

By Proposition 3.5.1 we see

𝔞 ∈ Supp_{𝕋^{(Nl)}}(\overline{L}(k − 2)).

By Proposition 4.3.2,

𝔞′ = 𝔞 ∩ 𝕋^{(Nlp)} ∈ Supp(((L′)^{p-new}/π(L′)^{p-new})(k − 2)),

and by Proposition 4.5.1 we deduce that either

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new})

or

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{l+3−k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}[k − 2]).
□

Corollary 4.5.3. Theorem 4.1.3 is true when 3 ≤ k ≤ l.

Proof. Note first that if 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}) then we may set k′ = k and we are finished. So we may assume that 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{l+3−k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}[k − 2]).

If 𝔞 is ordinary, we shall show that in fact this assumption must be false. Indeed, if 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{l+3−k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}[k − 2]), then there exists an eigenform g ∈ S_{l+3−k}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{𝔽}_{l}) whose associated maximal ideal under the twisted action of 𝕋^{(Nlp)} is 𝔞′. Hence the representation ρ_{g} attached to g satisfies

ρ_{g} ⊗ χ^{k−2} = ρ_{f},

where χ is the mod l reduction of the cyclotomic character. We have, by Theorem 3.6.1,

ρ_{f}|_{D_{l}} ≅ ( \begin{matrix}χ^{k−1}λ(ϵ(l)/a_{l}) & ∗ \\ 0 & λ(a_{l}) \end{matrix} )

and hence

ρ_{g}|_{D_{l}} ≅ ( \begin{matrix}χλ(ϵ(l)/a_{l}) & ∗ \\ 0 & χ^{2−k}λ(a_{l}) \end{matrix} ) .

But this latter representation is not irreducible (hence g must be ordinary) and cannot possess an unramified 1-dimensional submodule (hence g cannot be ordinary). Hence this representation contradicts Theorem 3.6.1. We deduce that in fact 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{k}(Γ_{1}(N)∩ Γ_{0}(p); 𝔽_{l})^{p-new}) if 𝔞 arises from an ordinary form, and also that we may take k′ = k in this case.

If f is not ordinary, then we use the observation that

\overline{V}_{l+k−3} → → I_{k−2} → → \overline{V}_{l+1−k}[k − 2]

via the maps α_{l+k−3} and β_{k−2} defined in [AS] and hence that if D/ℚ is the quaternion algebra with discriminant p, then

Supp_{𝕋^{(Nlp)}}(B(U_{1}(N); \overline{V}_{l+1−k})[k − 2]) ⊆ Supp_{𝕋^{(Nlp)}}(B(U_{1}(N); \overline{V}_{l+k−3})),

or equivalently,

Supp_{𝕋^{(Nlp)}}(S_{l+3−k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}[k − 2])
⊆Supp_{𝕋^{(Nlp)}}(S_{l+k−1}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}).

Hence in the case 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{l+3−k}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}[k − 2]), we may conclude that 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{l+k−1}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{𝔽}_{l})^{p-new}), set k′ = k + l − 1, and Theorem 4.1.3 is proved.

□

See the end of this chapter for some remarks about the non-ordinary case and the fact that k′ ≠ k in this case.

§4.6. The cases k = 2 and k = l + 1.

Proposition 4.6.1. Theorem 4.1.3 is true for k = 2.

Proof. This is proved in [DT1] but for completeness we shall give a sketch of how to modify our ideas to prove the result in this case. Note that things are not as difficult in this situation, for two reasons. Firstly, the abelian variety Jac(X_{1}(N)_{ℚ}) over ℚ has good reduction at l, and we can work with a model over ℤ_{l} which is an abelian scheme. Secondly, there is no problem about reducing to weight 2 and lifting back again, as we are in weight 2 already.

The two maps 1 and η : X_{1}(N; p)_{ℤ_{l}} → X_{1}(N)_{ℤ_{l}} induce maps between Jacobians Jac(X_{1}(N))_{ℤ_{l}} → Jac(X_{1}(N; p))_{ℤ_{l}} by Picard functoriality. The mod l reduction of these maps are just the maps between Jacobians of the corresponding irreducible curves over 𝔽_{l}. The map

1^{∗} + η^{∗} : H^{0}(X_{1}(N)_{𝔽_{l}}, Ω_{X_{1}(N)/𝔽_{l}})^{2} → H^{0}(X_{1}(N; p)_{𝔽_{l}}, Ω_{X_{1}(N;p)/𝔽_{l}})

is injective by the q-expansion principle, as can be seen by a calculation similar to the case i = 1 of Lemma 4.2.3. Hence by Proposition 2.3.4 the kernel K of

α : Jac(X_{1}(N)_{𝔽_{l}})^{2} → Jac(X_{1}(N; p)_{𝔽_{l}})

has multiplicative l-torsion.

If f is an eigenform giving rise to 𝔞 then there is a maximal ideal 𝔪 of a Hecke algebra H defined in [G] §12, and an l-divisible group G over ℤ_{l}, attached to f. Moreover, in [G] 12.8 a detailed analysis is made of G[𝔪]_{ℤ_{l}} in the case where f is ordinary. In particular, the special fibre of G[𝔪]_{ℤ_{l}} contains no local-local part, and Gal(\overline{ℚ}_{l}/ℚ_{l}) acts on the H/𝔪-vector space G^{0}[𝔪]_{ℤ_{l}}(\overline{ℚ}_{l}) via a character. If f is not ordinary, then it is shown in [E] Chapter 6 that the special fibre of G[𝔪]_{ℤ_{l}} is local-local. The analogue of Lemma 4.3.1, with R replaced by ℤ_{l}, holds, and all the techniques of the proof of Proposition 4.3.2 go through in this situation to show that 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{2}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}) and hence that we may take k′ = k = 2.

□

Proposition 4.6.2. Theorem 4.1.3 is true for k = l + 1.

Proof. Pick an eigenform f𝔞 ∈ S_{l+1}(Γ_{1}(N); \overline{𝔽}_{l}) corresponding to 𝔞. If f𝔞 is the mod l reduction of a characteristic 0 weight 2 form of level N then we may apply Proposition 4.6.1 to finish the proof. Hence we may assume that 𝔞 ∉ Supp(S_{2}(Γ_{1}(N); \overline{𝔽}_{l})), i.e. that the mod l eigenform f𝔞 has filtration l + 1.

By [G] Proposition 9.3 we can lift f𝔞 to an eigenform

F ∈ S_{2}(Γ_{1}(N) ∩ Γ_{0}(l); O_{\overline{ℚ}_{l}}),

which is new at l. Hence

𝔞 ∈ Supp(S_{2}(Γ_{1}(N) ∩ Γ_{0}(l); ℂ)^{l-new}).

Now say D is the quaternion algebra over ℚ ramified at l and ∞. We use the notation of §4.4, and briefly recall some of it. Let 𝔸_{f} be the finite adeles over ℚ, and let U_{1}(N) ⊆

(D⊗𝔸_{f})^{×} be the subgroup of ∏_{r} O_{D,r}^{×} consisting of matrices congruent to ( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} ) modulo

N, where for primes r dividing N we have identified O_{D,r} with M_{2}(ℤ_{r}). Define U_{1}(N; p) to

be the subgroup of U_{1}(N) consisting of matrices which are congruent to ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) modulo p.

By the Jacquet-Langlands Theorem we have that

𝔞 ∈ Supp(S_{2}^{D}(U_{1}(N); ℂ))

where S_{2}^{D}(U_{1}(N); ℂ) is the space S_{2}^{D}(U_{1}(N)) of weight 2 cusp forms of level U_{1}(N) over the quaternion algebra D, as defined in [DT1] §2. We want to deduce that 𝔞′ = 𝔞 ∩ 𝕋^{(Nlp)} is in the support of the space S_{2}^{D}(U_{1}(N; p); ℂ)^{p-new}, and this can be done using the technique of the proof of Theorem 1 of [DT1]. We cannot actually apply the theorem directly, because l divides the discriminant of D in our case. However, because k = 2 the proof of the theorem goes through. We again sketch the details.

Recall that if M is a ℤ_{l}-module with an action of O_{D,l}^{×} then we defined a space of functions B(U_{1}(N), M). In particular, if we let O_{D,l}^{×} act trivially on ℤ_{l}, ℚ_{l} and ℚ_{l}/ℤ_{l}, then we can consider spaces such as B(U_{1}(N); ℚ_{l}/ℤ_{l}). The maps 1 and η_{p} from B(U_{1}(N); ℚ_{l}/ℤ_{l}) to B(U_{1}(N) ∩ U_{0}(p); ℚ_{l}/ℤ_{l}) defined by

(1f)(g) = f(g)

and

(η_{p}f)(g) = f(η_{p}^{−1}gη_{p})

define a map α = 1⊕η_{p} : B(U_{1}(N); ℚ_{l}/ℤ_{l})^{2} → B(U_{1}(N)∩U_{0}(p); ℚ_{l}/ℤ_{l}). If (f_{1}, f_{2}) ∈ ker α, then we see that f_{1}(gu) = f_{1}(g) for all u ∈ U_{1}(N) and also for all u ∈ η_{p}U_{1}(N)η_{p}^{−1}. We deduce that f_{1}(gu) = f_{1}(g) for all u ∈ U_{1}(N). SL_{2}(ℚ_{p}), because SL_{2}(ℚ_{p}) is in the

subgroup of GL_{2}(ℚ_{p}) generated by GL_{2}(ℤ_{p}) and ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) GL_{2}(ℤ_{p}) ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )^{−1} . Moreover,

f_{1}(dg) = f_{1}(g) for all d ∈ D^{×}. By the strong approximation theorem, f_{1} is constant. Hence f_{2} is also constant, and ker α is Eisenstein. This proves the analogue of [DT1] Lemma 2.

The pairing ⟨, ⟩_{U_{1}(N)} : B(U_{1}(N); ℚ_{l}) × B(U_{1}(N); ℚ_{l}) → ℚ_{l} defined by the finite sum

⟨f_{1}, f_{2}⟩_{U_{1}(N)} = ∑_{g∈D^{×}\D_{f}^{×}/U_{1}(N)} f_{1}(g)f_{2}(g)

is the pairing we need to prove the analogue of Corollary 1 of [DT1]. Now the proof of Theorem 1 of [DT1] in the case disc(D) = l and k = 2 is immediate. We deduce that 𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{2}^{D}(U_{1}(N) ∩ U_{0}(p); ℂ)^{p-new}).

Once again by the Jacquet-Langlands Theorem we have that

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{2}(Γ_{1}(N) ∩ Γ_{0}(pl); ℂ)^{pl-new})

and hence a fortiori

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{2}(Γ_{1}(Nl) ∩ Γ_{0}(p); ℂ)^{p-new}).

So 𝔞′ ∈ Supp((L′)^{p-new}/π(L′)^{p-new})(l − 1) and hence by Proposition 4.5.1

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{l+1}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new})

or

𝔞′ ∈ Supp_{𝕋^{(Nlp)}}(S_{2}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}).

Hence setting k′ = l + 1 or 2 as appropriate gives us the result we require.

□

§4.7. Proof of Theorem 4.1.3 in general.

Proof of Theorem 4.1.3. Say k ≥ 2 and 𝔞 are as in Theorem 4.1.3. Then, after fixing an injection 𝕋^{(Nl)}/𝔞 ↪ \overline{𝔽}_{l}, we know that 𝔞 gives rise to an irreducible representation

ρ𝔞 : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{l}).

If χ is the mod l cyclotomic character, then by [AS] Corollary 3.6 one of the twists ρ𝔞 ⊗χ^{i}, 0 ≤ i ≤ l − 2, arises from an eigenform f_{1} ∈ S_{k_{1}}(Γ_{1}(N); \overline{𝔽}_{l}), where 2 ≤ k_{1} ≤ l + 1. Say 𝔞_{1} = 𝔞 ⊗ χ^{i}. Then 𝔞_{1} is the maximal ideal attached to f_{1}. We have proved Theorem 4.1.3 for representations arising from forms of weight k_{1} satisfying 2 ≤ k_{1} ≤ l + 1, and hence there is an eigenform g_{1} ∈ S_{k′_{1}}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{𝔽}_{l})^{p-new} such that

ρ_{g_{1}} ≅ ρ_{f_{1}}.

Then, applying Proposition 4.4.4, (l − 1 − i) times, there is an eigenform g ∈ S_{k′}(Γ_{1}(N) ∩ Γ_{0}(p); \overline{𝔽}_{l})^{p-new}, where k′ = k′_{1} + (l − 1 − i)(l + 1), giving rise to ρ_{𝔞}. The maximal ideal of 𝕋^{(Nlp)} attached to g is 𝔞′ = 𝔞 ∩ 𝕋^{(Nlp)}.

□

Note that Theorem 4.1.3 asserts that two representations, attached to forms of weight k and k′, are isomorphic. It unfortunately does not tell us that k = k′, a result that is present in many other results of this form ([R1],[D2],[DT1].) However, we can prove that k′ can be taken to be k in many cases, and we hope to prove that k′ = k will suffice in every case. For example, the proof of Corollary 4.5.3 shows that we can take k′ = k when f has weight k satisfying 3 ≤ k ≤ l and f is ordinary. However, when f is not ordinary, and 3 ≤ k ≤ l, then in Theorem 4.5.2 we show that the maximal ideal 𝔞′ attached to f is either in the support of S_{k}(Γ_{1}(N) ∩ Γ_{0}(p))^{p-new} or in the support of S_{l+3−k}(Γ_{1}(N)∩Γ_{0}(p))^{p-new}[k −2]. Now if g ∈ S_{l+3−k}(Γ_{1}(N)∩Γ_{0}(p))^{p-new} is a non-ordinary eigenform, then by [E] Proposition 3.3 we see that θ^{k−2}g has filtration k and gives rise to the representation we require. Unfortunately, we cannot at present show that this form is the mod l reduction of a characteristic 0 form which is new at p. One might hope to prove that θ^{k−2}g is p-new, and hence that k′ may be taken to equal k in Theorem 4.1.3 if 3 ≤ k ≤ l. A result of this form would follow from the analogue of various filtration results for mod l cusp forms, in the setting of modular forms on a definite quaternion algebra, and we are currently looking into proving analogues of these filtration results.

Chapter 5. Adding more primes to the level.

Summary. In this chapter, we generalise the results of Chapter 4 to Shimura curves and hence prove results analogous to those in Chapter 4, but for modular forms over an indefinite quaternion algebra. The observation that the theory of bad reduction of certain Shimura curves should be analogous to the bad reduction of modular curves is made in the introduction to [DR], together with a brief justification. We prove several instances of this observation, and then use ideas from chapter 4 to deduce the analogue of Theorem 1.1.1 of Chapter 1 for modular forms over an indefinite quaternion algebra. Using this and Theorem 1 of [DT1] we deduce Theorem 1.1.2 of Chapter 1. Note that in several cases we have omitted calculations which turned out to be completely analogous to the modular curve case, and we feel confident that the interested reader will be able to check any details omitted.

§5.1. Indefinite quaternion algebras.

Let D be an indefinite quaternion algebra over ℚ, and let d = disc(D). Then d is the product of a finite even number of distinct primes. Furthermore, assume d > 1, i.e. that D is not split. Fix once and for all a maximal order O_{D} of D, stable under the canonical involution of D. Fix also t ∈ O_{D} with t^{2} = d. Such t exist because ℚ(\sqrt{−d}) splits D.

For a prime p write O_{D,p} for O_{D} ⊗_{ℤ}ℤ_{p}. If p ∤ d then fix an isomorphism i_{p} : D⊗_{ℚ}ℚ_{p} → M_{2}(ℚ_{p}) with the following properties:

(i) i_{p}(O_{D,p}) = M_{2}(ℤ_{p})

(ii) If p is an odd prime, then i_{p}(t) = ( \begin{matrix}0 & 1 \\ −d & 0 \end{matrix} ) in M_{2}(ℤ_{p}).

It is not difficult to see that this can be achieved. Condition (ii) is not essential and we demand it merely for convenience. We also fix once and for all an isomorphism i_{∞} : D ⊗ ℝ → M_{2}(ℝ).

We wish to use the maps i_{p} to give us a notion of “congruence mod M” for M prime to d, but there is a slight caveat here, demonstrated by the following phenomenon. If p and q are two distinct primes, then the natural map

\begin{matrix}ℤ/qℤ & → ℤ/pqℤ \\ 1 & ↦ p \end{matrix}

and the natural map

\begin{matrix}ℤ/pℤ & → ℤ/pqℤ \\ 1 & ↦ q \end{matrix}

combine to give an isomorphism ℤ/pℤ × ℤ/qℤ → ℤ/pqℤ. However, this map is not in general the inverse of the natural isomorphism

\begin{matrix}ℤ/pqℤ & → ℤ/pℤ × ℤ/qℤ \\ 1 & ↦ (1, 1). \end{matrix}

So we explain our conventions of “natural map” carefully.

Let M be a positive integer. Write M = ∏_{j} p_{j}^{e_{j}}. There is a natural map

α : ∏_{p} O_{D,p} → → ∏_{j} O_{D,p_{j}} ⊗ ℤ/p_{j}^{e_{j}}ℤ ≅ ∏_{j} O_{D} ⊗ ℤ/p_{j}^{e_{j}}ℤ

and a natural map

β : O_{D} ⊗ ℤ/Mℤ → ∏_{j} O_{D} ⊗ ℤ/p_{j}^{e_{j}}ℤ,

given by the product of the surjections O_{D} ⊗ℤ/Mℤ → O_{D} ⊗ℤ/p_{j}^{e_{j}}ℤ. It is not hard to show that β is an isomorphism. The “natural map” t_{M} : ∏_{p} O_{D,p} → O_{D} ⊗ ℤ/Mℤ is defined to be β^{−1}α. The maps t_{M} have the property that if M | N then t_{M} is t_{N} followed by the natural surjection O_{D} ⊗ ℤ/Nℤ → O_{D} ⊗ ℤ/Mℤ.

If p is a prime not dividing d, then i_{p} induces isomorphisms O_{D}⊗ℤ/p^{n}ℤ ≅ M_{2}(ℤ/p^{n}ℤ) for n ≥ 0, and hence if M and d are coprime we have a diagram

O_{D} ⊗ ℤ/Mℤ  M_{2}(ℤ/Mℤ)
↓  ↓
∏_{j} O_{D} ⊗ ℤ/p_{j}^{e_{j}}ℤ ⟶ ∏_{j} M_{2}(ℤ/p_{j}^{e_{j}}ℤ)

where the vertical maps are the products of the natural surjections. All the maps in this diagram are isomorphisms, and hence we may identify O_{D} ⊗ ℤ/Mℤ with M_{2}(ℤ/Mℤ) via the diagram, and we call this the natural isomorphism O_{D} ⊗ ℤ/Mℤ → M_{2}(ℤ/Mℤ).

Finally, the composite of this map O_{D} ⊗ ℤ/Mℤ → M_{2}(ℤ/Mℤ) with the map t_{M} : ∏_{p} O_{D,p} → O_{D} ⊗ ℤ/Mℤ will be called the natural map u_{M} : ∏_{p} O_{D,p} → M_{2}(ℤ/Mℤ). The maps u_{M} are also well-behaved with respect to surjections, i.e. if M | N and N is coprime to d, then the surjection M_{2}(ℤ/Nℤ) → M_{2}(ℤ/Mℤ) composed with the map u_{N} is the map u_{M}.

§5.2. False elliptic curves.

Recall D is an indefinite quaternion algebra over ℚ, d = disc(D), and O_{D} is a fixed maximal order of D.

Let S be a scheme on which d is invertible. A false elliptic curve over S is a pair (A/S, i) where A/S is an abelian scheme of relative dimension 2, and i : O_{D} ↪ End_{S}(A) is an injective ring homomorphism. We will frequently abuse notation and talk of “the false elliptic curve A/S ”, or even just “the false elliptic curve A”. Strictly speaking we should really refer to “false elliptic curves for D”, but dropping the D will not cause confusion. Also, we shall only ever refer to false elliptic curves over ℤ[1/d]-schemes. Good references for false elliptic curves are [DT1], [B] and [BC]. If A/S is a false elliptic curve, and T is an S-scheme, the pullback A_{T} = A ×_{S} T can be naturally given the structure of a false elliptic curve over T.

Recall that we fixed t ∈ O_{D} such that t^{2} = −d. Let x ↦ x^{ι} denote the canonical involution of D. Then we can define an involution ∗ of D by x^{∗} = t^{−1}x^{ι}t, and we see that O_{D} is stable under ∗. Now by [DT1] Lemma 5 we see that if (A/S, i) is a false elliptic curve, then there is a unique principal polarisation on A such that if \overline{s} is a geometric point of S, then the corresponding Rosati involution on End(A\overline{s}) induces ∗ on O_{D}. Hence the false elliptic curves that we consider are canonically principally polarised.

If (A/S, i) and (B/S, j) are false elliptic curves, then by an isogeny of false elliptic curves we mean an isogeny π : A → B over S in the usual sense, such that j(x)∘π = π∘i(x) for all x ∈ O_{D}. If π : A → B is an isogeny of false elliptic curves, then π induces a dual isogeny π^{∨} : B^{∨} → A^{∨} which, because we have defined principal polarisations on A and B, induces a map π^{t} : B → A. One may check that this map is also an isogeny of false elliptic curves, and we refer to it as the dual isogeny to π. The composite π^{t} ∘ π : A → A is locally multiplication by an integer d, and if this integer is constant on S we call it the false degree of π. There is a canonical perfect pairing

⟨, ⟩_{π} : ker(π) × ker(π^{t}) → 𝔾_{m}.

In fact, there are two canonical perfect pairings, one being the inverse of the other. To fix our ideas, we shall again use the opposite of the definition in [O]. Lemma 1.1 of [O] then shows us that if δ ∈ D, P ∈ (ker π)(S) and Q ∈ (ker π^{t})(S), then ⟨P, j(δ)Q⟩_{π} = ⟨i(δ^{∗})P, Q⟩_{π} in 𝔾_{m}(S).

We will now explain how to attach various (naïve) level structures to false elliptic curves, analogous to the level structures that one can define on elliptic curves. Let M be a positive integer prime to d. A (naïve) full level M structure on a false elliptic curve (A/S, i) is an isomorphism

α : (O_{D} ⊗ ℤ/Mℤ)_{S}⟶^{≅}A[M]

of schemes which preserves the left action of O_{D}. Here (O_{D} ⊗ ℤ/Mℤ)_{S} is the constant group scheme over S associated to the finite group O_{D} ⊗ ℤ/Mℤ. Note that the existence of a full level M structure on A/S implies that M is invertible on S.

If δ ∈ O_{D}⊗ℤ/Mℤ then right multiplication by δ induces a morphism of group schemes

r_{δ} : (O_{D} ⊗ ℤ/Mℤ)_{S} → (O_{D} ⊗ ℤ/Mℤ)_{S}

which preserves the left action of O_{D}. Moreover, if δ ∈ (O_{D} ⊗ ℤ/Mℤ)^{×} then r_{δ} is an isomorphism. Hence if δ ∈ (O_{D} ⊗ ℤ/Mℤ)^{×} and α is a full level M structure on a false elliptic curve A, then α ∘ r_{δ} is also a full level M structure on A. The map α ↦ α ∘ r_{δ} defines a left action of (O_{D} ⊗ ℤ/Mℤ)^{×} on the set of full level M structures on A.

Let H ⊆ (O_{D} ⊗ ℤ/Mℤ)^{×} be a subgroup, let S be a ℤ[1/Md]-scheme, and let (A/S, i) be a false elliptic curve. We will now define a level H structure on A. Consider the contravariant functor F_{A/S,M} from S-schemes to sets, sending T/S to the set of full level M structures on A_{T} . Then F_{A/S,M} is represented by a closed subscheme of A[M], and is an étale (because formally étale) (O_{D} ⊗ ℤ/Mℤ)^{×}-torsor. For each T, there is a left action of (O_{D} ⊗ ℤ/Mℤ)^{×}, and hence of H, on F_{A/S,M}(T) as defined above. If H\F_{A/S,M}(T) denotes the orbit space, then it is easy to check that T ↦ H\F_{A/S,M}(T) is a functor from S-schemes to sets. Let F_{A/S,H} be its sheafification with respect to the étale topology. Then F_{A/S,H} is represented by a quotient of the S-scheme representing F_{A/S,M} and is again a finite étale covering of S. A level H structure on (A/S, i) is an element of F_{A/S,H}(S).

Recall we defined a natural map t_{M} : ∏_{p} O_{D,p} → O_{D} ⊗ ℤ/Mℤ and this induces a homomorphism ∏_{p} O_{D,p}^{×} → (O_{D}⊗ℤ/Mℤ)^{×}. Let U ⊆ ∏_{p} O_{D,p}^{×} be the preimage of H under this homomorphism. It is not difficult to check that the definition of a level H structure depends only on U, in the following sense. Let M_{1} and M_{2} be two positive integers prime to d, with M_{1} | M_{2}. Let H_{1} ⊆ (O_{D} ⊗ ℤ/M_{1}ℤ)^{×} be a subgroup, and let H_{2} be its preimage in (O_{D} ⊗ ℤ/M_{2}ℤ)^{×} under the natural surjection. Then the notion of an H_{1} structure and an H_{2} structure coincide for false elliptic curves over ℤ[1/M_{2}d]-schemes. Hence it makes sense to talk about the functor F_{A/S,U} and also of level U structures on a false elliptic curve A/S, for subgroups U ⊆ ∏_{p} O_{D,p}^{×} arising in this way.

Finally we record how level U structures may be pushed forward in some cases. Say (A/S, i) and (B/S, j) are false elliptic curves, and ψ : A → B is an isogeny of false elliptic curves. Say H ⊆ GL_{2}(ℤ/Mℤ) is a subgroup, and ϕ is a level H structure on A/S. Assume moreover that the false degree of ψ is coprime to M. Then ψ induces an isomorphism A[M] → B[M] and hence composition with ψ induces an isomorphism of functors F_{A/S,M} → F_{B/S,M}. Furthermore, this isomorphism preserves the action of O_{D} and hence induces an isomorphism F_{A/S,H} ≅ F_{B/S,H}, and hence a bijection ψ_{∗} from the level H structures on A/S and the level H structures on B/S. In particular, ψ_{∗}ϕ is a level H structure on B/S.

§5.3. Shimura curves.

As before, 𝔸_{f} is the finite adeles over ℚ, and D_{f} = D ⊗ 𝔸_{f}. Let M be a positive integer prime to d. We have a natural map u_{M} : ∏_{p} O_{D,p} → M_{2}(ℤ/Mℤ), which induces a natural map u_{M} : ∏_{p} O_{D,p}^{×} → GL_{2}(ℤ/Mℤ). We now make the following three definitions.

Let U_{0}(M) denote the preimage of \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℤ/Mℤ)|c = 0 \} under u_{M}.

Let U_{1}(M) denote the preimage of \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℤ/Mℤ)|c = 0 and d = 1 \} under u_{M}.

Let U_{M} be the kernel of the map u_{M}.

We wish to consider a general open compact subgroup U of ∏_{p} O_{D,p}^{×}. However, sometimes for theorems to be true, U must satisfy certain properties. To simplify matters slightly, we shall put some blanket assumptions on U which will always be true in the applications but which will sometimes not be needed in the proofs of the theorems. So from now on, U shall stand for a compact open subgroup of ∏_{p} O_{D,p}^{×} with the following three properties:

(i) det(U) = ℤ̂^{×} = ∏_{p} ℤ_{p}^{×} (ii) U = ∏_{p|d} O_{D,p}^{×} × U^{d}, where U^{d} is the projection of U onto ∏_{p∤d} O_{D,p}^{×}. (iii) U ⊆ U_{1}(N) for some N ≥ 5 (prime to d, by (ii)).

Briefly, here are the reasons that these assumptions are here. (i) is convenient because it implies that the complex curve attached to U (see below) is connected. (ii) makes sure that we never consider level structures at primes dividing d, and (iii) is a smallness criterion which we shall need for later representability results.

Define U_{∞} ⊆ (D⊗ℝ)^{×} to be the stabiliser of i under the natural action of (D⊗ℝ)^{×} ≅ GL_{2}(ℝ) on H^{±}, the union of the upper and lower half planes. Then there is a complex curve X^{D}(U)_{ℂ} whose ℂ-valued points are naturally in bijection with the set

D^{×}\(D ⊗ 𝔸)^{×}/U.U_{∞}.

Our aim is to study certain integral models of these curves. To do this is will be convenient to introduce the language of algebraic stacks, because this is the language of many of the theorems in the literature, although ultimately we shall prove that all the stacks we consider are in fact schemes. One reference for algebraic stacks is [CF], and others are mentioned therein.

Let U be a compact open subgroup of ∏_{p} O_{D,p}^{×}, satisfying (i)–(iii) above. There are positive integers M prime to d such that U_{M} ⊆ U, where U_{M} is the kernel of the natural map ∏_{p} O_{D,p}^{×} → → (O_{D} ⊗ ℤ/Mℤ)^{×}. Let M_{U} be the smallest such integer. Let X^{D}(U) be the following category. An object of X^{D}(U) is a false elliptic curve (A/S/ℤ[1/M_{U}d], i_{A}) equipped with a level U structure α ∈ F_{A/S,U}(S), and a morphism from ((A/S, i_{A}), α) to ((B/T, i_{B}), β) in this category is a commutative diagram

\begin{matrix}A & ⟶ & B \\ ↓ & & ↓ \\ S & ⟶ & T \end{matrix}

such that • The induced morphism A → B_{S} = B ×_{T} S is an isomorphism,

• The induced map a : End_{T} (B) → End_{S}(A) satisfies i_{A} = a ∘ i_{B}.

• α is the image of β under the natural map F_{B/S,U}(T) → F_{B_{S}/S,U}(S) = F_{A/S,U}(S). There is a morphism from X^{D}(U) to the category of ℤ[1/M_{U}d]-schemes, sending

((A/S, i_{A}), α) to S.

Theorem 5.3.1. X^{D}(U) is an algebraic stack over ℤ[1/M_{U}d]. Moreover, the morphism X^{D}(U) → ℤ[1/M_{U}d] is proper and smooth of relative dimension 1.

Proof. This is proved in [B], section 14. The representability result is deduced from a theorem of M. Artin.

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We will now use the smallness criterion (iii) on U to show that X^{D}(U) is in fact the algebraic stack associated to a scheme. For this we shall need

Lemma 5.3.2. Let N ≥ 5 be an integer prime to d, and let k be an algebraically closed field of either characteristic 0 or of characteristic p prime to Nd. Let (A/k, i) be a false elliptic curve over k, and let α be a U_{1}(N)-structure on A/k. Then the only automorphism of (A/k, i) which fixes α is the identity.

Proof. Because k is an algebraically closed field, the U_{1}(N)-structure α can be thought

of as an orbit of full level N structures under the action of the group H = \{( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) ∈

GL_{2}(ℤ/Nℤ)}. Let β : (O_{D} ⊗ ℤ/Nℤ)_{k} → A[N] be an element of this orbit. We must show that if θ is an automorphism of (A/k, i) such that θ ∘ β = β ∘ r_{δ} for some δ ∈ H, then θ is the identity. So let θ be such an automorphism.

Let ψ = θ − 1. Then ψ is an endomorphism of (A/k, i). Let G = ker(ψ). Now G is a group scheme of finite type over k, with an action of O_{D}. This action induces an action of O_{D} on G^{0}, the identity component of G, and hence on H = (G^{0})^{red}, the reduced scheme associated to G^{0}. We see that G/H is a finite group scheme over k, and that H is an abelian variety. Moreover, we have an action of O_{D} on H. If H were 1-dimensional, then it would be an elliptic curve and this would be a contradiction, because there are no ring homomorphisms from O_{D} to R if R is End_{k}(E) for some elliptic curve E (To see this, tensor up by ℚ and observe that there are no maps (sending 1 to 1) from D to ℚ, to an imaginary quadratic extension of ℚ, or to a definite quaternion algebra over ℚ.). Hence H is either zero or 2-dimensional. If H is 2-dimensional, then H = G and ψ = 0, which implies θ = 1 and we are finished. So assume for a contradiction that H = 0. Then G is finite and ψ is an isogeny. If ψ^{t} is the transpose of this isogeny, then the composite ψ^{t} ∘ ψ is multiplication by a positive integer n, the false degree of ψ. Moreover, θ is an automorphism and hence θ^{t} ∘ θ is the identity. We have

n = ψ^{t}ψ = (θ − 1)^{t}(θ − 1)
= θ^{t}θ − θ^{t} − θ + 1
= 2 − (θ^{t} + θ).

Hence (θ^{t} + θ) is multiplication by an integer, namely m = 2 − n. We deduce that θ satisfies the equation X^{2} − mX + 1 = 0 in End_{k}(A).

Now θ is an automorphism of the abelian variety A/k, commuting with the action of O_{D}. One can check that this implies that it preserves the canonical principal polarisation of [DT1] Lemma 5. By [Mil 1] proposition 17.5, this implies that θ has finite order in End_{k}(A). Say θ has order f. Then set C = ℤ[θ] ⊆ End_{k}(A), the ring generated by θ in End_{k}(A). Now C ⊗ℚ must be a quotient of ℚ[T]/(T^{f} −1), and the fact that θ^{2}−mθ+1 = 0 and that C ≠ 0 implies that T^{2} − mT + 1 and T^{f} − 1 must have a common factor in ℚ[T]. Hence one of the roots of T^{2} − mT + 1 = 0 is a root of unity and hence |m| ≤ 2. But by definition, m = 2 − n and hence 1 ≤ n ≤ 4 because n is a positive integer.

The k-valued point ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) of (O_{D} ⊗ ℤ/Nℤ)_{k} is mapped to a k-valued point of

A by β. Let P be this k-valued point. We observe that β = θ^{t}θβ = θ^{t}βr_{δ} and hence that θ^{t}β = βr_{δ^{−1}}. Hence mβ = θβ + θ^{t}β = β ∘ (r_{δ} + r_{δ^{−1}}) = β ∘ (r_{δ+δ^{−1}}). Applying

these maps to the k-valued point ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) of (O_{D} ⊗ ℤ/Nℤ)_{k} yields mP = 2P and hence

nP = 0 ∈ A[N](k). Because β is an isomorphism we deduce that n ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) = 0 in

M_{2}(ℤ/Nℤ) and hence that N divides n. This is a contradiction because N ≥ 5.

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Corollary 5.3.3. If U ⊆ ∏_{p} O_{D,p}^{×} is open and compact, and satisfies properties (i)–(iii) above, then X^{D}(U) is a regular scheme, and X^{D}(U) → ℤ[1/M_{U}d] is projective.

Proof. Firstly we show that X^{D}(U) is the stack associated to an algebraic space. To do this, it suffices to show that the objects of the category X^{D}(U) have no non-trivial automorphisms. Assume for a contradiction that A/S is an abelian scheme with a level U structure α, and there is an automorphism θ ≠ 1 preserving α. By [DR] II 1.14 there is a geometric point k → S of S such that θ_{k} : A_{k} → A_{k} is not the identity. But this contradicts Lemma 5.3.2, because U ⊆ U_{1}(N) for some N ≥ 5, and hence X^{D}(U) is an algebraic space.

Now ℤ[1/M_{U}d] is regular, and the morphism X^{D}(U) → ℤ[1/M_{U}d] is smooth of relative dimension 1. Hence by [SGA3] II Proposition 3.1 we see that X^{D}(U) is a regular algebraic space. It follows that X^{D}(U) → ℤ[1/M_{U}d] is quasi-projective, and hence by [Kn] (II.7.6) we see that X^{D}(U) is a scheme. Finally, the map X^{D}(U) → ℤ[1/M_{U}d] is quasiprojective and proper, hence projective by [Kn] II.7.8.

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Finally, we record

Proposition 5.3.4. If U satisfies (i)–(iii) above, then the fibres of X^{D}(U) → ℤ[1/M_{U}d] are geometrically irreducible.

Proof. By the theory over ℂ, the generic fibre of X^{D}(U) is geometrically connected. By Zariski’s connectedness theorem, all the fibres of X^{D}(U) → ℤ[1/M_{U}d] are geometrically connected. By [B] §16, Corollary, they are geometrically unibranch and hence geometrically irreducible.

□

§5.4. New structures at l.

We now briefly analyse the l-torsion in false elliptic curves, where l is a prime not dividing d. Let A/S be a false elliptic curve over a ℤ[1/d]-scheme S. Then the l-divisible group A[l^{∞}] attached to A inherits an action of O_{D} and hence of O_{D} ⊗_{ℤ} ℤ_{l} ≅ M_{2}(ℤ_{l}). If

we set ẽ = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ∈ M_{2}(ℤ_{l}) then A[l^{∞}] splits as ker(1 − ẽ) × ker(ẽ). Moreover, ker(1 − ẽ)

and ker(ẽ) are isomorphic, because 1 − ẽ and ẽ are conjugate in M_{2}(ℤ_{l}). If S = k is an algebraically closed field, then we can completely classify the possibilities for ker (1 − ẽ), as follows.

If the characteristic of k is not equal to l, then A[l^{∞}] ≅ (ℚ_{l}/ℤ_{l})^{4} and hence ker(1−ẽ) ≅ (ℚ_{l}/ℤ_{l})^{2}.

If the characteristic of k is equal to l, then there are two possibilities. A is an abelian variety and it is a classical result that the l-divisible group A[l^{∞}] is isomorphic to (μ_{l^{∞}})^{r} × (ℚ_{l}/ℤ_{l})^{r} × Γ, where Γ/k is a local-local l-divisible group. Because we have seen that A[l^{∞}] ≅ G × G for some l-divisible group G, we deduce that either r = 2 and G = μ_{l^{∞}} ×ℚ_{l}/ℤ_{l}, or r = 0 and G is the unique one-parameter formal Lie group over k of height 2. In particular, in the former case we have A[l^{∞}] ≅ (E[l^{∞}]^{2}) for E/k any ordinary elliptic curve, and in the latter case A[l^{∞}] ≅ (E[l^{∞}]^{2}) for E/k any supersingular elliptic curve. Note that in either case there is an elliptic curve E/k with A[l^{∞}] ≅ (E[l^{∞}]^{2}), a fact which will be very useful to us later, because it will enable us to deduce facts about the local rings of the Shimura curves we are studying from facts about the local rings of modular curves, facts which are already documented in the literature.

If A/k is a false elliptic curve over any field of characteristic l, then we have just seen above that the l-divisible group of A\overline{k} is isomorphic to E[l^{∞}]^{2} for E/\overline{k} some elliptic curve. We shall call A/k ordinary or supersingular depending on whether E/\overline{k} is ordinary or supersingular.

We shall now consider some much more subtle moduli problems. As before, l is a prime not dividing d, and U ⊆ ∏_{p} O_{D,p}^{×} is a compact open subgroup with the properties mentioned above. We have seen that X^{D}(U) is a scheme over ℤ[1/M_{U}d]. We shall now “build” on this scheme by adding some structures at l, analogous to structures already considered on elliptic curves.

Definition 5.4.1. A Γ_{0}(l)-structure on a false elliptic curve A/S is an isogeny of false elliptic curves A/S → B/S of false degree l.

Remark. We have imposed no “cyclicity” condition in this definition, for simplicity. We can do this because l is prime. Compare the definitions of a Γ_{0}(l) structure in [DR] and in [KM], and also [KM] Corollary 6.8.7.

It is sometimes helpful to have an alternative, equivalent definition of a Γ_{0}(l)-structure, which we shall now describe. If A/S is a false elliptic curve, then A[l] is a finite S-group

scheme with an action of O_{D} ⊗ 𝔽_{l} ≅ M_{2}(𝔽_{l}). If e = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ∈ M_{2}(𝔽_{l}), and we write

G_{1} = ker(1 − e : A[l] → A[l]) and G_{2} = ker(e : A[l] → A[l]), then A[l] ≅ G_{1} × G_{2}.

If ψ : A → B is an isogeny of false elliptic curves over S, with false degree l, then ker ψ is a finite locally free subgroup of A[l] of rank l^{2}, which is fixed by the action of M_{2}(𝔽_{l}). Hence ker ψ ≅ K_{1} × K_{2} ⊆ G_{1} × G_{2}, where K_{1} is the kernel of 1 − e : ker ψ → ker ψ and K_{2} is the kernel of e on ker ψ. Note that K_{1} is finite and flat, locally free of rank l, and isomorphic to K_{2}.

Conversely, if K_{1} is an arbitrary finite flat subgroup scheme of G_{1} which is locally free of rank l, then setting K_{2} to be the subgroup scheme of G_{2} which becomes identified with K_{1} under the isomorphism of G_{1} with G_{2}, we see that K_{1} × K_{2} is an M_{2}(𝔽_{l})-invariant subgroup of A[l] of rank l^{2} and ψ : A → A/(K_{1} × K_{2}) is an isogeny of abelian surfaces. Moreover, A/(K_{1} × K_{2}) can be given the structure of a false elliptic curve in such a way that ψ becomes an isogeny of false elliptic curves of false degree l, as follows. If d ∈ O_{D}, then ψ∘i(d) kills K_{1}×K_{2}, and hence must factor through ψ on the right, i.e. must be equal to j(d) ∘ ψ for some map j(d) : A/(K_{1} × K_{2}) → A/(K_{1} × K_{2}). Then j gives A/(K_{1} × K_{2}) the structure of a false elliptic curve.

We deduce that to give a Γ_{0}(l)-structure on A is equivalent to giving a finite flat subgroup K_{1} of ker(1 − e : A[l] → A[l]) which is locally free of rank l.

It is not difficult to see that if l is invertible on S then to give a Γ_{0}(l) structure on

A/S is to give an H-structure on A/S, where H ⊆ \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(𝔽_{l})|c = 0\} and an H-

structure is defined as in §5.2. Hence if l does not divide M_{U}, and if U_{0}(l) is the subgroup of elements of U in the preimage of H under the natural map U → GL_{2}(ℤ/lℤ) then to give a U_{0}(l)-structure on a false elliptic curve over a ℤ[1/M_{U}dl]-scheme is to give both a level U structure and a Γ_{0}(l)-structure.

We now turn to the some other structures, analagous to structures already defined on

elliptic curves. Recall that e = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ∈ M_{2}(𝔽_{l}) and if A/S is a false elliptic curve then

e acts on A[l]. Recall also from [KM] that if K is a finite flat group scheme over S which is locally free of rank n, then a point P ∈ K(S) is called a generator of K if the n points P, 2P, 3P, …, nP form a “full set of sections” for K ([KM] 1.8).

Let A/S be a false elliptic curve, where S is a ℤ[1/d]-scheme as usual. Also recall that l is a prime not dividing d.

A Γ_{1}(l)-structure on A/S is

• π : A → B an isogeny of false elliptic curves, of false degree l,

• P ∈ K_{1}(S) a generator of K_{1} = ker(1 − e : ker π → ker π).

A balanced Γ_{1}(l)-structure on (A/S, i) is the following collection of data:

• π : A → B an isogeny of false elliptic curves, of false degree l,

• π^{t} : B → A the dual isogeny.

• P ∈ K_{1}(S) a generator of K_{1} = ker(1 − e : ker π → ker π),

• P′ ∈ K′_{1}(S) a generator of K′_{1} = ker(1 − e : ker π^{t} → ker π^{t}).

Now assume that S is a ℤ[1/d][ζ_{l}]-scheme, where ζ_{l} is a fixed primitive lth root of unity. We define a canonical balanced Γ_{1}(l)-structure on (A/S, i) to be the following collection of data:

• π : A → B an isogeny of false elliptic curves, of false degree l and kernel K,

• π^{t} : B → A the dual isogeny, with kernel K′,

• P ∈ K_{1}(S) a generator of K_{1} = ker(1 − e : K → K),

• P′ ∈ K′_{1}(S) a generator of K′_{1} = ker(1 − e : K′ → K′),

such that ⟨P, P′⟩_{π} = ζ_{l}, the fixed lth root of unity in R. Here ⟨P, P′⟩_{π} is the canonical pairing between ker π and ker π^{t}.

Key remark. If S is a ℤ[1/d]-scheme, and A/S and B/S are false elliptic curves, such that A[l] ≅ B[l] as O_{D}-group schemes, then there is an induced isomorphism between the set of Γ_{0}(l) (resp. Γ_{1}(l), resp. balanced Γ_{1}(l), resp. canonical balanced Γ_{1}(l) if S/ℤ[1/d][ζ_{l}]) structures on A and on B. Moreover, if E/S is a (genuine) elliptic curve, and A[l] ≅ E[l]^{2} as O_{D}-group schemes, where here O_{D} acts via the natural action of O_{D} ⊗𝔽_{l} ≅ M_{2}(𝔽_{l}) on E[l]^{2}, then we have an induced isomorphism between the set of Γ_{0}(l) (resp. Γ_{1}(l), resp. etc) structures on A, as defined above, and the set of Γ_{0}(l) (resp. Γ_{1}(l), resp. etc) structures on E/S as defined in [KM]. The only structure for which this is not clear given the results of [KM] chapter 1 is the Γ_{0}(l) structure, because of the cyclicity condition in [KM] for Γ_{0}(l) structures. But in the theory of elliptic curves, all l-isogenies are cyclic by [KM] Corollary 6.8.7.

§5.5. Analysis of the new structures.

We now wish to prove many results about these structures at l. All the results here are analogues of results proved in [DR]. Moreover, in many cases the details of the proofs are also very similar.

Let S_{0} be a base scheme, and let F be a contravariant functor from false elliptic curves over S_{0}-schemes to the category of sets. For example, S_{0} = ℤ[1/d] and F is the functor defined on objects by F(A/S) = {isomorphism classes of Γ_{0}(l)-structures}, or S_{0} = ℤ[1/d][ζ_{l}] and F : A/S ↦ {isomorphism classes of canonical balanced Γ_{1}(l)- structures}. We say that F is relatively representable if, given any S/S_{0} and any fixed false elliptic curve A/S, the functor from S-schemes to sets sending T/S to F(A_{T}) is representable. We now prove some relative representability results.

Proposition 5.5.1. If S_{0} = ℤ[1/d], then the functor F is relatively representable for F = Γ_{0}(l), F = Γ_{1}(l) and F =balanced Γ_{1}(l). If S_{0} = ℤ[1/d][ζ_{l}], then the canonical balanced Γ_{1}(l) functor is also relatively representable.

Proof. Let A/S/ℤ[1/d] be a false elliptic curve. The result for F = Γ_{0}(l) is an easy consequence of the theory of the Hilbert scheme. Hence there is S′/S and ϖ : A_{S′} → B a Γ_{0}(l)-structure on A_{S′} which is universal for this functor, i.e. such that every Γ_{0}(l)- structure on every extension A_{T} of A arises as a pullback of this one. Hence it is easy to see that the functor from S-schemes to sets sending T/S to the set of pairs consisting of a Γ_{0}(l)-structure π, and a homomorphism α from ℤ/lℤ to (ker(e : ker π → ker π))(T), is representable by ker(e : ker ϖ → ker ϖ). The relative representability of the Γ_{1}(l)-structure follows from this, because there is a closed subscheme of this scheme which is universal for the statement “α is a generator” by [KM] 1.9.1.

The relative representability of the balanced Γ_{1}(l) functor can be deduced in a similar way.

Finally, if S is a ℤ[1/d][ζ_{l}]-scheme, and A/S is a false elliptic curve, and S′′ represents the functor of balanced Γ_{1}(l) structures on A/S, then there is a closed subscheme of S′′ which is universal for the statement “the structure is canonical”.

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Corollary 5.5.2. If U is compact and open satisfying (i)–(iii) as usual, and l does not divide M_{U}, then the functor from ℤ[1/M_{U}d]-schemes to sets sending S to the set of (isomorphism classes of) false elliptic curves over S equipped with a level U structure and a Γ_{0}(l)-structure is representable by a scheme X^{D}(U, Γ_{0}(l)).

Proof. Apply the relative representability result for Γ_{0}(l) to the universal false elliptic curve over X^{D}(U).

□

Corollary 5.5.3. If U is as above, and l is a prime not dividing M_{U}, then the functor from ℤ[1/M_{U}d][ζ_{l}]-schemes to sets sending S to the set of isomorphism classes of false elliptic curves over S, equipped with a level U structure and a canonical balanced Γ_{1}(l)-structure, is representable by a scheme X^{D}(U, Bal.can Γ_{1}(l))/ℤ[1/M_{U}d][ζ_{l}].

Proof. Apply the relative representability result for canonical balanced Γ_{1}(l) to the universal false elliptic curve over the pullback X^{D}(U)_{ℤ[1/M_{U} d][ζ_{l}]} of X^{D}(U).

□

We now show that these new moduli problems are in fact extensions of naïve moduli problems discussed earlier.

Lemma 5.5.4.

(i) If S/ℤ[1/dl] is a scheme, and A/S is a false elliptic curve, then to give a Γ_{0}(l)-structure on A is to give a naïve U_{0}(l)-structure on A, where U_{0}(l) is the compact open subgroup of

D_{f}^{×} consisting of elements which are congruent to ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) mod l.

(ii) If S/ℤ[1/dl][ζ_{l}] is a scheme, and A/S is a false elliptic curve, then to give a canonical balanced Γ_{1}(l)-structure on A is to give a naïve U_{1}(l)-structure on A, where U_{1}(l) is the compact open subgroup of D_{f}^{×} consisting of elements which are congruent to ( \begin{matrix}a & b \\ 0 & 1 \end{matrix} )

mod l.

Proof.

(i) We can replace S by an étale surjective cover of S, and hence without loss of generality we may assume that A[l] ≅ ((ℤ/lℤ)^{4})_{S}. Then to the full level l structure α : (O_{D} ⊗

ℤ/lℤ)_{S} → A[l] we associate the subgroup α(( \begin{matrix}0 & ∗ \\ 0 & ∗ \end{matrix} )) of A[l]. It is easily checked that this

sets up the required bijection between Γ_{0}(l)-structures and U_{0}(l)-structures. It can be seen similarly that to give a U_{1}(l)-structure on A/S is to give a Γ_{1}(l)-structure

on A/S, the point of exact order l being α(( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} )).

(ii) By the previous remark it suffices to show that every Γ_{1}(l)-structure on A/S can be uniquely extended to a canonical balanced Γ_{1}(l)-structure. We see this as follows. Let π : A → B and P ∈ ker(1 − e : ker π → ker π) be a Γ_{1}(l)-structure on A. Then the fact that the pairing ⟨, ⟩_{π} : ker π × ker π^{t} → μ_{l} is perfect and that P ≠ 0 implies that there exists P_{1} ∈ ker(π^{t})(S) such that ⟨P, P_{1}⟩_{π} = ζ_{l} ∈ μ_{l}(S). Let P′ = eP_{1}. Then ⟨P, P′⟩ = ⟨P, eP_{1}⟩ = ⟨e^{∗}P, P_{1}⟩. But e = e^{∗} and hence e^{∗}P = P and ⟨e^{∗}P, P_{1}⟩ = ⟨P, P_{1}⟩ = ζ_{l} as required.

Hence Γ_{1}(l) structures can be extended. Because l is invertible on S we quickly deduce that they extend uniquely thus: If Q is any other S-valued point of ker(1 − e : ker π^{t} → ker π^{t}) then Q = (P′)^{n} for some n and hence ⟨P, Q⟩ = (ζ_{l})^{n}. Hence ⟨P, Q⟩ = ζ_{l} implies that n is 1 mod l and hence that Q = P′.

□

As in [DR] we use the suffix ^{h} to denote the ordinary (at l) locus of a moduli problem. So for example X^{D}(U, Γ_{0}(l))^{h} is the scheme obtained by removing the supersingular points of X^{D}(U, Γ_{0}(l)) in the fibre at l.

The following theorem summarises the main results which we shall need about the scheme X^{D}(U, Γ_{0}(l)).

Theorem 5.5.5.

(i) The scheme X^{D}(U, Γ_{0}(l)) is smooth over ℤ[1/M_{U}d] away from the supersingular points in characteristic l. (ii) X^{D}(U, Γ_{0}(l))/ℤ[1/M_{U}d] is proper. (iii) The map c : X^{D}(U, Γ_{0}(l)) → X^{D}(U) forgetting the Γ_{0}(l) structure is finite. (iv) c is flat, and X^{D}(U, Γ_{0}(l)) is a regular scheme.

(v) The fibre X^{D}(U, Γ_{0}(l))_{𝔽_{l}} of X^{D}(U, Γ_{0}(l)) is composed of two irreducible components, both isomorphic to X^{D}(U)_{𝔽_{l}}, which cross transversally at the supersingular points.

Remark. All these results can be extracted from the proofs of the analogous results in [DR] V.1 for modular curves. We also use the techniques of [KM] chapter 5.

Proof.

(i) Let A(U) denote the functor from ℤ[1/M_{U}d]-schemes to sets, sending S to the set of false elliptic curves over S equipped with a level U structure and a subgroup scheme which locally for the étale topology is isomorphic to (μ_{l})^{2}. By the theory of the Hilbert scheme this functor is representable by an étale (because formally étale) cover of X^{D}(U). Now define B(U) to be the functor from ℤ[1/M_{U}d]-schemes to sets, sending S to the set of false elliptic curves over S equipped with a level U structure and a subgroup locally isomorphic to (ℤ/lℤ)^{2}. If (A/S, i) is a false elliptic curve with a level U structure α and a subgroup G of A[l] locally isomorphic to (μ_{l})^{2}, then A/G can also be given the structure of a false elliptic curve over S, and the subgroup A[l]/G is isomorphic to (ℤ/lℤ)^{2}. This induces a functor from A(U) to B(U) which is easily checked to be an isomorphism. Hence B(U) is also an étale cover of X^{D}(U). Moreover, X^{D}(U, Γ_{0}(l))^{h} is naturally the union of A(U) and B(U), and hence X^{D}(U, Γ_{0}(l)) is smooth away from the supersingular points. (ii) This follows from the valuative criterion of properness and the fact that false elliptic curves have potentially good reduction (see [B] section 6). One need only check that if A/O is an abelian scheme over a discrete valuation ring, and π is a Γ_{0}(l) structure on the generic fibre of A, that π extends uniquely to a Γ_{0}(l) structure on A. This follows from the fact that A is a Néron model of its generic fibre. (iii) One checks using the calculations on [DR] p239 that the fibres of c are finite, simply by checking the three cases

(a) A/k with k of characteristic not equal to l

(b) char(k) = l and A/k ordinary

(c) char(k) = l and A/k supersingular. Thus c is proper with finite fibres and is hence finite. (iv) Here, rather than modifying proofs in [DR] and [KM] into this setting, we instead actually use results of [KM] to deduce results in our case. In particular, [KM] have to work quite hard to get their Theorem 6.6.1, deducing it from analogous results for Γ_{1}(l) structures. However, we do not have to redo the results of [KM] chapters 5 and 6 in our setting, because we can use [KM] Theorem 6.6.1 itself. Firstly here is a brief summary of what we shall do.

The conditions that we have to check (regularity and flatness) are conditions on the local rings of X^{D}(U) and X^{D}(U, Γ_{0}(l)). Firstly we reduce the problem to having to check these conditions only on the local rings of closed points with residue field of characteristic l. Then we show that, after strict henselisation and completion (which does not affect the properties we need to check), these latter local rings become isomorphic to the local rings of points on modular curves, where flatness and regularity have already been proved in [DR] or [KM]. Hence we deduce flatness and regularity in our case. Now here are the details.

Consider the open subset Z of X^{D}(U) consisting of the points y ∈ X^{D}(U) such that, for all points x of X^{D}(U, Γ_{0}(l)) lying above y, the local ring O_{x} is regular, and flat over O_{y}. We must show that Z = X^{D}(U). Now because X^{D}(U) is of finite type over ℤ[1/M_{U}d], we need only prove that Z contains every closed point of X^{D}(U). For points of residue characteristic not equal to l this is a consequence of the results of §5.3, where a Γ_{0}(l) structure can be interpreted as a naïve U_{0}(l) structure. Hence we are reduced to considering points of residue characteristic l.

Hence we must show that if y ∈ X^{D}(U) is a closed point of residue characteristic l, and x ∈ X^{D}(U, Γ_{0}(l)) lies above it, then the local ring O_{x} is regular, and flat over O_{y}.

Let W be the Witt vectors of \overline{𝔽}_{l}. For a local ring A, let A^{h.s.} denote its strict henselisation, and Â its completion. By [KM] pp133-134, the condition on y that we must show is the condition that the scheme

(X^{D}(U, Γ_{0}(l)) ×_{ℤ[1/M_{U} d]} W) ×_{(X^{D}(U)×W)} Spec(Ô_{X^{D}(U),y}^{h.s.})  (∗)

is a regular scheme, flat over Ô_{X^{D}(U),y}^{h.s.}. Now let A_{0}/\overline{𝔽}_{l} denote the false elliptic curve underlying y, viewed as an \overline{𝔽}_{l}-valued point of X^{D}(U) (after fixing some embedding of the residue field of y into \overline{𝔽}_{l}). By [B] §12 (3) the deformation functor of A_{0} to Artin local W - algebras with residue field \overline{𝔽}_{l} is effectively pro-represented by a false elliptic curve 𝔸/W [[T]]. Now Ô_{X^{D}(U),y}^{h.s.} ≅ W [[T]], and the scheme (∗) above is the scheme relatively representing Γ_{0}(l) structures on 𝔸/W [[T]]. The trick is that a Γ_{0}(l) structure on 𝔸/W [[T]] only depends on the l-divisible group attached to 𝔸/W [[T]] (and in fact only on the l-torsion of this group). This l-divisible group is isomorphic to Γ × Γ for Γ the l-divisible group attached to the universal formal deformation 𝔼/W [[T]] of E_{0}/\overline{𝔽}_{l}, where E_{0}/\overline{𝔽}_{l} is any elliptic curve such that E_{0}[l^{∞}]^{2} ≅ A_{0}[l^{∞}]. Moreover, to give a Γ_{0}(l) structure on 𝔸/W [[T]] is to give a Γ_{0}(l) structure on 𝔼/W [[T]] in the sense of [KM]. Hence the scheme (∗) is actually equal to the scheme denoted P_{𝔼/W [[T]]} in [KM] p134, for P = [l−Isog] = [Γ_{0}(l)] in the notation of [KM]. Now by [KM], Theorem 6.6.1, this scheme is indeed a regular scheme, flat over W [[T]], and hence Z does indeed contain every closed point with residue characteristic l and we are done.

(v) If A/S/𝔽_{l} is a false elliptic curve, then the Frobenius morphism F : A → A^{(l)} is an isogeny of false elliptic curves, of false degree l. Its dual, V : A^{(l)} → A is also of false degree l. The maps

Φ_{1} : (A, ϕ) ↦ (A, ϕ, F)

and

Φ_{2} : (A, ϕ) ↦ (A^{(l)}, F_{∗}ϕ, V)

both induce maps X^{D}(U)_{𝔽_{l}} → X^{D}(U, Γ_{0}(l))_{𝔽_{l}}. Hence we have a map

Φ_{1} ⨿ Φ_{2} : X^{D}(U)_{𝔽_{l}} ⨿ X^{D}(U)_{𝔽_{l}} → X^{D}(U, Γ_{0}(l))_{𝔽_{l}}.

If c : X^{D}(U, Γ_{0}(l)) → X^{D}(U) is the forgetful map, then cΦ_{1} is the identity on X^{D}(U)_{𝔽_{l}}, and cΦ_{2} is the map (A, ϕ) ↦ (A^{(l)}, F_{∗}ϕ).

Define w : X^{D}(U, Γ_{0}(l)) → X^{D}(U, Γ_{0}(l)) thus. If A/S is a false elliptic curve over a ℤ[1/M_{U}d]-scheme, with a level U structure ϕ and an isogeny ψ : A → B of false degree l, then w(A, ϕ, ψ) = (B, ψ_{∗}ϕ, ψ^{t}) where ψ_{∗}ϕ is the push-forward of ϕ by ψ as defined in §5.2. Note that Φ_{2} = wΦ_{1}.

The two maps Φ_{1} and Φ_{2} are closed immersions, because they have left inverses. Moreover, the only subgroups of (μ_{l} × ℤ/lℤ)/\overline{𝔽}_{l} are μ_{l} and ℤ/lℤ, and hence

Φ_{1} ⨿ Φ_{2} : X^{D}(U)_{𝔽_{l}}^{h} ⨿ X^{D}(U)_{𝔽_{l}}^{h} → X^{D}(U, Γ_{0}(l))_{𝔽_{l}}^{h}

is an isomorphism. Hence the images of Φ_{1} and Φ_{2} are the two irreducible (by Proposition 5.3.4) components of X^{D}(U, Γ_{0}(l))_{𝔽_{l}}. Finally we must show that the intersection of these two components at the supersingular points is transversal. One way of seeing this is as follows: we have a map c : X^{D}(U, Γ_{0}(l)) → X^{D}(U) forgetting the Γ_{0}(l) structure. The map cw is hence also a map from X^{D}(U, Γ_{0}(l)) to X^{D}(U). Now one sees that the tangent vectors at a supersingular point in characteristic l are distinct, because one is killed by d(c) and not by d(wc), whereas the other is killed by d(wc) and not by d(c).

□

Before we embark on analysing the structure of X^{D}(U, Bal.can Γ_{1}(l)) we have to define one final structure, which only makes sense for false elliptic curves over 𝔽_{l}-schemes.

If S is a scheme over 𝔽_{l}, then the absolute Frobenius map F_{abs} : S → S gives us by pullback a diagram

A^{(l)} ⟶ A
↓ ↓
S ⟶^{F_{abs}} S

and hence in the usual way an S-morphism F : A → A^{(l)}. The endomorphisms i(x) of A, for x ∈ D, induce endomorphisms of A^{(l)}, and these endomorphisms define the structure of a false elliptic curve on A^{(l)}, with respect to which F becomes an isogeny of false elliptic curves. The isogeny F has false degree l, and the isogeny dual to F is denoted V. We have that ker(V) splits up as H_{1} ×_{S} H_{2}, with H_{1} = ker(1 − e : ker V → ker V) and H_{2} = ker(e : ker V → ker V) as usual. An Igusa structure on (A/S/𝔽_{l}, i) is a point P ∈ H_{1}(S) which is a generator of H_{1}. Note that a naïve level M structure ϕ : (O_{D} ⊗ ℤ/Mℤ)_{S} → A[M] on A induces a naïve level M structure ϕ^{(l)} = F ∘ ϕ on A^{(l)}, analogously to Construction 3.1.1, and hence we see that naïve level U structures can be pushed forward also.

Proposition 5.5.6. Igusa structures are relatively representable, i.e. if A/S/𝔽_{l} is a false elliptic curve, then the functor from S-schemes to sets sending T/S to the set of isomorphism classes of Igusa structures on A_{T} is representable.

Proof. Indeed, the functor is represented by a closed subscheme of ker(1 − e : ker V → ker V) where V is the Verschiebung on A/S.

□

As a corollary we see that if l ∤ M_{U} then there is a scheme X^{D}(U, Ig(l)) classifying isomorphism classes of false elliptic curves over 𝔽_{l}-schemes equipped with a naïve level U structure and an Igusa structure at l. Analogously to the classical case, X^{D}(U, Ig(l)) → X^{D}(U)_{𝔽_{l}} is étale over the ordinary locus and totally ramified over the supersingular points. As a corollary, one deduces that X^{D}(U, Ig(l)) is connected. Finally X^{D}(U, Ig(l)) is regular—we can see this by reducing to the classical case as in Theorem 5.5.5 (iv)—and hence irreducible.

The main theorem of this section is the following. Recall that we have shown in Corollary 5.5.3 that if U is a compact open subgroup of ∏_{p} O_{D,p}^{×} satisfying the usual conditions (i)–(iii), then the functor sending an ℤ[1/M_{U}d][ζ_{l}]-scheme S to the set of false elliptic curves (A/S, i) equipped with a naïve level U structure and a canonical balanced Γ_{1}(l)-structure is representable by a scheme X^{D}(U, Bal.can Γ_{1}(l))/ℤ[1/M_{U}d][ζ_{l}].

Theorem 5.5.7.

(i) X^{D}(U, Bal.can Γ_{1}(l))^{h} → ℤ[1/M_{U}d][ζ_{l}] is smooth of relative dimension 1. (ii) X^{D}(U, Bal.can Γ_{1}(l)) → ℤ[1/M_{U}d][ζ_{l}] is proper.

(iii) The natural map X^{D}(U, Bal.can Γ_{1}(l)) → X^{D}(U, Γ_{0}(l))_{ℤ[1/M_{U} d][ζ_{l}]} is finite, and hence X^{D}(U, Bal.can Γ_{1}(l)) → X^{D}(U)_{ℤ[1/M_{U} d][ζ_{l}]} is also finite. (iv) The scheme X^{D}(U, Bal.can Γ_{1}(l)) is regular. If X^{D}(U, Bal.can Γ_{1}(l))_{ℤ_{l}[ζ_{l}]} denotes the pullback X^{D}(U, Bal.can Γ_{1}(l)) ×_{ℤ[1/M_{U} d]} ℤ_{l}[ζ_{l}] then X^{D}(U, Bal.can Γ_{1}(l))_{ℤ_{l}[ζ_{l}]} → ℤ_{l}[ζ_{l}] is flat.

(v) The fibre X^{D}(U, Bal.can Γ_{1}(l))_{𝔽_{l}} is composed of two irreducible components, both isomorphic to X^{D}(U, Ig(l)), and which cross transversally at the supersingular points. (vi) The pullback X^{D}(U, Bal.can Γ_{1}(l))_{ℤ_{l}[ζ_{l}]} of the scheme X^{D}(U, Bal.can Γ_{1}(l)) to ℤ_{l}[ζ_{l}] is an amenable scheme, in the sense of Definition 2.4.2.

Proof. Many of the ideas here are similar to those in the proof of Theorem 5.5.5.

(i) It suffices to prove that X^{D}(U, Bal.can Γ_{1}(l))^{h} → X^{D}(U)^{h} is étale. Recall A(U) and B(U) defined in the course of the proof of Theorem 5.5.5. Let U(U) (resp. V(U)) be the functor on ℤ[1/M_{U}d][ζ_{l}]-schemes sending a scheme S to the set of isomorphism classes of false elliptic curves A/S equipped with a level U structure and a canonical balanced Γ_{1}(l)- structure (π, π^{t}, P, P′) such that ker π is locally isomorphic to μ_{l}^{2} (resp. (ℤ/lℤ)^{2}). Then the natural map U(U) → A(U)_{ℤ[1/M_{U} d][ζ_{l}]} (resp. V(U) → B(U)_{ℤ[1/M_{U} d][ζ_{l}]}) is finite and étale, and X^{D}(U, Bal.can Γ_{1}(l)) is the union of U(U) and V(U), glued over U(U)[1/l] ≅ V(U)[1/l] where ℤ/lℤ and μ_{l} become canonically isomorphic via the map 1 ↦ ζ_{l}. (ii) Again this follows from the valuative criterion of properness, once one checks that canonical balanced Γ_{1}(l) structures extend uniquely from the generic fibre to the whole of a discrete valuation ring, after base extension if necessary. More precisely, the theorem of potentially good reduction in [B] section 6 reduces us to showing the following: if K is a field and a ℤ[1/d][ζ_{l}]-algebra, and K is equipped with a discrete valuation, O_{K} is the integers of K, and A/O_{K} is a false elliptic curve, then a canonical balanced Γ_{1}(l)- structure on A_{K} extends uniquely to a canonical balanced Γ_{1}(l) structure on A. That the structure extends uniquely to a balanced Γ_{1}(l)-structure is immediate from the Néron property of A/O_{K}. Now one observes that the pairings defined in [O] are compatible with base extension, and hence that the extended structure is still canonical. (iii) It is an easy check that X^{D}(U, Bal.can Γ_{1}(l)) → X^{D}(U, Γ_{0}(l))_{ℤ[1/M_{U} d][ζ_{l}]} is proper with finite fibres and is hence finite. (iv) As in Theorem 5.5.5 we can read this off from the corresponding results on modular curves. The regularity of X^{D}(U, Bal.can Γ_{1}(l)) can be deduced from the regularity of the classical balanced Γ_{1}(l) structure, and the regularity of this latter structure is proved in [KM] Chapter 5. Now all the local rings of X^{D}(U, Bal.can Γ_{1}(l))_{ℤ_{l}[ζ_{l}]} are integral domains in which l is not a zero divisor, and hence they are all torsion-free ℤ_{l}[ζ_{l}]-modules and are hence flat ℤ_{l}[ζ_{l}]-modules. Hence X^{D}(U, Bal.can Γ_{1}(l))_{ℤ_{l}[ζ_{l}]} → ℤ_{l}[ζ_{l}] is flat.

(v) The generic fibre of X^{D}(U, Bal.can Γ_{1}(l)) is isomorphic to X^{D}(U ∩ U_{0}(l))_{ℚ} × ℚ(ζ_{l}), where X^{D}(U ∩ U_{0}(l))_{ℚ} is the canonical model defined by Shimura. We can identify the special fibre at l of X^{D}(U, Bal.can Γ_{1}(l)) with the union of two copies of X^{D}(U, Ig(l))_{𝔽_{l}} by giving two morphisms from X^{D}(U, Ig(l))_{𝔽_{l}} to X^{D}(U, Bal.can Γ_{1}(l))_{𝔽_{l}} in the following way. The morphisms are induced by the two morphisms of moduli problems

j_{1} : (A/S, i, ϕ, P ∈ A^{(l)}(S)) ↦ (A/S, i, ϕ, F, V, 0, P)

and

j_{2} : (A/S, i, ϕ, P ∈ A^{(l)}(S)) ↦ (A^{(l)}/S, i, F_{∗}ϕ, V, F, P, 0).

Here ϕ is a naïve level U structure on A/S. Note that

j_{1} ⨿ j_{2} : X^{D}(U, Ig(l)) ⨿ X^{D}(U, Ig(l)) → X^{D}(U, Bal.can Γ_{1}(l))_{𝔽_{l}}

is surjective on \overline{𝔽}_{l}-valued points, by checking the two cases. If A/\overline{𝔽}_{l} is an ordinary false elliptic curve, then the finite group scheme A[l] is isomorphic to (ℤ/lℤ × μ_{l})^{2} and hence at least one of P and P′ must be the zero section, generating ker(1 − e : ker F → ker F). If A/\overline{𝔽}_{l} is supersingular, then F = V and P = P′ = 0 is the only canonical balanced Γ_{1}(l)- structure on A and we see that this structure is in the image of both j_{1} and j_{2}. Moreover, j_{1} and j_{2} are certainly monomorphisms by their definitions, and hence closed immersions because they are proper, being X^{D}(U)-maps between finite flat X^{D}(U)_{𝔽_{l}}-schemes, after choosing appropriate maps down to X^{D}(U). We deduce that the special fibre at l of X^{D}(U, Bal.can Γ_{1}(l)) has two irreducible components, which intersect at the supersingular points. One can again check that the intersection at the supersingular points is transversal by using the analogous result for the modular curve case: if A/k is a supersingular false elliptic curve over an algebraically closed field of characteristic l, then the spectrum of the complete local ring of X^{D}(U, Bal.can Γ_{1}(l))_{𝔽_{l}} at A/k is just the scheme relatively representing canonical balanced Γ_{1}(l) structures on 𝔸/k[[T]], where 𝔸/k[[T]] is the universal deformation of A/k to a complete local k-algebra with residue field k. However, this scheme is equal to the scheme relatively representing canonical balanced Γ_{1}(l) structures on 𝔼/k[[T]] where 𝔼/k[[T]] is the universal deformation of any supersingular elliptic curve E/k to a complete local k-algebra with residue field k, and this latter representing scheme is the spectrum of the completed local ring of X(Γ_{1}(N), Bal.can Γ_{1}(l))_{𝔽_{l}} for any N ≥ 5 prime to l. Hence the fact that X(Γ_{1}(N), Bal.can Γ_{1}(l))_{𝔽_{l}} is the union of two Igusa curves crossing transversally at the supersingular points implies that the crossings in X^{D}(U, Bal.can Γ_{1}(l))_{𝔽_{l}} are transversal too. (vi) This is immediate from (i)–(v).

□

Note the similarity between this theorem and theorem V 2.12 of [DR]. Proofs of analogous results for totally real fields other than ℚ have been given by Jarvis in [J] and by Carayol in [C1].

If we set U = U_{1}(N) for some N ≥ 5 then X^{D}(U, Bal.can Γ_{1}(l)) is the analogue of X_{1}(Nl)_{R}. We have seen that these two curves have very similar geometric properties. We shall prove other more arithmetic results about X^{D}(U, Bal.can Γ_{1}(l)) in the following sections, results which are analogous to the arithmetic properties which X_{1}(Nl) has.

§5.6. Results about these curves.

To emphasise the analogy between what we have been doing in the previous sections and the analysis of modular curves in chapters 3 and 4, we shall introduce some more notation.

As always, N ≥ 5 is a positive integer, l is a prime not dividing N, and Nl and d are coprime. Let p be a prime not dividing Nld. We shall now revert to the notation analogous to that used in chapters 3 and 4, by introducing the following definitions.

X_{1}^{D}(N)_{ℤ_{l}} := X^{D}(U_{1}(N))_{ℤ_{l}},
X_{1}^{D}(Nl)_{ℤ_{l}[ζ_{l}]} := X^{D}(U_{1}(N), Bal.can Γ_{1}(l))_{ℤ_{l}[ζ_{l}]},
X_{1}^{D}(N; p)_{ℤ_{l}} := X^{D}(U_{1}(N) ∩ U_{0}(p))_{ℤ_{l}},
X_{1}^{D}(Nl; p)_{ℤ_{l}[ζ_{l}]} := X^{D}(U_{1}(N) ∩ U_{0}(p), Bal.can Γ_{1}(l))_{ℤ_{l}[ζ_{l}]},
I_{1}^{D}(N)_{𝔽_{l}} := X^{D}(U_{1}(N), Ig(l))_{𝔽_{l}},
I_{1}^{D}(N; p)_{𝔽_{l}} := X^{D}(U_{1}(N) ∩ U_{0}(p), Ig(l))_{𝔽_{l}}.

We now proceed to prove various results about these curves, analogous to the case of modular curves covered in Chapters 3 and 4. As in Chapter 3, we shall frequently omit the suffix ℤ_{l}[ζ_{l}] on Shimura curves and the suffix 𝔽_{l} on Igusa curves. Moreover, we shall frequently think of U_{1}(N)-structures as e_{N}-invariant points of exact order N (where

e_{N} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ∈ O_{D} ⊗ ℤ/Nℤ ≅ M_{2}(ℤ/Nℤ)) and U_{0}(p)-structures as isogenies of false

degree p.

There is one aspect of the theory of these Shimura curves where the analogy with modular curves is less than perfect. There is no need for a theory of compactification in this setting, because the Shimura curves over ℂ arising from indefinite non-split quaternion algebras are already compact Riemann surfaces. Hence there are no cusps in this setting. Whilst saving us the details of compactification, we do lose out in that there is no obvious notion of a q-expansion. Hence we have to do more work to prove the analogue of Lemma 4.2.3. Moreover, this latter result is not in fact quite true in this setting, due essentially to the fact that the analogue of the section a of H^{0}(I_{1}(N)_{𝔽_{l}}, ω) which did not affect our calculations in the modular curve case because it did not vanish at the cusps, does constitute an obstruction in this setting.

We shall explain this more explicitly. The natural map π : I_{1}^{D}(N)_{𝔽_{l}} → X_{1}^{D}(N)_{𝔽_{l}} corresponds to the morphism forgetting the Igusa structure of moduli problems. Note that the projection π : I_{1}^{D}(N)_{𝔽_{l}} → X_{1}^{D}(N)_{𝔽_{l}} is étale away from the supersingular points, and is totally ramified of degree l − 1 at the supersingular points. In [DT1] §4 an invertible sheaf ω = ω_{X_{1}^{D}(N)} is defined on X_{1}^{D}(N), analogous to the sheaf ω_{X_{1}(N)} on X_{1}(N). Define ω_{I_{1}^{D}(N)} to be π^{∗}ω_{X_{1}^{D}(N)}. Let ss denote the divisor of supersingular points on I_{1}^{D}(N), and let s be the degree of this divisor, so s is the number of points in I_{1}^{D}(N)(\overline{𝔽}_{l}) (or equivalently in X_{1}^{D}(N)(\overline{𝔽}_{l})) corresponding to supersingular false elliptic curves over \overline{𝔽}_{l}.

Proposition 5.6.1. The invertible sheaf ω_{I_{1}^{D}(N)} on I_{1}^{D}(N) has degree s. We have

Ω_{I_{1}^{D}(N)}^{1} ≅ ω_{I_{1}^{D}(N)}^{⊗2}((l − 2)ss).

Proof. From [DT1] Lemma 6 we have 2s = (l − 1) deg Ω_{X_{1}^{D}(N)/𝔽_{l}}, and from [DT1] Lemma 7 we deduce ω_{X_{1}^{D}(N)}^{⊗2} ≅ Ω_{X_{1}^{D}(N)/𝔽_{l}}. Hence

\begin{aligned}2s & = (l − 1) deg Ω_{X_{1}^{D}(N)} \\ & = (l − 1)(2 deg ω_{X_{1}^{D}(N)}) \end{aligned}

and so deg ω_{I_{1}^{D}(N)} = (l − 1) deg ω_{X_{1}^{D}(N)} = s.

Now by the Riemann-Hurwitz formula, we have

Ω_{I_{1}^{D}(N)} ≅ π^{∗}Ω_{X_{1}^{D}(N)}((l − 2)ss)
≅ π^{∗}(ω_{X_{1}^{D}(N)}^{⊗2})((l − 2)ss)
≅ ω_{I_{1}^{D}(N)}^{⊗2}((l − 2)ss).
□

There is a theory of Hecke operators for these Shimura curves, analogous to the classical theory. We shall say more about these Hecke operators later, but it is convenient at this stage to mention that (ℤ/lℤ)^{×} acts naturally on I_{1}^{D}(N) because it acts naturally on Igusa structures, via d ∈ (ℤ/lℤ)^{×} sending an Igusa structure P to dP. We denote this action as usual by ⟨d⟩_{l}. This action, as in the modular curve case, induces an action of (ℤ/lℤ)^{×} on H^{0}(I_{1}^{D}(N), ω_{I_{1}^{D}(N)}). We shall write ⟨d⟩_{l}^{∗} for the action of d ∈ (ℤ/lℤ)^{×} on H^{0}(I_{1}^{D}(N), ω_{I_{1}^{D}(N)}).

Theorem 5.6.2. There is a non-zero global section a of ω_{I_{1}^{D}(N)} with simple zeros at precisely the supersingular points. We have ⟨d⟩_{l}^{∗}a = da for all d ∈ (ℤ/lℤ)^{×}.

Proof. We define a analogously to [KM] Section 12.8. Firstly we construct, given a false elliptic curve (A/S, i) over S/𝔽_{l} with an Igusa structure P ∈ ker(V |_{A^{(l)}})(S), an element of H^{0}(A, Ω_{A/S}), thus. The point P gives rise, by duality, to a map

ϕ_{P} : ker(F |_{A}) → 𝔾_{m}.

The invariant differential dX/X on 𝔾_{m} gives rise by pullback to an invariant differential ϕ_{P}^{∗} (dX/X) on ker(F), which is the reduction of a unique invariant differential ν on A. This construction gives us, for any (A, i) over S/𝔽_{l} with an Igusa structure P, an element of H^{0}(A, Ω_{A/S}) = H^{0}(S, π_{∗}Ω_{A/S}), and applying e to this section gives us the section of H^{0}(S, ω) we require. If S = I_{1}^{D}(N) with its universal false elliptic curve (A, i, ϕ, P), then this construction defines the global section a which we seek. Note that if A/\overline{𝔽}_{l} is supersingular then P = 0 and hence the map ϕ_{P} is 0, so a vanishes at every supersingular point. On the other hand, if A/\overline{𝔽}_{l} is ordinary, then the construction gives us a non-zero element of H^{0}(S, ω_{A/S}) and hence a is not the zero section of ω_{I_{1}^{D}(N)}.

To show that a has a simple zero at each supersingular point, we use a counting argument. We have shown that a has zeros at every supersingular point, and by Proposition 5.6.1 the degree of ω_{I_{1}^{D}(N)} is equal to s, the number of supersingular points. Hence a must have a simple zero at each supersingular point.

If d ∈ (ℤ/lℤ)^{×} then the endomorphism ⟨d⟩ sends (A, i, P) to (A, i, dP). The morphism ϕ_{dP} is equal to dϕ_{P} . Hence the section of Ω_{A/S} arising from dP is d times the section arising from P. We deduce that ⟨d⟩^{∗}a = da.

□

We have seen in Proposition 5.6.1 that Ω_{I_{1}^{D}(N)/𝔽_{l}} ≅ ω_{I_{1}^{D}(N)/𝔽_{l}}^{⊗2}((l − 2)ss). We can

deduce from this that the element a^{⊗2} of H^{0}(I_{1}^{D}(N), ω_{I_{1}^{D}(N)}^{⊗2}), which has zeros of order 2 at each supersingular point, gives rise to a holomorphic differential, which we shall also call a^{⊗2}, on I_{1}^{D}(N), with zeros of order l at each supersingular point, and no other zeros. Note that ⟨d⟩_{l}^{∗}a^{⊗2} = d^{2}a^{⊗2}.

We now give a brief description of the maps 1, η, 1_{i} and η_{i} in this setting, which is totally analogous to the modular curve case. Firstly, we must make some remarks about our structure at p. Our definition of a naïve U_{0}(p)-structure above is equivalent to the usual definition of an isogeny of false degree p. Hereafter we shall regard a U_{0}(p)-structure on A as an isogeny ψ : A → A_{1}. Again we shall write R for ℤ_{l}[ζ_{l}].

Say (A, i, ϕ, ψ, π, π^{t}, P, P′) is a false elliptic curve (A, i) over a scheme S/R, equipped with a U_{1}(N)-structure ϕ (now thought of as a ℤ/Nℤ-structure ϕ : ℤ/Nℤ → ker(1 − e_{N} :

A[N] → A[N])(S), where e_{N} is congruent to ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) mod N), a U_{0}(p)-structure ψ (now

thought of as an isogeny ψ : A → A_{1} of false degree p and commuting with the actions of O_{D}), and a canonical balanced Γ_{1}(l)-structure (π, π^{t}, P, P′). The functor forgetting the Γ_{0}(p)-structure defines a morphism

1 : X_{1}^{D}(Nl; p)_{R} → X_{1}^{D}(Nl)_{R}.

We now briefly describe the analogue of Construction 3.1.1 in this setting. Say ψ : (A, i) → (A_{1}, i_{1}) is an isogeny of false degree p. If ϕ is a full level N structure on A, then ϕ_{1} = ψ∘ϕ is a full level N structure on A_{1}. Hence a U_{1}(N)-structure on (A, i) induces a U_{1}(N)-structure on A_{1}.

Now ψ induces an isomorphism A[l] ≅ A_{1}[l] and it is not difficult to construct, from a canonical balanced Γ_{1}(l)-structure (π, π^{t}, P, P′) on (A, i), a canonical balanced Γ_{1}(l)- structure (π_{1}, π_{1}^{t}, P_{1}, P′_{1}) on (A_{1}, i_{1}), such that P_{1} = ψ(P).

These two constructions of structures on A_{1} give rise to a morphism of moduli problems sending (A, i, ψ, ϕ, π, π^{t}, P, P′) to (A_{1}, i_{1}, ϕ_{1}, π_{1}, π_{1}^{t}, P_{1}, P′_{1}), and hence to a morphism of schemes

η : X_{1}^{D}(Nl; p)_{R} → X_{1}^{D}(Nl)_{R}.

The morphisms j_{1} and j_{2} in the proof of Theorem 5.5.7 decompose the special fibre of X_{1}^{D}(Nl)_{R} into two irreducible curves, and we shall write Y_{i} for the image of j_{i}. Then j_{i} can be thought of as an isomorphism of I_{1}^{D}(N)_{𝔽_{l}} with Y_{i}. As in the classical case, we define an action of (ℤ/Nℤ)^{×} and of (ℤ/lℤ)^{×} on X_{1}^{D}(Nl)_{R} and this induces actions on both Y_{1} and Y_{2}. Again we see that j_{2} commutes with these actions, but j_{1} inverts the action of (ℤ/lℤ)^{×}.

The analogues of j_{1} and j_{2} for the curve X_{1}^{D}(Nl; p)_{R} decompose the special fibre of this curve as the union of two curves X_{1} and X_{2}, both isomorphic to I_{1}^{D}(N; p)_{𝔽_{l}}, and it can be easily checked analogously to the modular curve case that both 1 and η restricted to the special fibre of X_{1}(Nl; p)_{R} send the curve X_{i} to the curve Y_{i} for 1 ≤ i ≤ 2. After identifying X_{i} and Y_{i} with Igusa curves, we can realise the maps 1_{i} and η_{i}, the restrictions of 1 and η to X_{i}, as morphisms of moduli problems, and it is easily checked that these morphisms are analogous to the morphisms of moduli problems occurring in Lemma 4.2.1. For example,

η_{1} : I_{1}^{D}(N; p) → I_{1}^{D}(N)

is the map sending (A/S/𝔽_{l}, i, ϕ, P, ψ) to (A_{1}/S/𝔽_{l}, i_{1}, ψ ∘ ϕ, nψ^{(l)}(P)). We can now state the analogue of Lemma 4.2.3.

Lemma 5.6.3. If (ν_{1}, ν_{2}) is an element of the kernel of

1_{i}^{∗} + η_{i}^{∗} : H^{0}(Y_{i}, Ω_{Y_{i}/𝔽_{l}})^{2} → H^{0}(X_{i}, Ω_{X_{i}/𝔽_{l}}),

then ν_{1} and ν_{2} are both multiples of a^{⊗2}, after Y_{i} has been identified with I_{1}^{D}(N) via j_{i}.

Proof. We define an equivalence relation on the \overline{𝔽}_{l}-valued points of Y_{i} by identifying it with I_{1}^{D}(N)_{𝔽_{l}} and using the analogue of the equivalence relation used in Lemmas 8 and 9 of [DT1]. Explicitly, if τ : (A, i) → (B, j) is an isogeny of false elliptic curves, of false degree p^{2r}, then we have (A, i, p^{r}ϕ, p^{r}P) ∼ (B, j, τ ∘ ϕ, τ^{(l)}(P)). Note that this does not look quite like the equivalence relation of [DT1], but this is because our structures at N are maps in the opposite direction to [DT1].

The key property that this equivalence relation has is that if (ν_{1}, ν_{2}) is in the kernel of 1_{i}^{∗} + η_{i}^{∗} and ν_{2} ≠ 0, then the set of points in I_{1}^{D}(N)(\overline{𝔽}_{l}) where ν_{2} vanishes is a union of equivalence classes. We recall from [DT1] that if (A/\overline{𝔽}_{l}, i, ϕ, P) is an ordinary false elliptic curve with added structure, then its equivalence class is infinite. We can deduce this from [DT1] Lemma 8 by noting that if (A_{1}, i_{1}, ϕ_{1}, P_{1}) ∼ (A_{2}, i_{2}, ϕ_{2}, P_{2}) with respect to the equivalence relation defined here, then (A_{1}, i_{1}, ϕ_{1}) ∼ (A_{2}, i_{2}, ϕ_{2}) with respect to the equivalence relation of [DT1]. Hence we deduce that ν_{2} has no zeros at any point classifying an ordinary false elliptic curve.

Note that if f : X → Y is an unramified morphism of curves sending x to y, then ord_{y}(ω) = ord_{x}(f^{∗}ω) for any non-zero differential ω. By the analogue of [DT1] Lemma 9 (see also [DT2], footnote on page 268), remembering that any supersingular false elliptic curve over \overline{𝔽}_{l} only has one Igusa structure, namely P = 0, we see that all the equivalence classes of supersingular false elliptic curves are Gal(\overline{𝔽}_{l}/𝔽_{l})-conjugate. We deduce that if ν_{2} ≠ 0 then it must have a zero of the same order at every supersingular point. We see that the degree of Ω_{I_{1}^{D}(N)_{𝔽_{l}}}^{1} is ls, by Proposition 5.6.1. Now because ν_{2} is a global section of this sheaf which vanishes at every supersingular point to the same order, we see that ν_{2} must vanish to order l at each supersingular point. Hence, dividing by a^{⊗2} gives an element of H^{0}(I_{1}^{D}(N)_{𝔽_{l}}, O) with no zeros, i.e. a constant. Hence ν_{2} is a constant multiple of a^{⊗2}. A similar argument shows that ν_{1} is also a multiple of a^{⊗2}.

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Corollary 5.6.4. If j ≢ ±2 mod (l − 1) then the maps

1_{i}^{∗} + η_{i}^{∗} : H^{0}(Y_{i}, Ω_{Y_{i}/𝔽_{l}})^{2}(j) → H^{0}(X_{i}, Ω_{X_{i}/𝔽_{l}})(j)

are injective.

Proof. This follows immediately from the fact that action of (ℤ/lℤ)^{×} on the differential a^{⊗2} is given by ⟨d⟩_{l}a^{⊗2} = d^{2}a^{⊗2}, and the fact that the isomorphisms j_{i} : I_{1}^{D}(N)_{𝔽_{l}} → Y_{i} either preserve or invert the action of (ℤ/lℤ)^{×}.

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Remark. Because the kernel of 1_{i}^{∗} + η_{i}^{∗} is essentially evaluated in Lemma 5.6.3, it does not seem unreasonable to expect our methods to give some information about the cases j ≡ ±2 mod (l − 1), which correspond to the cases of weight 4 or l − 1. However, because weights k = 4 and k = l − 1 are treated in [DT1] and [DT2] respectively, and our main contribution is the case k = l, we shall merely omit these cases in what follows.

Because the curves X_{1}^{D}(Nl)_{R} and X_{1}^{D}(Nl; p)_{R} are amenable, we can deduce from Theorem 2.4.7 a result about the kernel of a map between the Néron models of the Jacobians of the generic fibres of these curves. More explicitly, if we set X = X_{1}^{D}(Nl; p)_{R} and Y = X_{1}^{D}(Nl)_{R}, then the two maps 1^{∗} and η^{∗} from Jac(Y_{ℚ_{l}(ζ_{l})}) to Jac(X_{ℚ_{l}(ζ_{l})}) induce two maps between the Néron models J_{Y} → J_{X} and hence two maps between the identity components of these schemes. If K is the kernel of 1^{∗} + η^{∗} on these identity components, then we deduce as a corollary of Corollary 5.6.4, using Theorem 2.4.7, the following.

Corollary 5.6.5. The finite group scheme K[l](j) has multiplicative special fibre, if j ≢ ±2 mod (l − 1).

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Now we consider the analogue of Theorem 3.6.3. Firstly, the briefest of words about Hecke operators.

The curve X_{1}^{D}(N) comes equipped with correspondences T_{r} and S_{r} for primes r not dividing Nld, and diamond operators ⟨d⟩_{N}, defined analogously to the correspondences on X_{1}(N). The curve I_{1}^{D}(N)_{𝔽_{l}} also comes equipped with correspondences with these names, and in addition the diamond operators ⟨d⟩_{l}. These correspondences induce an action of 𝕋^{(Nld)} on H^{0}(X_{1}^{D}(N), ω^{⊗k}), H^{0}(I_{1}^{D}(N), Ω_{I_{1}^{D}(N)/𝔽_{l}}) and so on. Similarly, there are correspondences T_{r} and S_{r} on the curve X_{1}^{D}(Nl)_{ℚ_{l}(ζ_{l})}, for all primes r not dividing Nld. If r divides Nl, then there is a correspondence U_{r} on X_{1}^{D}(Nl)_{ℚ_{l}(ζ_{l})}. There are natural actions of (ℤ/Nℤ)^{×} and (ℤ/lℤ)^{×} on X_{1}^{D}(Nl)_{ℚ_{l}(ζ_{l})} and hence we can define H to be the subalgebra of End(Jac(X_{1}^{D}(Nl)_{ℚ_{l}(ζ_{l})})) generated by the endomorphisms corresponding to these correspondences.

We now turn to the analogue of [KM] Theorem 12.8.8, and our Proposition 3.5.1. Theorem 5.6.6. If 1 ≤ j ≤ l − 1, and k = l + 1 − j, then there is an isomorphism

H^{0}(X_{1}^{D}(N)_{𝔽_{l}}, ω^{⊗k}) ≅ H^{0}(I_{1}^{D}(N), Ω_{I_{1}^{D}(N)/𝔽_{l}})(j).

Proof. We simply mimic the proof given for modular curves in [G]. We will give the details. If f ∈ H^{0}(X_{1}^{D}(N)_{𝔽_{l}}, ω^{⊗k}) and π : I_{1}^{D}(N)_{𝔽_{l}} → X_{1}^{D}(N)_{𝔽_{l}} is the projection, then π^{∗}f ∈ H^{0}(I_{1}^{D}(N)_{𝔽_{l}}, ω^{⊗k}) and hence π^{∗}f/a^{k−2} is a meromorphic section of ω_{I_{1}^{D}(N)_{𝔽_{l}}}^{⊗2} which is regular away from the supersingular points, and has poles of order at most k − 2 ≤ l − 2 at the supersingular points. Hence because Ω_{I_{1}^{D}(N)} ≅ ω_{I_{1}^{D}(N)}^{⊗2}((l − 2)ss) we see

that π^{∗}f/a^{k−2} can be thought of as an element of H^{0}(I_{1}^{D}(N), Ω_{I_{1}^{D}(N)})(j). Conversely, if ν ∈ H^{0}(I_{1}^{D}(N), Ω_{I_{1}^{D}(N)})(j), then ν can be thought of as a meromorphic section of ω_{I_{1}^{D}(N)}^{⊗2}

with poles of order at most l−2 at the supersingular points, and hence a^{k−2}ν is a section of ω_{I_{1}^{D}(N)}^{⊗k}, invariant under the action of (ℤ/lℤ)^{×}, and with poles of order at most l − k at the

supersingular points. Hence a^{k−2}ν descends to X_{1}^{D}(N) and in particular the order of the pole of a^{k−2}ν at any supersingular point must be a multiple of l − 1. Because l − k < l − 1 we see that a^{k−2}ν is in fact holomorphic at each supersingular point and hence descends to an element f of H^{0}(X_{1}^{D}(N), ω^{⊗k}). These constructions are inverses of each other and establish the isomorphism.

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Analogously to the modular curve case, this isomorphism twists the Hecke action. Note that the sketch of the proof of this fact in the modular curve case, given in Chapter 3, did not involve q-expansions. Note also that the calculation in [G] Proposition 3.15 is quite delicate and we must check that all the ingredients are there in our case. If A is a false elliptic curve, we have defined a principal polarisation on A. Hence the perfect duality between H_{DR}^{1}(A/S) and H_{DR}^{1}(A^{∨}/S) induces a perfect pairing on H_{DR}^{1}(A/S). Moreover, the statement on [G] p455 about how this pairing behaves for elliptic curves with respect to isogenies holds in the case of false elliptic curves, if one uses the notion of false degree instead of degree. We discover, for primes r not dividing Ndl, that T_{r} acting on the space H^{0}(X_{1}^{D}(N)_{𝔽_{l}}, ω^{⊗k}) is identified with ⟨r⟩_{l}^{−1}T_{r} = r^{l−1−j}T_{r} acting on the space H^{0}(I_{1}^{D}(N), Ω_{I_{1}^{D}(N)/𝔽_{l}})(j).

Let L_{D} be the space of regular differentials

L_{D} = H^{0}(X_{1}^{D}(Nl)_{R}, Ω_{X_{1}^{D}(Nl)_{R}/R})

for X_{1}^{D}(Nl)_{R}. An element of \overline{L}_{D} = L_{D}/πL_{D} = H^{0}(X_{1}(Nl)_{𝔽_{l}}, Ω_{X_{1}^{D}(Nl)_{𝔽_{l}} /𝔽_{l}}) can be thought of as a meromorphic differential ν_{i} on each fibre Y_{i} of X_{1}^{D}(Nl)_{𝔽_{l}}, 1 ≤ i ≤ 2, regular at all ordinary points, and having at worst simple poles at the supersingular points, such that Res_{s}(ν_{1}) + Res_{s}(ν_{2}) = 0 for all supersingular points s. Moreover, (ℤ/lℤ)^{×} acts trivially on the supersingular points of X_{1}^{D}(Nl)_{𝔽_{l}} and hence if \overline{L}_{D}(j) is the subspace of \overline{L}_{D} where (ℤ/lℤ)^{×} acts by ⟨d⟩_{l}x = d^{j}x, we see that for j ≢ 0 mod (l − 1) we have

\overline{L}_{D}(j) ≅ H^{0}(X_{1}, Ω_{X_{1}/𝔽_{l}})(j) ⊕ H^{0}(X_{2}, Ω_{X_{2}/𝔽_{l}})(j).

For A an R-algebra, define S_{k}^{D}(Γ_{1}(N); A) to be H^{0}(X_{1}^{D}(N)_{A}, ω^{⊗k}). Using Theorem 5.6.6 and the fact that the isomorphism j_{1} : I_{1}^{D}(N)_{𝔽_{l}} → Y_{1} inverts the action of (ℤ/lℤ)^{×} whereas the isomorphism j_{2} : I_{1}^{D}(N)_{𝔽_{l}} → Y_{2} preserves it, we deduce that there is an isomorphism of vector spaces

\overline{L}_{D}(j) ≅ S_{j+2}^{D}(Γ_{1}(N); 𝔽_{l}) ⊕ S_{l+1−j}^{D}(Γ_{1}(N); 𝔽_{l}).

Moreover, we can check how the Hecke operators behave on both sides, because we know how they behave with respect to the isomorphism of Theorem 5.6.6. We find that, analogously to the modular curve case (Proposition 3.5.1), we have

Proposition 5.6.7. If 1 ≤ j ≤ l − 2, we have an isomorphism of 𝕋^{(dl)}-modules

\overline{L}_{D}(j) ≅ S_{j+2}^{D}(Γ_{1}(N); 𝔽_{l}) ⊕ S_{l+1−j}^{D}(Γ_{1}(N); 𝔽_{l})[j].
□

Now say 3 ≤ k ≤ l and f ∈ S_{k}^{D}(Γ_{1}(N); \overline{𝔽}_{l}) is an eigenform. By the Jacquet-Langlands Theorem, the system of eigenvalues of 𝕋^{(Nld)} attached to f arises in S_{k}(Γ_{1}(N) ∩ Γ_{0}(d); \overline{𝔽}_{l}) and hence there is a mod l Galois representation ρ_{f} attached to f.

As in the modular curve case, there is F ∈ L_{D}⊗\overline{ℚ}_{l} with the property that the reduction of the system of eigenvalues attached to F is the system of eigenvalues attached to f. In this way, we associate to f a maximal ideal 𝔪 of the ring H ⊆ End(Jac(X_{1}^{D}(Nl)_{ℚ_{l}(ζ_{l})})) defined just before Theorem 5.6.6. As in Chapter 3, we associate an l-divisible group G to f. Note that again G has good reduction over R, because the automorphisms ⟨d⟩_{l} act non-trivially on F. From now on we will assume that ρ_{f} is irreducible and not D_{l}-scalar.

Theorem 5.6.8. The conclusions of Theorem 3.6.3 all hold for G and G[𝔪].

Proof. One checks that the formula for the operator U_{l} on the abelian variety part of the special fibre of J_{X_{1}^{D}(Nl)}^{0} is the same as that given in [E] for the modular curve case. Then one observes that the geometry is the same in this new situation, and the operators ⟨d⟩_{l} act trivially on the supersingular points. Finally, one checks that the proofs in [G] and [E] referred to in the proof of Theorem 3.6.3 carry over immediately.

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Note that we are not asserting any of the mod l multiplicity one results proved in [G] and [E]. Indeed they will not be true in this setting.

§5.7. The main theorem.

Theorem 5.7.1. Let d be the product of a non-zero even number of distinct primes. Let N ≥ 5 be an integer prime to d, and let k be an integer such that 2 ≤ k ≤ l + 1. Let l ≥ 5 be a prime not dividing Nd. Say f ∈ S_{k}(Γ_{1}(N) ∩ Γ_{0}(d); \overline{𝔽}_{l})^{d-new} is an eigenform for 𝕋^{(Nd)}, and say the representation ρ_{f} attached to f is irreducible and not D_{l}-scalar. Say p is a prime not dividing Nld such that

p(trρ_{f}(Frob_{p}))^{2} = (1 + p)^{2}detρ_{f}(Frob_{p}) ∈ \overline{𝔽}_{l}.

Then there exists either

g_{1} ∈ S_{k}(Γ_{1}(N) ∩ Γ_{0}(dp))^{dp-new}

such that ρ_{f} ≅ ρ_{g_{1}}, or

g_{2} ∈ S_{l+3−k}(Γ_{1}(N) ∩ Γ_{0}(dp))^{dp-new}

such that ρ_{f} ≅ ρ_{g_{2}} ⊗ χ^{k−2}.

In either case, there exists k′ ≥ 2 and g ∈ S_{k′}(Γ_{1}(N)∩Γ_{0}(dp))^{dp-new} such that ρ_{f} ≅ ρ_{g}. We will give a sketch of the proof of this theorem. Note that the techniques used are

precisely those used to prove the analogous statement when d = 1, which we have proved already as Theorem 4.1.2. Many of the ingredients of the proof have been assembled in §5.2.

Corollary 5.7.2. Let l > 3 be a prime. Let ρ be an irreducible modular mod l representation arising from a form of level N ≥ 5 prime to l, and weight k with 2 ≤ k ≤ l+1. Suppose moreover that ρ is not D_{l}-scalar. Let d be a positive square-free integer, prime to Nl. Suppose that for all primes p dividing d, we have

p(trρ_{f}(Frob_{p}))^{2} = (1 + p)^{2}detρ_{f}(Frob_{p}) ∈ \overline{𝔽}_{l}.

Then either ρ arises from a form in S_{k}(Γ_{1}(N) ∩ Γ_{0}(d))^{d-new} or ρ ⊗ χ^{l+1−k} arises from a form in S_{l+3−k}(Γ_{1}(N) ∩ Γ_{0}(d))^{d-new}.

Proof. We proceed by induction on the number of primes dividing d. If d = p then this is just Theorem 4.5.2. In general, pick a prime p dividing d. We may assume the theorem is true for d/p. Then either [DT1] Theorem 1 and the Jacquet-Langlands Theorem (if the number of prime divisors of d/p is odd) or Theorem 5.7.1 above (if this number is even) applied to the form asserted to exist by the result for d/p completes the proof.

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Now we shall remove the upper bound on the weight.

Corollary 5.7.3. Let l > 3 be a prime. Let ρ be an irreducible modular mod l representation arising from a form of level N ≥ 5 prime to l and weight k ≥ 2. Suppose moreover that ρ is not D_{l}-scalar. Let d be a positive square-free integer, prime to Nl. Suppose that for all primes p dividing d, we have

p(trρ_{f}(Frob_{p}))^{2} = (1 + p)^{2}detρ_{f}(Frob_{p}) ∈ \overline{𝔽}_{l}.

Then ρ arises from a form in S_{k′}(Γ_{1}(N) ∩ Γ_{0}(d))^{d-new}.

Proof. Some twist of ρ arises from a form f of weight k_{1} satisfying 2 ≤ k_{1} ≤ l + 1. If d is the product of an odd number of primes, then we proceed as follows. Applying Corollary 5.7.2 to f gives us a form in either S_{k_{1}}(Γ_{1}(N) ∩ Γ_{0}(d))^{d-new} or S_{l+3−k_{1}}(Γ_{1}(N) ∩ Γ_{0}(d))^{d-new} which gives rise to some twist of ρ. Now using Proposition 4.4.4 we can untwist the representation and we are done.

If d is the product of a non-zero even number of primes, then pick a prime p dividing d. Then d/p is the product of an odd number of primes, so by the preceding paragraph the corollary is true for d/p. Now apply [DT1] Theorem 1 once more.

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Now we shall remove the condition N ≥ 5.

Corollary 5.7.4. Let l > 3 be a prime. Let ρ be an irreducible modular mod l representation arising from a form of level N prime to l and weight k ≥ 2. Suppose moreover that ρ is not D_{l}-scalar. Let d be a positive square-free integer, prime to Nl. Suppose that for all primes p dividing d, we have

p(trρ_{f}(Frob_{p}))^{2} = (1 + p)^{2}detρ_{f}(Frob_{p}) ∈ \overline{𝔽}_{l}.

Then ρ arises from a form in S_{k′}(Γ_{1}(N) ∩ Γ_{0}(d))^{d-new}.

Proof. This is done precisely as in [DT1] Lemma 11 and the remarks following it. An auxiliary prime q is added to the level such that Nq ≥ 5, and such that no lift of ρ can have q dividing its conductor.

□

From Corollary 5.7.4 we deduce

Theorem 5.7.5. If ρ is irreducible and modular, and M is a possible level for ρ, and moreover ρ is not D_{l}-scalar, then ρ is modular of level M.

Proof. The proof of this theorem in the case where ρ arises from a form of weight at most l − 2 is in [DT1] Chapter 1, and uses a result called Theorem B. Using our Corollary 5.7.4 as a replacement for Theorem B, their argument gives us Theorem 5.7.5.

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We now explain the proof of Theorem 5.7.1. If 2 ≤ k < l − 1 then this result is proved in [DT1], and if k = l − 1 then it is proved in [DT2]. If k = l + 1 then we can apply the ideas of the proof of Proposition 4.6.2, using the quaternion algebra D with discriminant dl, and the argument goes straight through. If k = l we mimic the argument of the proof of Theorem 4.5.2. Let 𝔞 ⊆ 𝕋^{(Nld)} be the maximal ideal attached to the form f. By the Jacquet-Langlands Theorem we have 𝔞 ∈ Supp_{𝕋^{(Nld)}}(S_{k}^{D}(Γ_{1}(N); 𝔽_{l})) where D is the quaternion algebra with discriminant d. By Proposition 5.6.7 we have

𝔞 ∈ Supp_{𝕋^{(Nld)}}(\overline{L}(j)),

where j = k − 2. The ideas of the proof of Proposition 4.3.2 go through in this setting, and if we set

L′_{D} = H^{0}(X_{1}^{D}(Nl; p), Ω_{X_{1}^{D}(Nl;p)/R})

and

(L′_{D})^{p-new} = L′_{D} ∩ ((L′_{D} ⊗ ℚ_{l}(ζ_{l}))^{p-new}),

and define 𝔞′ = 𝔞∩𝕋^{(Nldp)}, then we deduce, using Corollary 5.6.5 and the fact that j ≢ ±2 mod l − 1, that

𝔞′ ∈ Supp_{𝕋^{(Nldp)}}((L′_{D})^{p-new}/π(L′_{D})^{p-new})(j).

From this, we conclude using the proof of Proposition 4.5.1 that either

𝔞′ ∈ Supp_{𝕋^{(Nldp)}}(S_{k}^{D}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new})

or that

𝔞′ ∈ Supp_{𝕋^{(Nldp)}}(S_{l+3−k}^{D}(Γ_{1}(N) ∩ Γ_{0}(p); 𝔽_{l})^{p-new}[k − 2]).

By the Jacquet-Langlands Theorem once more, we deduce that either

𝔞′ ∈ Supp_{𝕋^{(Nldp)}}(S_{k}(Γ_{1}(N) ∩ Γ_{0}(dp); 𝔽_{l})^{dp-new})

or

𝔞′ ∈ Supp_{𝕋^{(Nldp)}}(S_{l+3−k}(Γ_{1}(N) ∩ Γ_{0}(dp); 𝔽_{l})^{dp-new}[k − 2]).

The eigenforms corresponding to 𝔞′ are the forms we seek.

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Chapter 6. On modular mod 2 Galois representations.

Summary. We examine a particular family of indecomposable, but not irreducible, mod 2 representations and show that they arise as the “naïve” mod 2 reduction of the 2-adic representation associated to some newform of weight 1 and level equal to the conductor of the representation up to a power of 2.

§6.1. Introduction.

We begin by describing the process of “naïve” reduction mentioned in the summary. If f = ∑_{n=1}^{∞} c_{n}q^{n} is a characteristic zero newform of level N and weight k, then it gives rise to a compatible series of 2-dimensional l-adic representations of Gal(\overline{ℚ}/ℚ). Now fix a prime l.

Because ρ_{f,l} is unramified outside a finite set of primes, it can be shown that the image of ρ_{f,l} may be conjugated into GL_{2}(K), where K/ℚ_{l} is a finite extension. Now because Gal(\overline{ℚ}/ℚ) is compact, the representation ρ_{f,l} must be conjugate to a representation whose image lies in GL_{2}(O_{K}), where O_{K} is the ring of integers of K. We say that a representation ρ is integral if its image lies in GL_{2}(O_{K}) for some finite extension K of ℚ_{l}. If ρ is integral, we may reduce it modulo the maximal ideal of O_{K} to get a representation ρ̃: Gal(\overline{ℚ}/ℚ) → GL_{2}(k) where k, the residue field of K, is a finite field of characteristic l. If ρ is integral and conjugate to ρ_{f,l}, we call ρ̃ a naïve reduction of ρ_{f,l}, or just a reduction of ρ_{f,l}. An important thing to note here is that ρ̃ may not be the unique reduction of ρ_{f,l}, if ρ̃ is not irreducible. More precisely, ρ̃ is continuous with respect to the discrete topology, and hence factors through Gal(L/ℚ) where L is now a finite extension of ℚ, unramified outside Nl, but knowing traces and determinants of Frobenius elements for every p not dividing Nl does not uniquely determine ρ̃, because ρ̃ may not be irreducible. In fact, the isomorphism class of the reduction ρ̃ of an integral model ρ of ρ_{f,l} depends on more than just the isomorphism class of ρ_{f,l}, although all the possible reductions have the same semisimplification, by the Brauer-Nesbitt Theorem, as any two reductions have the same trace and determinant.

In this chapter, we call a representation ρ̃: Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{l}) modular if it arises as the naïve reduction of some ρ_{f,l}. We say that ρ̃ is modular of level N and weight k if the corresponding newform f has level N and weight k. Note that if ρ̃ is irreducible then this coincides with the usual definition of modular. However, we shall see later that non-semi-simple representations may also be modular, if we use this definition.

If ρ̃ is a reducible, mod l representation, then, analogous to the case when ρ̃ is irreducible, one can ask whether ρ̃ is modular, and if so then for which pairs {N, k} is ρ̃ modular of level N and weight k? Note that ρ̃ may well be modular of many weights and levels. For ρ̃ to be modular we certainly require ρ̃ to be continuous, and for l ≠ 2 we must have ρ̃ odd. If ρ̃ is the direct sum of two 1-dimensional representations then we can lift ρ̃ back to a reducible l-adic representation which will arise from an appropriate Eisenstein series. Wiles raised the question as to what could be said in the case when ρ̃ is indecomposable but not irreducible, i.e. after conjugation is a representation whose image is upper triangular in GL_{2}(\overline{𝔽}_{l}) but not diagonalisable. The purpose of this chapter is to look at some examples of indecomposable but not irreducible representations, to discuss which are modular, and which weights and levels occur. In all that follows, we shall be considering the case when l = 2.

§6.2. Δ.

It is not too difficult to see that, unsurprisingly, the conjectures that Serre made for the weight, character and level of the newform f giving rise to an irreducible mod l representation ρ̃ must be modified in the reducible case. In the irreducible case, Serre conjectures that ρ̃ will be modular of level N = cond(ρ̃), the conductor of the representation, which is coprime to l by definition. He also conjectures that in the irreducible case, the weight will be no larger than l^{2} − 1 if l is odd, and no larger than 4 if l = 2. This cannot be so in the reducible case however, as is easy to see. For example, if L is a quadratic extension of ℚ, unramified outside 2 (i.e. L = ℚ(\sqrt{2}), ℚ(\sqrt{−1}) or ℚ(\sqrt{−2}) ), then the representation

Gal(\overline{ℚ}/ℚ) → → Gal(L/ℚ) ⟶^{∼} B(𝔽_{2}),

where B(k) is the upper triangular matrices in GL_{2}(k), has trivial conductor, and yet cannot arise from a newform of level 1 and weight at most 4 as there are none. We can however show that these three representations are modular of level 1 using the following result of Swinnerton-Dyer ([Sw-D], appendix):

Theorem 6.2.1. If Δ is the newform of level 1 and weight 12, then the image of the corresponding 2-adic representation ρ_{Δ,2} is conjugate to the following group G_{0} to be defined below.

Let θ, ϕ and ψ be the three non-trivial characters of (ℤ/8ℤ)^{×} thus:

1 3 5 7
θ 1 1 −1 −1
ϕ 1 −1 1 −1
ψ 1 −1 −1 1

Now set

G_{0} = \{σ = ( \begin{matrix}1 + 2^{7}A & 2^{4}B \\ 2^{5}C & 1 + 2D \end{matrix} ) ∈ GL_{2}(ℤ_{2})| Properties 1 to 4 below hold.\}

1) A, B, C, D ∈ ℤ_{2},

2) Set d = det(σ). Then B and C are both even if d ≡ ±1 mod 8, and both odd if d ≡ ±3 mod 8,

3) B + Cd ≡ \frac{d^{2}+3−4ψ(d)}{2} mod 16,

4) A ≡ \frac{d+2θ(d)−3ϕ(d)}{8}(3d + 10θ(d) − 3ϕ(d)) + \frac{3}{2}(1 − ψ(d)) − 2C^{2} mod 64.

(Note that in all cases these fractions are well-defined elements of ℤ_{2}, because, for example, any d ∈ ℤ_{2}^{×} satisfies 8|(d + 2θ(d) − 3ϕ(d)). )

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We can hope to read off from this explicit result precisely which representations ρ̃: Gal(\overline{ℚ}/ℚ) → B(𝔽_{2}) are naïve reductions of the 2-adic representation attached to Δ.

We see immediately that reducing G_{0} mod 2 gives us the identity in GL_{2}(𝔽_{2}), and deduce that the trivial 2-dimensional representation is modular. Of course, this representation is not irreducible, and so by the Brauer-Nesbitt Theorem no integral conjugate of ρ_{Δ,2} can reduce to an irreducible mod 2 representation. In fact, any integral conjugate of ρ_{Δ,2} will reduce to a mod 2 representation whose semisimplification is the trivial 2-dimensional representation. However there may be (and indeed are) other integral conjugates of ρ_{Δ,2} which reduce to give non-trivial representations of Gal(\overline{ℚ}/ℚ). We see immediately that we cannot expect too many other representations to GL_{2}(𝔽_{2}) as reductions of the 2-adic representation attached to Δ. For Δ is of level 1, so all the mod 2 reductions of ρ_{Δ,2} will be unramified away from 2. If the reduced representation has image contained in GL_{2}(𝔽_{2}), then it must be conjugate to a mod 2 representation whose image is contained in B(𝔽_{2}),

namely, the matrices ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). Hence, if the reduced representation is not

trivial, it must factor through Gal(L/ℚ), where L is a quadratic extension of ℚ unramified outside 2. There are only three of these, and all three can arise, as is shown by the following theorem.

Theorem 6.2.2. If L is a quadratic extension of ℚ which is unramified outside 2 (i.e. L = ℚ(i), ℚ(\sqrt{2}) or ℚ(\sqrt{−2}) ), then the representation Gal(\overline{ℚ}/ℚ) → → Gal(L/ℚ) →^{∼} B(𝔽_{2}) (which has conductor 1) is modular of level 1 and weight 12, i.e. it arises from Δ.

Proof. We seek appropriate integral models for the representation ρ_{Δ,2} : Gal(\overline{ℚ}/ℚ) → GL_{2}(ℤ_{2}). We may assume that the image of ρ_{Δ,2} is G_{0}. If L is a quadratic extension of ℚ, unramified outside 2, then we would like to find an element τ ∈ GL_{2}(\overline{ℚ}_{2}) such that τ^{−1}G_{0}τ = G_{L} ⊆ GL_{2}(ℤ_{2}) with the property that the mod 2 reduction of G_{L} is the two upper triangular matrices in GL_{2}(𝔽_{2}) and such that the reduction of τ^{−1}ρ_{Δ,2}(σ)τ for σ ∈ Gal(\overline{ℚ}/ℚ) is trivial iff σ fixes L pointwise.

The easiest to find is L = ℚ(\sqrt{2}). Note that if we set G_{1} = τG_{0}τ^{−1} with τ =

( \begin{matrix}1 & 0 \\ 0 & 16 \end{matrix} ), and write ρ_{1} = τ(ρ_{Δ,2})τ^{−1}, then G_{1} is the image of ρ_{1} and consists of the

matrices ( \begin{matrix}1 + 2^{7}A & B \\ 2^{9}C & 1 + 2D \end{matrix} ) such that properties 1 to 4 of Theorem 6.2.1 hold. Moreover,

the mod 2 reduction of G_{1} is contained in B(𝔽_{2}), and the kernel of this mod 2 representation

is the elements of Gal(\overline{ℚ}/ℚ) whose images under ρ_{Δ,2} are of the form ( \begin{matrix}1 + 2^{7}A & 2^{4}B \\ 2^{5}C & 1 + 2D \end{matrix} )

with B even. However, by condition 2 of the definition of G_{0}, B is even precisely when the determinant of this latter matrix is congruent to ±1 mod 8. Say σ is a Frobenius element for an odd prime p. We know that det(ρ_{Δ,2}(Frob_{p})) = p^{11}, and so if ρ_{Δ,2}(Frob_{p}) =

( \begin{matrix}1 + 2^{7}A & 2^{4}B \\ 2^{5}C & 1 + 2D \end{matrix} ) then B is even iff p^{11} ≡ ±1 mod 8 iff p ≡ ±1 mod 8, and hence for

p ≡ ±3 mod 8 we must have B odd, which implies that ρ̃_{1} is non-trivial and there is a quadratic extension L of ℚ such that ρ̃_{1} is the map Gal(\overline{ℚ}/ℚ) → → Gal(L/ℚ) ⟶^{∼} B(𝔽_{2}). Moreover, in this extension L/ℚ, the Frobenius element in the Galois group for an odd prime p is trivial (i.e. p splits) iff p ≡ ±1 mod 8, i.e. iff (\frac{2}{p}) = 1, and so L = ℚ(\sqrt{2}) and ρ̃_{1} factors through Gal(ℚ(\sqrt{2})/ℚ).

Next, set G_{2} = βG_{1}β^{−1} and ρ_{2} = βρ_{1}β^{−1} where β = ( \begin{matrix}4 & 1 \\ 8 & 4 \end{matrix} ).

Observe that

( \begin{matrix}4 & 1 \\ 8 & 4 \end{matrix} ) ( \begin{matrix}1 + 2^{7}A & B \\ 2^{9}C & 1 + 2D \end{matrix} ) ( \begin{matrix}4 & 1 \\ 8 & 4 \end{matrix} )^{−1}
= ( \begin{matrix}1 + 2^{8}A − 4B + 2^{8}C − 2D & −2^{6}A + 2B − 2^{6}C + D \\ 2^{9}A − 8B + 2^{10}C − 8D & 1 − 2^{7}A + 2B − 2^{8}C + 4D \end{matrix} )
≡ ( \begin{matrix}1 & D \\ 0 & 1 \end{matrix} ) mod 2

and det ( \begin{matrix}1 + 2^{7}A & B \\ 2^{9}C & 1 + 2D \end{matrix} ) ≡ 1 + 2D mod 4, from which we deduce that D is even

precisely when the determinant is congruent to 1 mod 4. Hence by a similar argument to that for ρ̃_{1}, we deduce that ρ̃_{2} factors through the quadratic extension of ℚ in which odd primes split iff they are congruent to 1 mod 4, and this extension is ℚ(i).

Finally, set G_{3} = γG_{1}γ^{−1} and ρ_{3} = γρ_{1}γ^{−1} where γ = ( \begin{matrix}2 & 1 \\ 0 & 2 \end{matrix} ). Observe that

\begin{aligned}( \begin{matrix}2 & 1 \\ 0 & 2 \end{matrix} ) ( \begin{matrix}1 + 2^{7}A & B \\ 2^{9}C & 1 + 2D \end{matrix} ) ( \begin{matrix}2 & 1 \\ 0 & 2 \end{matrix} )^{−1} & = ( \begin{matrix}1 + 2^{7}A + 2^{8}C & −2^{6}A + B − 2^{7}C + D \\ 2^{9}C & 1 − 2^{8}C + 2D \end{matrix} ) \\ & ≡ ( \begin{matrix}1 & B + D \\ 0 & 1 \end{matrix} ) mod 2 \end{aligned}

and we have already observed that if ρ_{Δ,2}(Frob_{p}) = ( \begin{matrix}1 + 2^{7}A & 2^{4}B \\ 2^{5}C & 1 + 2D \end{matrix} ) , then B is even

precisely when p ≡ ±1 mod 8 and D is even precisely when p ≡ 1 mod 4, hence B + D is even precisely when p ≡ 1 or 3 mod 8, from which we can deduce that ρ̃_{3} factors through Gal(ℚ(\sqrt{−2})/ℚ). This proves the theorem.

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Note that the representations ρ, ρ_{1}, ρ_{2} and ρ_{3} are all integral and isomorphic, but their mod 2 reductions are all distinct.

We now consider the more general question of precisely which quadratic extensions L of ℚ are such that the representation Gal(\overline{ℚ}/ℚ) → → Gal(L/ℚ) →^{∼} B(𝔽_{2}) is modular. This is the subject of the remainder of this chapter.

§6.3. Mod 2 representations.

Lemma 6.3.1. If E/ℚ is a modular elliptic curve, then the mod 2 Galois representation afforded by the 2-torsion points of E is a modular representation of level Cond(E) and weight 2.

Proof. Indeed, if f is the modular form associated to E, then the mod 2 representation attached to f is realised on the vector-space E[2](\overline{ℚ}).

□

From this observation, we can deduce the following

Proposition 6.3.2. If L = ℚ(\sqrt{m}) is a quadratic extension of the rationals, with m squarefree, then the representation Gal(\overline{ℚ}/ℚ) → → Gal(L/ℚ) →^{∼} B(𝔽_{2}) is modular of weight 2.

Proof. Let E be the elliptic curve Y^{2} = X(X − \sqrt{m})(X + \sqrt{m}) = X^{3} − mX. Because E has complex multiplication, we know that E is modular. The mod 2 representation on E[2](\overline{ℚ}) is hence modular. However, the set E[2](\overline{ℚ}) consists of the origin O, and the points (0, 0), (\sqrt{m}, 0), (−\sqrt{m}, 0). Hence any element of Gal(\overline{ℚ}/ℚ) which fixes L = ℚ(\sqrt{m}) pointwise acts trivially on this 𝔽_{2}-module, and so the mod 2 representation factors through Gal(L/ℚ) and the non-trivial element of this Galois group permutes the two non-rational 2-division points thus giving us the required representation.

□

Note that the elliptic curve occurring in the proof of this proposition has multiplicative reduction at all primes dividing m and so has conductor some multiple of m^{2}. However, the conductor of the representation Gal(ℚ(\sqrt{m})/ℚ) →^{∼} B(𝔽_{2}) is |m| or |m/2| depending on whether m is odd or even. A natural question to ask, analogous to the case when ρ̃ is irreducible, is whether there is a form of level equal to the conductor of ρ̃ which gives rise to ρ̃. Note that this is true when m = −1 or m = ±2 because we have shown that the newform Δ of level 1 gives rise to the representation in these three cases. For general squarefree m, we shall prove the following.

Theorem 6.3.3. If m is a square-free integer, then the representation

Gal(\overline{ℚ}/ℚ) → → Gal(ℚ(\sqrt{m})/ℚ)⟶^{≅}B(𝔽_{2})

is modular of weight 1 and level N where N = 2^{r}|m|, r some non-negative integer. The key ingredients are the following two theorems:

Theorem 6.3.4. If K/ℚ is an imaginary quadratic extension of the rationals, and θ : 𝔸_{K}^{×}/K^{×} → ℂ^{×} is an algebraic Grössencharacter, with infinity type θ_{∞} : ℂ^{×} → ℂ^{×} the map z ↦ z^{1−k} for some positive integer k, and furthermore if θ ∘ c ≠ θ, where c is the involution on 𝔸_{K}^{×} corresponding to complex conjugation, then θ gives rise to a compatible system of l-adic representations θ^{G} : Gal(\overline{K}/K) → \overline{ℚ}_{l}^{×}, and the corresponding induced 2dimensional representations Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{ℚ}_{l}) are modular, i.e. arise from a newform f of weight k. Moreover, the level of f is N = ∏_{p} p^{a(p)}, where a(p) is defined thus:

a(p) = \{ \begin{matrix}a(θ_{v}) + a(θ_{v^{c}}) & if p = vv^{c} is split in K \\ 2a(θ_{v}) & if p = v is inert in K \\ a(θ_{v}) + a(ϵ) & if p = v^{2} is ramified in K \end{matrix}

where here a(θ_{v}) is the conductor of the character θ|_{K_{v}^{×}} and a(ϵ) is the conductor of the quadratic character of Gal(\overline{ℚ}_{p}/ℚ_{p}) which determines K_{v}/ℚ_{p}.

□

Theorem 6.3.5. If K/ℚ is a real quadratic extension of the rationals, and θ : 𝔸_{K}^{×}/K^{×} → ℂ^{×} is an algebraic Grössencharacter, with infinity type θ_{∞} : ℝ_{1}^{×} × ℝ_{2}^{×} → ℂ^{×} the map (x, y) ↦ sgn(y), then θ gives rise to a compatible system of l-adic representations θ^{G} : Gal(\overline{K}/K) → \overline{ℚ}_{l}^{×} and the corresponding induced 2-dimensional representations Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{ℚ}_{l}) are modular of weight 1, i.e. arise from a newform f of weight 1. Moreover, the level of f is N = ∏_{p} p^{a(p)}, where a(p) is defined as in Theorem 6.3.4:

a(p) = \{ \begin{matrix}a(θ_{v}) + a(θ_{v^{c}}) & if p = vv^{c} is split in K \\ 2a(θ_{v}) & if p = v is inert in K \\ a(θ_{v}) + a(ϵ) & if p = v^{2} is ramified in K \end{matrix}

where, as before, a(θ_{v}) is the conductor of the character θ|_{K_{v}^{×}} and a(ϵ) is the conductor of the quadratic character of Gal(\overline{ℚ}_{p}/ℚ_{p}) which determines K_{v}/ℚ_{p}.

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Theorem 6.3.4 is easily deducible from Miyake [Miy] Theorem 4.8.2, although there is also a classical proof involving theta functions. Theorem 6.3.5, its real quadratic analogue, is similarly a corollary of Theorem 4.8.3 of [Miy].

§6.4. The case m < 0.

In this section, we prove Theorem 6.3.3 in the case m < 0. We start with a lemma. Say K is a quadratic extension of ℚ_{2}, with ring of integers O_{K}. There is a natural action of Gal(K/ℚ_{2}) on O_{K}^{×} and hence on the quotient M = O_{K}^{×}/ ± 1.

Lemma 6.4.1. There is a submodule N of M, invariant under this action, for which the quotient M/N is finite, and the Galois action on this quotient is non-trivial. Moreover, N can be taken to be ((1 + π^{5}O_{K}) · (±1))/ ± 1, where π is a uniformiser for O_{K}.

Proof. There are only seven quadratic extensions of ℚ_{2}, as can be seen from local class field theory, and so we can do a case by case check. Note that any quadratic extension of ℚ_{2} is of the form ℚ_{2}(\sqrt{m}) for some m ∈ ℤ_{2} and moreover we may choose m of valuation 0 or 1. Furthermore, if m is chosen to be of this form, and n ≡ m mod 16, then n has a square root in ℚ_{2}(\sqrt{m}) by Hensel’s lemma. Hence we need only consider squarefree integers m between −8 and 8, and we quickly deduce that the only quadratic extensions of ℚ_{2} are ℚ_{2}(\sqrt{m}) for m = −1, ±2, ±3 and ±6. Now it is just a case of checking these seven cases. As an illustrative example, we sketch the details for m = −1. In this case, K = ℚ_{2}(i), O_{K} = ℤ_{2}[ i], π = 1 + i and if we set M = O_{K}^{×}/ ± 1 and N = ((1 + π^{5}O_{K}) · (±1))/ ± 1 we see that a set of coset representatives for M/N is {1, 3, i, 2 + i, 4 + i, 1 + 2i, 2 − i, 1 − 2i}. The group M/N is abelian of order 8, and the Galois action, sending a + bi to a − bi mod N, is non-trivial. The other cases are similar.

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Using this and Theorem 6.3.4, we can now deduce the case m < 0 of Theorem 6.3.3.

Theorem 6.4.2. If m < 0 is squarefree, then the representation

ρ : Gal(\overline{ℚ}/ℚ) → → Gal(ℚ(\sqrt{m})/ℚ) →^{∼} B(𝔽_{2})

is modular of weight 1 and level N = −2^{r}m, for some r ≤ 7.

Proof. Set k = 1 and K = ℚ(\sqrt{m}). We wish to define an appropriate θ : 𝔸_{K}^{×}/K^{×} → ℂ^{×}, and then apply Theorem 6.3.4. Firstly, observe that K^{×}Ô_{K}^{×}ℂ^{×} ⊆ 𝔸_{K}^{×} with quotient group the class group of K. Let H = K^{×}Ô_{K}^{×}ℂ^{×}. We will firstly define θ on this subgroup H. There are four cases.

a) m ≡ 1 mod 8.

Then 2 splits in K. Pick a prime v of K lying above 2. Then H has a subgroup J = K^{×} ∏_{w≠v} O_{K_{w}}^{×}ℂ^{×} and H = JO_{K_{v}}^{×}. So

H/J ≅ O_{K_{v}}^{×}/(O_{K_{v}}^{×} ∩ J)
≅ O_{K_{v}}^{×}/{units of K}
=_{defn} M.

Because m ≡ 1 mod 8 we have K_{v} = ℚ_{2}, and we define a homomorphism from M to ℂ^{×} thus: the units of K are just ±1, so we have M ≅ ℤ_{2}^{×}/ ± 1 → → (ℤ/8ℤ)^{×}/ ± 1 ≅ ±1 ⊆ ℂ^{×}. This homomorphism induces a homomorphism θ : H → → H/J = M → ℂ^{×}, which satisfies θ ∘ c ≠ θ, as v^{c} ≠ v.

b) m ≢ 1 mod 8, and m ≠ −1, −3.

Then let v be the unique prime of K lying above 2, and define J = K^{×} ∏_{w≠v} O_{K_{w}}^{×}ℂ^{×} as above and note that as before, H = JO_{K_{v}}^{×} and

H/J ≅ O_{K_{v}}^{×}/(O_{K_{v}}^{×} ∩ J)
≅ O_{K_{v}}^{×}/{units of K}
= O_{K_{v}}^{×}/{±1} (as m ≠ −1, −3)
=_{defn} M.

Note that [K_{v} : ℚ_{2}] = 2. By Lemma 6.4.1, we can pick some finite quotient of M on which the Galois involution acts non-trivially. Now pick an element g of this finite quotient such that the Galois action sends g to h ≠ g. Define any homomorphism from this finite quotient to ℂ^{×} such that the images of g and h are distinct—of course, this can be done because of the orthogonality relations on the characters of the quotient evaluated at g and h. We can extend this character to a character of M, which induces a character θ of H satisfying θ ∘ c ≠ θ.

c) m = −3

Here we can give an explicit construction of θ. Defining v to be the prime of K above 2, and J as before, we see that H/J ≡ O_{K_{v}}^{×}/{units of K} and the units of K are a cyclic group of order 6, the 6th roots of unity in K. Now, we can check that O_{K_{v}}^{×}/{μ_{6}} → → (O_{K_{v}}^{×}/(1 + 𝔪_{v}^{3}))/{μ_{6}} where 𝔪_{v} is the maximal ideal of O_{K_{v}}, and that this quotient is a finite group of order 8 with a non-trivial Galois action, so we can proceed as in b) to define θ : H → ℂ^{×} satisfying θ ∘ c ≠ θ.

d) m = −1

As in c) we just proceed explicitly. If v is the prime of K above 2, then we define a suitable θ factoring through the quotient (O_{K_{v}}^{×}/(1 + 𝔪_{v}^{5}))/(±1, ±i).

Note that in all four cases, θ factors through a finite abelian 2-group, and hence the image of θ is contained in the 2-power roots of unity in ℂ^{×}.

We can now extend θ to a homomorphism from 𝔸_{K}^{×} to the 2-power roots of unity in ℂ^{×} by the following lemma.

Lemma 6.4.3. If G is an abelian group, H ⊆ G is a subgroup of finite index, and θ is a homomorphism from H to the 2-power roots of unity in ℂ^{×}, then θ extends to a homomorphism from G to the 2-power roots of unity in ℂ^{×}.

Proof. Write G/H ≅ C_{a_{1}} × C_{a_{2}} × … × C_{a_{s}} and pick generators γ_{1}, γ_{2}, …, γ_{s} for these cyclic groups. Next pick preimages g_{1}, g_{2}, …, g_{s} of γ_{1}, γ_{2}, …, γ_{s} in G, and note that we have already defined θ(g_{j}^{a_{j}}) because g_{j}^{a_{j}} is an element of H. Hence θ(g_{j}^{a_{j}}) is a 2^{r}-th root of unity for some r, and at least one a_{j}-th root of this is a 2^{r+s}-th root of unity, where 2^{s} is the highest power of 2 dividing a_{j}. Define θ(g_{j}) to be any one of these 2^{r+s}-th roots of unity. Doing this for all j extends θ to a homomorphism from G to the 2-power roots of unity in ℂ^{×}.

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Applying this with G = 𝔸_{K}^{×} and H = K^{×}Ô_{K}^{×}ℂ^{×} gives us a homomorphism θ from 𝔸_{K}^{×} to ℂ^{×} with image contained in the 2-power roots of unity. Moreover, θ is trivial on K^{×} and has trivial infinity type. This θ gives rise to a 1-dimensional representation of Gal(\overline{K}/K) into the ring of integers of some local field Φ which has residue field 𝔽_{2} and is totally ramified at 2. Theorem 6.3.4 says that the induced 2-dimensional representation of Gal(\overline{ℚ}/ℚ) into GL_{2}(O_{Φ}) is modular, and the image of this representation is included

in the normaliser of the diagonal subgroup of GL_{2}(O_{Φ}), namely the matrices ( \begin{matrix}a & b \\ c & d \end{matrix} )

in GL_{2}(O_{Φ}) for which a = d = 0 or b = c = 0. Clearly a = d = 0 implies b ≡ c ≡ 1 mod 𝔪_{Φ}, and b = c = 0 implies a ≡ d ≡ 1 mod 𝔪_{Φ}, where 𝔪_{Φ} is the maximal ideal of O_{Φ}, and so the reduced mod 2 representation is a homomorphism from Gal(\overline{ℚ}/ℚ) onto

\{( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \} ⊆ GL_{2}(𝔽_{2}). This representation factors through Gal(K/ℚ), and

can be conjugated into an upper triangular representation, observing that

( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} )^{−1} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) in GL_{2}(𝔽_{2}),

and so any matrix congruent to ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) in GL_{2}(ℤ_{2}) will conjugate our 2-adic represen-

tation into one which reduces to the mod 2 representation we require.

Finally we must check the level of this representation. By Theorem 6.3.4 we see that if N is the level of the corresponding newform, then any odd prime p not dividing m will not divide N, because K is unramified at p and θ is unramified at all primes above p. For odd primes p dividing m, we have that p ramifies in K but θ is unramified at the prime v of K above p, and the quadratic character ϵ determining K_{v}/ℚ_{p} can only be tamely ramified, from which it follows that p divides N exactly once. For 2 the situation is more complex, and we have omitted the explicit calculations describing the ramification of θ at 2, but it turns out that 2 will divide N, and that the highest power of 2 that can divide N is 2^{7}, which occurs when m ≡ 2 or 3 mod 4.

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§6.5. The case m > 0.

In this section we finish the proof of Theorem 6.3.3, by proving it in the case m > 0. We start with a lemma.

Lemma 6.5.1. If K is an extension of ℚ_{2} of degree 1 or 2, O_{K} is the ring of integers of K, and u ∈ O_{K}^{×} is not a root of unity, then there is a continuous homomorphism θ from O_{K}^{×} to the 2-power roots of unity such that θ(−1) = −1 and θ(u) = 1.

Proof. Analogously to Lemma 6.4.1, we go by cases. The easiest are K = ℚ_{2}(\sqrt{m}), where m = 1, ±2, 3 or ±6. In these cases, O_{K}^{×} is a pro-2-group and −1 has no square root in O_{K}^{×}. We pick an appropriately large power π^{n} of a uniformiser π such that in the quotient G = O_{K}^{×}/(1 + π^{n}O_{K}) we still have that −1 is not a square and that the image of u in this quotient is not equal to the image of −1. Write ũ for the image of u in G, and ˜−1 for the image of −1. Now G is a finite 2-group, and so we see that ˜−1 is not in the subgroup generated by ũ, because ũ must have order some power of 2, and ˜−1 has order exactly 2 and is neither ũ nor a square. Hence the map G → → G/ < ũ > sends ˜−1 to an element of order exactly 2, and hence some character of G/ < ũ > sends ˜−1 to −1 and this induces the required continuous homomorphism of O_{K}^{×}. The cases K = ℚ_{2}(\sqrt{−1}) (here −1 has a square root but no fourth root) and K = ℚ_{2}(\sqrt{−3}) (here O_{K}^{×} is not a pro-2-group but after quotienting out by the cube roots of unity it is) are hardly more difficult.

□

From this we deduce the following theorem.

Theorem 6.5.2. If m > 1 is squarefree, then the representation

ρ : Gal(\overline{ℚ}/ℚ) → → Gal(ℚ(\sqrt{m})/ℚ) →^{∼} B(𝔽_{2})

is modular of weight 1 and level N = 2^{r}m for some non-negative integer r (note that in this case we have no bound on r).

Proof. Set K = ℚ(\sqrt{m}) in Theorem 6.3.5. We will define a Grössencharacter θ : 𝔸_{K}^{×}/K^{×} → ℂ^{×}. As in §6.4, we firstly define θ on the subgroup H = K^{×}Ô_{K}^{×}ℝ_{1}^{×}ℝ_{2}^{×}. Pick a prime v of K lying above 2 and define J = K^{×} ∏_{w≠v} O_{K_{w}}^{×}ℝ_{1}^{×}. Then H = JO_{K_{v}}^{×}ℝ_{2}^{×} and so H/J ≅ (O_{K_{v}}^{×}ℝ_{2}^{×})/{units of K} where the units of K are embedded diagonally into the product O_{K_{v}}^{×}ℝ_{2}^{×}. It is well-known that the units of K are of the form {±1} × u^{ℤ} where u is a fundamental unit for K. By changing the sign of u if necessary, we may assume that the image of u in ℝ_{2} is positive. By the previous lemma there is a homomorphism from O_{K_{v}}^{×} to the 2-power roots of unity in ℂ which sends u to 1 and −1 to −1, and multiplying this by the homomorphism ℝ_{2}^{×} → {±1} sending x to sgn(x) gives us a homomorphism θ : O_{K_{v}}^{×}ℝ_{2}^{×} → ℂ^{×} which contains the units of K, embedded diagonally, in its kernel and hence induces a homomorphism H/J → ℂ^{×} and finally H → ℂ^{×} having image in the 2-power roots of unity. By Lemma 6.4.3, we can extend θ to a Grössencharacter on 𝔸_{K}^{×} having image in the 2-power roots of unity and everything now works exactly as in the case m < 0, except that in this case we have no bound on the ramification of θ at v because we do not know how badly the character in the lemma above will ramify.

□

Theorem 6.4.2 and Theorem 6.5.2 together prove Theorem 6.3.3.

Chapter 7. On the number fields generated by coefficients of newforms.

Summary. Let K be the splitting field of the characteristic polynomial of the Hecke operator T_{2} acting on the d-dimensional space of cusp forms of level 1 and weight k, with coefficients in ℂ. We show, for various values of k, that the Galois group Gal(K/ℚ) is the full symmetric group on d symbols. As a corollary, we construct eigenforms whose eigenvalues generate extensions of ℚ which are contained in no soluble extension of ℚ, answering a question of Ramakrishnan. This chapter is a slightly expanded version of [Bu].

Let k ≥ 2 be a fixed even integer and S_{k} the complex vector space of cusp forms of weight k for the full modular group SL_{2}(ℤ). Say the dimension of S_{k} is d. Let χ_{k} be the characteristic polynomial of the Hecke operator T_{2} acting on S_{k}, and let K/ℚ be the splitting field of χ_{k}.

The Galois group Gal(K/ℚ) acts faithfully on the d roots of χ_{k}, and hence we can identify Gal(K/ℚ) with a subgroup of the symmetric group Σ_{d} on d symbols. We restrict now to the case where k = 12l for some prime l. In this case, the dimension d of S_{k} is equal to l.

Theorem 7.1.1. If l ∈ {2, 3, 5, 7, 11, 13, 17, 19} and k = 12l then χ_{k} is irreducible and if K is its splitting field over ℚ then Gal(K/ℚ) ≅ Σ_{l}, the full symmetric group.

Remarks.

i) If l ≤ 7 then the Galois group of the splitting field of an irreducible polynomial of degree l can be evaluated using standard computer algebra packages like PARI-GP or MAPLE. For l > 7 the packages available at present, to our knowledge, are unable to deal with polynomials of degree l, so one has to use a trick.

ii) One has dim_{ℂ}S_{k} ≤ 7 for 2 ≤ k ≤ 98, k ≠ 96, and in these cases the referee* has calculated Gal(K/ℚ), using the cited computer algebra packages, and has observed that it is isomorphic to the full symmetric group.

Ramakrishnan has asked whether anything can be said about the number fields generated by coefficients of newforms, and in particular whether they are always contained in a soluble extension of ℚ.

Corollary 7.1.2. If l ∈ {2, 3, 5, 7, 11, 13, 17, 19} and k = 12l, then for any cuspidal eigenform f = Σa_{n}q^{n} of weight k for SL_{2}(ℤ) with a_{1} = 1, the field L_{f} generated over ℚ by the a_{n} has degree l over ℚ and the Galois closure K of L_{f} over ℚ satisfies Gal(K/ℚ) ≅ Σ_{l}.

Proof of corollary. If f = f_{1}, f_{2}, …, f_{l} are the distinct normalised eigenforms of level 1 and weight k, then Gal(\overline{ℚ}/ℚ) acts on these eigenforms (via the action on q-expansions) and the action is determined by the action of Gal(\overline{ℚ}/ℚ) on the coefficients of q^{2} of these eigenforms, as the coefficients of q^{2} are all distinct, being the l roots of χ_{k}. Hence we see that any element of Gal(\overline{ℚ}/ℚ) fixing a_{2} must fix f and deduce that L_{f} = ℚ(a_{2}). Because χ_{k} is irreducible by the theorem, we have that ℚ(a_{2}) has degree l over ℚ and its Galois closure is the splitting field of χ_{k}.

* A shorter version of this chapter has been submitted to the Journal of Number Theory, and this comment refers to the referee’s report on the paper.

□

The proof of the theorem uses the following lemma.

Lemma 7.1.3. Let l ∈ ℤ be a prime, and let P ∈ ℤ[X] be a monic irreducible polynomial of degree l, with splitting field K over ℚ. Say q is a prime such that the mod q reduction \overline{P} ∈ 𝔽_{q}[X] of P satisfies

\overline{P} = ∏_{i=0}^{r} h_{i}

for distinct irreducible polynomials h_{i} in 𝔽_{q}[X] with the following properties: i) The degree of h_{0} is 2 ii) The degree of h_{i} is odd, for 1 ≤ i ≤ r. Then Gal(K/ℚ) ≅ Σ_{l}. Proof. Let G be Gal(K/ℚ) identified as a subgroup of Σ_{l}. Now P has distinct roots

modulo q and hence if ℘ is a prime of K above q, there is a well-defined element Frob_{℘} ∈ G, the Frobenius element at ℘. Moreover, if d_{i} is the degree of h_{i}, we see that Frob_{℘} is the product of r +1 disjoint cycles of lengths d_{0}, d_{1}, …, d_{r}. Hence if t = ∏_{i=1}^{r} d_{i} then (Frob_{℘})^{t} is a transposition in G. So the transitive subgroup G of Σ_{l} contains a transposition. This forces G to be the whole of Σ_{l} as can be seen thus: Put an equivalence relation ∼ on the roots of P by setting a ∼ b if either a = b or the transposition (a, b) is an element of G. Then because the action is transitive on the roots, all equivalence classes have the same size, and because l is a prime there can be either 1 or l of them. But there is some transposition in G, and hence there is only 1 equivalence class and so G contains all transpositions and is thus Σ_{l}.

□

Proof of Theorem 7.1.1. Clearly it suffices to show that the characteristic polynomial χ_{k} of T_{2} acting on S_{k} for k = 12l, l ∈ {2, 3, 5, 7, 11, 13, 17, 19} satisfies the conditions of the lemma for some q. Note that we have reduced the problem to calculating the factorisation of χ_{k} modulo various primes (as χ_{k} can be proved to be irreducible in this way). The first (to our knowledge) calculations of these characteristic polynomials χ_{k}, for k ≤ 158, were done by Maeda in [Ma].

We now demonstrate the technique we used to calculate χ_{k} modulo many primes, without actually calculating χ_{k}. Set E_{0} = 1, and for t > 0 define

E_{t} = 1 + \frac{2}{ζ(1 − t)} ∑_{n≥1} σ_{t−1}(n)q^{n},

the normalised Eisenstein series of level 1 and weight t. Let Δ be the normalised cusp form of level 1 and weight 12. Set e_{j} = E_{12(r−j)} · Δ^{j} for j = 1, …, r. Note that {e_{1}, e_{2}, …, e_{l}} is a basis for S_{k}, the space of cusp forms of level 1 and weight k, and that e_{j} = q^{j} +(higher powers of q).

The explicit action of T_{2} on q-expansions enables us to explicitly calculate the matrix of T_{2} with respect to this basis. For l = 2 this calculation can be done by hand, as follows.

If l = 2 then k = 24, S_{k} has dimension 2, and has for a basis the two forms e_{1} and e_{2}, where

e_{1} = E_{12} · Δ = (1 + \frac{65520}{691}(q + (1 + 2^{11})q^{2} + (1 + 3^{11})q^{3} + …))(q − 24q^{2} + 252q^{3} + …)

= q + \frac{48936}{691}q^{2} + \frac{132852132}{691}q^{3} + \frac{8400219328}{691}q^{4} + …

and

e_{2} = Δ^{2} = (q − 24q^{2} + 252q^{3} − …)^{2}
= q^{2} − 48q^{3} + 1080q^{4} + ….

By the standard explicit formulae giving the q-expansion of e_{i}|T_{2} from the q-expansion of e_{i}, we see that

e_{1} | T_{2} = \frac{48936}{691}q + \frac{8401634496}{691}q^{2} + … = \frac{48936}{691}e_{1} + \frac{9807557760000}{477481}e_{2}

and

e_{2} | T_{2} = q + 1080q^{2} + … = e_{1} + \frac{697344}{691}e_{2},

the latter two equalities obtained by comparing coefficients of q and q^{2}. So with respect to the basis {e_{1}, e_{2} }, T_{2} is represented by the matrix

( \begin{matrix}48936/691 & 9807557760000/477481 \\ 1 & 697344/691 \end{matrix} ) ,

which has characteristic polynomial χ_{24} = X^{2} − 1080X − 20468736. Note that this polynomial is irreducible over ℚ, and so the theorem is proved for l = 2.

For l > 2, the relevant matrices are easily calculated by computer. The program in appendix A, written for Mathematica, calculates the matrix (λ_{ij}) representing the operator T_{2} with respect to the basis {e_{1}, …, e_{r} }, calling it wq[k], where here k must be a positive integer congruent to 0 mod 12, and defines chi[k,X] as the characteristic polynomial of this matrix, and chimodp[k,p,T] as the factorisation of this characteristic polynomial mod p, with a few exceptions on the prime p, as outlined below.

Once the matrix wq[k] is calculated, one can very quickly factorise its characteristic polynomial modulo many primes (by firstly reducing the matrix modulo p, then calculating the characteristic polynomial modulo p and factorising). In this way, the factorisation of χ_{k} modulo p can be determined very quickly for many primes p. For each l, we must find a prime p such that χ_{k} is irreducible modulo p, and another prime q such that χ_{k} satisfies the conditions of Lemma 7.1.3 modulo q. In each case, such primes were found, and the results are tabulated below, finishing the proof. Note that we have restricted to l > 7, because it is in these cases that one cannot use the usual computer algebra packages. Note also that this technique cannot be used to calculate χ_{k} modulo p when p is one of the finitely many primes dividing the denominator of one of the entries of wq[k]. Note that in the case l = 2 above, the only prime dividing any denominator is 691. In fact, in general any prime dividing a denominator of an entry of wq[k] can be shown to be an irregular prime.

l  prime  Complete factorisation of χ_{12l} modulo this prime.
11  q = 37  (X^{2} + 30X + 34)(X + 4)(X + 6)(X + 11)(X + 14)

(X + 15)(X + 21)(X + 26)(X + 31)(X + 33)

11 p = 479 X^{11} + 189X^{10} + 343X^{9} + 43X^{8} + 424X^{7} + 323X^{6}+
+58X^{5} + 100X^{4} + 131X^{3} + 307X^{2} + 192X + 133
13 q = 29 (X^{2} + 24X + 26)(X + 9)(X + 11)(X + 13)(X + 14)
(X + 16)(X + 17)(X + 22)(X + 24)(X + 26)(X + 27)(X + 28)
13 p = 353 X^{13} + 287X^{12} + 288X^{11} + 304X^{10} + 252X^{9}+
+76X^{8} + 139X^{7} + 218X^{6} + 62X^{5}+

+350X^{4} + 249X^{3} + 299X^{2} + 307X + 73

17 p = 263 X^{17} + 123X^{16} + 97X^{15} + 194X^{14} + 30X^{13} + 60X^{12}+
+99X^{11} + 2X^{10} + 94X^{9} + 209X^{8} + 203X^{7} + 157X^{6}+

+46X^{5} + 8X^{4} + 83X^{3} + 209X^{2} + 204X + 4

17 q = 317 (X^{2} + 123X + 261)(X^{15} + 91X^{14} + 66X^{13} + 205X^{12}+
+71X^{11} + 191X^{10} + 77X^{9} + 43X^{8} + 295X^{7} + 28X^{6}+
+168X^{5} + 253X^{4} + 18X^{3} + 54X^{2} + 186X + 4)
19 q = 53 (X^{2} + 41X + 6)(X + 13)(X + 17)(X + 21)(X + 23)
(X + 37)(X + 45)(X + 46)(X + 49)(X^{3} + 47X^{2} + 7X + 21)
(X^{3} + 35X^{2} + 36X + 52)(X^{3} + 18X^{2} + 35X + 25)
19 p = 251 X^{19} + 186X^{18} + 5X^{17} + 86X^{16} + 71X^{15} + 237X^{14} + 15X^{13}+
+145X^{12} + 113X^{11} + 30X^{10} + 155X^{8} + 162X^{7} + 70X^{6}+
+89X^{5} + 241X^{4} + 188X^{3} + 52X^{2} + 217X + 199
□

Appendix.

This is the MATHEMATICA program used to calculate the table in Chapter 7.

BeginPackage["HeckeOperatorCharPoly`"]
chi::usage = "chi[k,X] returns the characteristic polynomial of the second
Hecke operator T2 acting on the space of cusp forms of level 1 and weight k,
where k is a positive integer congruent to 0 mod 12."
chimodp::usage = "chimodp[k,p,X] returns the characteristic polynomial of the
second Hecke operator T2 acting on the space of cusp forms of level 1 and
weight k, factorised mod p, where k is a positive integer congruent to 0 mod
12 and p is a 'good' prime, i.e. not dividing the numerator of Zeta[1-12i],
i=1,2,...,k-1."
Begin["`Private`"]
chimodp[k_, p_, l_] :=
  Block[{fc, fp, sg}, fc = infacts[k]; fp = Mod[fc[[1]], p];
    If[fp == 0, "Bad Prime", Block[{ifp, tes, i},
      ifp = PowerMod[fp, -1, p]; tes = Mod[Mod[fc[[2]], p]*ifp, p];
       sg = If[Mod[k, 24] == 0, 1, -1];
       Factor[sg*CharacteristicPolynomial[tes, l], Modulus -> p]]]]
infacts[k_] := infacts[k] =
   Block[{fc}, fc = Product[Numerator[Zeta[1 - i]], {i, 12, k - 12, 12}]^2;
     {fc, wq[k]*fc}]
wq[k_] := wq[k] =
   Block[{qw, qwo}, qw = tt[k];
     qwo = Table[qw[[1]][[ii]][[jj]], {ii, k/12}, {jj, k/12}];
     qw[[2]] . Inverse[qwo]]
tt[k_] := Block[{u}, u = ma[k];
    Return[{u, Table[u[[y]][[2*z]] +
        If[Mod[z, 2] == 0, 2^(k - 1)*u[[y]][[z/2]], 0], {y, k/12}, {z, k/12}]}
      ]]
ma[k_] := Block[{t}, t = ba[k];
    Table[Coefficient[t[[f]], q^g], {f, k/12}, {g, k/6}]]
ba[k_] := Block[{d}, d = del[k/6]; Table[d^j*ei[k - 12*j, k/6], {j, k/12}]]
del[a_] := Series[q*Product[(1 - q^h)^24, {h, a}], {q, 0, a}]

ei[k_, a_] := 1 + (2*Sum[DivisorSigma[k - 1, i]*q^i, {i, a}])/Zeta[1 - k]
chi[k_, l_] := Block[{sg}, sg = If[Mod[k, 24] == 0, 1, -1];
    sg*CharacteristicPolynomial[wq[k], l]]
End[(* "`Private`" *)]
EndPackage[]

Notes on the program: del[a] and ei[k,a] are Δ and E_{k} respectively, expanded up to the term in q^{a}. ba[k] is the basis of cusp forms of weight k. ma[k] is this basis in table form, and tt[k] is the result of applying T_{2} to the elements of this basis, also in table form. wq[k] is the matrix representing T_{2}. Calculating wq[k] is reasonably fast—e.g. wq[132] was calculated in about a minute on a Sun SPARCstation 10. However, a typical entry of wq[132] is the quotient of 2 coprime integers, both of which may have well over 400 digits, which makes the calculation of the characteristic polynomial chi[132,X] very slow. However, after multiplying up wq[k] by an appropriate factor fc to get a matrix with integer entries, one can reduce mod p for any prime p not dividing fc, and then calculate the characteristic polynomial mod p. This calculation is very quick and has the advantage that wq[k] only need be calculated once. Unfortunately, this technique has the disadvantage that the calculation will fail if p divides the factor fc. However, it is easily shown that fc can be taken to be ∏_{j=1}^{(k/12)−1}(Numerator of ζ(1 − 12j))^{2} which is the product of certain irregular primes. It has been pointed out to us by the referee* that a more subtle choice of basis (see for example [La2] Chapter X, Theorem 4.4) would have avoided this problem.

The program was checked by modifying it so that it evaluated the matrices representing T_{n} for n ≥ 2, and checking that for a fixed k, the T_{n} were commuting, and T_{2}T_{3} = T_{6}, T_{4} = (T_{2})^{2} − 2^{k−1}, and so on.

The table in chapter seven can be calculated using the function chimodp. For example, the input

In[2]:= chimodp[132,37,X]

produces the result

Out[2]= (4 + X) (6 + X) (11 + X) (14 + X) (15 + X) (21 + X) (26 + X) (31 + X)
                            2
>    (33 + X) (34 + 30 X + X )

which is the first line of the table.

Note that the matrix wq[132] representing the endomorphism T_{2} takes about a minute to calculate, but is stored after this calculation, so the factorisation of χ_{132} modulo p for hundreds of primes can be done very quickly after this calculation has terminated.

* A shorter version of this chapter has been submitted to the Journal of Number Theory, and this comment refers to the referee’s report on the paper.

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