Modular parametrizations at primes of bad reduction.

Bas Edixhoven February 6, 2001

Contents

1 Introduction. 2 2 Modular curves, modular forms and Hecke operators. 3 3 Fourier expansions of new forms at various cusps. 10 4 A stable model for \overline{M}(Γ(p), Γ_{1}(N)) at p. 18 5 A stable model for \overline{M}(Γ_{0}(p^{2}), Γ_{1}(N))a at p. 25 6 Divisors and leading terms of new forms. 29 7 Modular parametrizations. 36 8 Manin constants. 37 9 More on parametrizations. 38

1 Introduction.

2

2 Modular curves, modular forms and Hecke operators.

2.1 Modular curves. The aim of this section is to explain how we use the arithmetic moduli theory of elliptic curves in what follows. We also discuss how the moduli theory of Drinfelds level structures on elliptic curves, as exposed in [19], extends to the moduli theory of the generalized elliptic curves of [11]. The delicate results in this section concerning the cusps in the “bad” characteristics will not really be used in the later sections.

2.1.1 Definition. For n a positive integer we define the following categories fibered in groupoids over the category (Sch) of schemes:

M(Γ(n)): objects are pairs (E/S, α), where E/S is an elliptic curve and α: (Z/nZ)^{2} → E(S) is a Drinfeld basis of E[n] (see [19] 3.1). Morphisms are cartesian squares compatible with the α’s.

M(Γ_{1}(n)): the objects are pairs (E/S, α), where α: Z/nZ → E(S) is a Z/nZ-structure on E (see [19] 3.2).

M(Γ_{0}(n)): (E/S, G), where G is a cyclic closed subgroup scheme of E[n] of rank n (see [19] 3.4).

It is well known that M := M(Γ(1)) is an algebraic stack for the etale topology on (Sch), of finite type, separated and smooth over Spec(Z) (for example see [11] III, Thm. 2.5). Let P denote Γ(n), Γ_{1}(n) or Γ_{0}(n). Then the obvious morphism M(P) → M is relatively representable, finite and flat and makes M(P) into an algebraic stack, regular and of dimension two, by [19], Thm. 5.1.1.

We can compactify M by adding a chart at infinity as in [15] I, §4 and IV, §5.5. For example one might take the degenerate elliptic curve over Z[t, (1 + 2^{6}3^{3}t)^{−1}] defined by:

y^{2} + xy = x^{3} − 2^{2}3^{2}tx − t,

which has c_{4} = 1 + 2^{6}3^{3}t, c_{6} = −c_{4}, Δ = t(1 + 2^{6}3^{3}t)^{2} and j = t^{−1} + 2^{6}3^{3} ([11] VI.1.6). This gives an algebraic stack \overline{M} which is proper and smooth over Spec(Z) and contains M as an open substack. We can then obtain compactifications \overline{M}(P) of the M(P) by normalizing \overline{M} in M(P) ([10] page 104). This procedure of compactifying is essentially the one followed in [19]. What happens at ∞ can be studied using the Tate curve ([19] 8.8).

In [11] Deligne and Rapoport give a moduli interpretation for \overline{M}: it is equivalent to the category of generalized elliptic curves whose fibres are irreducible. They also give a moduli interpretation for the \overline{M}(P) over Z[1/n], and in some cases over Z. Obvious candidates for moduli interpretations for the \overline{M}(P) are the categories \overline{M}(P)′ defined as follows. For P = Γ(n), Γ_{1}(n) or Γ_{0}(n) the objects of \overline{M}(P)′ are pairs (E/S, α), where E/S is a generalized elliptic curve and α is a Drinfeld basis (Z/nZ)^{2} → E^{reg}(S), or a Z/nZ-structure Z/nZ → E^{reg}(S), or a cyclic subgroup scheme of rank n of E^{reg}[n], such that in each case α meets all irreducible components of all geometric fibres of E/S. The \overline{M}(P)′ are stacks for the fpqc topology because the relevant descent data are effective; use [11] III, Lemma 2.1 and show that the level structures descend too.

2.1.2 Proposition. Let n be a positive integer and let P denote Γ(n), Γ_{1}(n) or Γ_{0}(n). For P = Γ(n) and for P = Γ_{1}(n) the obvious morphism M(P) → \overline{M}(P)′ extends to an isomorphism \overline{M}(P) ˜→ \overline{M}(P)′. The morphism M(Γ_{0}(n)) → \overline{M}(Γ_{0}(n))′ extends to an isomorphism \overline{M}(Γ_{0}(n)) → \overline{M}(Γ_{0}(n))′ if and only if n is square free. In all cases \overline{M}(P) is regular.

Proof. Let us first prove the assertions concerning Γ_{0}(n). For n square free one uses [11] V, Thm. 1.6. If n is not square free we can write n = p^{a+b}m with p prime, p |̸m, a ≥ b > 0. We claim that \overline{M}(Γ_{0}(n)) is not an algebraic stack because it has objects with infinitesimal automorphisms. Indeed, it is not hard to check that the standard p^{a}-gon over \overline{F}_{p}, equipped with a \overline{M}(Γ_{0}(n))-structure, has automorphism functor isomorphic to the group scheme μ_{p^{a}} × Z/2Z

over \overline{F}_{p}.

The proofs in the two remaining cases P = Γ(n) and P = Γ_{1}(n) will be given in four steps. Two of these steps depend on certain computations concerning Tate curves that will be done in §2.2. Step 1: the diagonal \overline{M}(P)′ ⟶ \overline{M}(P)′ × \overline{M}(P)′ is representable. Let (E_{1}/X, α_{1}) and (E_{2}/X, α_{2}) be in \overline{M}(P)′. By [11] III, Thm. 2.5 the functor Isom_{X}(E_{1}, E_{2}) is representable. The functor Isom_{X}((E_{1}, α_{1}), (E_{2}, α_{2})) is represented by the closed subscheme of Isom_{X}(E_{1}, E_{2}) defined by the compatibility of the universal isomorphism with α_{1} and α_{2}. Step 2: there exists an etale surjective \overline{X} → \overline{M}(P)′ with \overline{X} of finite type over Z, regular and everywhere of dimension two. Using Prop. 2.2.1 and [11] III, Thm. 2.3 (including the two remarks following it) it is straightforward to verify that \overline{M}(P)′ is an algebraic stack. Step 3: the diagonal \overline{M}(P)′ ⟶ \overline{M}(P)′ × \overline{M}(P)′ is finite. Let X → \overline{M}(P)′ × \overline{M}(P)′ be etale surjective. We have to show that Y := Isom_{X}(p_{1}^{∗}(E, α), p_{2}^{∗}(E, α)) → X is finite. Because Y → X is quasi-finite it suffices to prove that it is proper. For that we use the valuative criterion for properness. Because Y → \overline{M}(P)′ is etale we only need to consider morphisms Spec(D) → X, D a complete discrete valuation ring, under which the image of the generic point is in the interior of X. Then use [11] IV, Prop. 1.6. Step 4: M(P) → \overline{M}(P)′ extends to an isomorphism \overline{M}(P) ˜→ \overline{M}(P)′. Let \overline{Y} → \overline{M} and \overline{X} → \overline{M}(P)′ be etale surjective. Let \overline{X}′ := \overline{Y} ×\overline{M} \overline{X}. Then \overline{X}′ → \overline{M}(P)′ is also etale surjective. Let Y, X and X′ denote the pullbacks of \overline{Y}, \overline{X} and \overline{X}′ to M. Let Z := Y ×_{M} M(P) and let \overline{Z} → \overline{Y} be the normalization of Z → Y. So by definition \overline{Z} → \overline{M}(P) is etale surjective. Because \overline{X}′ is a chart for \overline{M}(P)′ it is regular, hence normal. Also \overline{X}′ → \overline{Y} is everywhere dominant, so it factors uniquely as \overline{X}′ → \overline{Z} → \overline{Y}. It follows from Prop. 2.2.1 that \overline{X}′ → \overline{Z} is etale surjective. So now we have two charts \overline{X}′ → \overline{M}(P) and \overline{X}′ → \overline{M}(P)′. The projections from \overline{X}′ ×_{\overline{M}(P)} \overline{X}′ to \overline{X}′ are etale and the morphism \overline{X}′ ×_{\overline{M}(P)} \overline{X}′ → \overline{X}′ ×\overline{X}′ is finite. The same holds for \overline{X}′ ×_{\overline{M}(P)′} \overline{X}′, hence these two fibre products are equal (both being the normalization of \overline{X}′ × \overline{X}′ in X′ ×_{M(P)} X′ → X′ × X′). It is clear that the two “compositions” in the groupoid structures coincide. □

2.2 Computations with Tate curves. In this section we prove some results that are needed in the proof of Prop. 2.1.2. Let \overline{G}_{m}^{q}/q^{Z} denote the Tate curve over Z[[q]] as constructed in [11] VII, §1.10; it is a generalized elliptic curve whose restriction to Spec(Z[[q]]/(q)) is the standard 1-gon. The restriction of \overline{G}_{m}^{q}/q^{Z} to Z((q)) is the elliptic curve denoted Tate(q) in [19] §8.8. By definition, \overline{G}_{m}^{q}/q^{Z} induces a formally etale morphism Spec(Z[[q]]) → \overline{M}. Let n ≥ 1 and let P denote Γ(n), Γ_{1}(n) or Γ_{0}(n). By construction (see [19] §8.11), Spec(Z[[q]])×\overline{M}\overline{M}(P) is the normalization of Spec(Z[[q]]) in Spec(Z((q))) ×_{M} M(P).

2.2.1 Proposition. Let α denote the universal level-P structure that \overline{G}_{m}^{q}/q^{Z} acquires over Spec(Z((q))) ×_{M} M(P). The object (\overline{G}_{m}^{q}/q^{Z}, α) of M(P) extends uniquely to an object over Spec(Z[[q]]) ×\overline{M} \overline{M}(P). The induced morphism Spec(Z[[q]]) ×\overline{M} \overline{M}(P) → \overline{M}(P)′ is formally etale for P = Γ(n) and for P = Γ_{1}(n).

Proof. The uniqueness is clear. Apart from this the proof is a long computation that, in order to save space, we will give only for P = Γ_{1}(n).

Let us first compute Spec(Z((q))) ×_{M} M(Γ_{1}(n)). As in [19] §8.7 let T [n] denote the ntorsion subgroup scheme of Tate(q) over Z((q)) (see also [11] VII §1.13). Then T [n] is given as the extension

0 ⟶ μ_{n} ⟶ T [n] ⟶ Z/nZ ⟶ 0

for which the inverse image of b, 0 ≤ b < n, is the μ_{n}-torsor given by the nth roots of q^{b}. It follows that T [n], as a scheme, is the disjoint union of the schemes T_{b} = Spec(Z((q))[X]/(X^{n} − q^{b})), with 0 ≤ b < n. The scheme Spec(Z((q))) ×_{M} M(Γ_{1}(n)) is the closed subscheme of T [n]_{T [n]} = T [n] ×_{Z((q))} T [n] over which the universal point (i.e., the diagonal) defines a Z/nZstructure. Let 0 ≤ b < n and let P denote the tautological point in T [n]_{T_{b}}(T_{b}). Let d be the order of b in Z/nZ, let m = n/d and let a = b/m. It is now easy to prove that for P to be a Z/nZ-structure is the same as for dP to be a Z/mZ-structure, i.e., for dP to be a primitive mth root of unity ([19] Thm. 1.12.9). We find (compare [19] Thm. 13.6.4):

Spec(Z((q))) ×_{M} M(Γ_{1}(n)) = ∐_{gcd(a,d)=1^{0<a<d|n}} Spec(Z((q))[X]/Φ_{m}(X^{d}q^{−a}))  (2.2.2)

We may check the statements in Prop. 2.2.1 separately for each term of the disjoint union in 2.2.2. These terms are permuted by the action of (Z/nZ)^{∗} on M(Γ_{1}(n)) and we only need to consider one term in each orbit. Since (Z/nZ)^{∗} → (Z/dZ)^{∗} is surjective we may restrict our atttention to terms with a = 1. Let d|n be given. We must compute the normalization of Z[[q]] in Z((q))[X]/Φ_{m}(X^{d}q^{−1}). Let ζ_{m} denote the image of X^{d}q^{−1} in the last ring; by definition ζ_{m} is a primitive mth root of unity. We get an isomorphism

Z((q))[X]/Φ_{m}(X^{d}q^{−1}) ˜ ⟶ Z[ζ_{m}]((X)) q ↦ ζ_{m}^{−1}X^{d}

It is clear that the normalization of Z[[q]] in Z[ζ_{m}]((X)) is equal to Z[ζ_{m}][[X]]. In particular the normalization is regular. By [11] VII, §1.4 and Construction 1.14, \overline{G}_{m}^{q}/q^{Z} and \overline{G}_{m}^{X}/q^{Z} are isomorphic over Z[ζ_{m}]((X)) and the tautological Γ_{1}(n)-structure α on \overline{G}_{m}^{q}/q^{Z} corresponds to the point X on \overline{G}_{m}^{X}/q^{Z}. It follows that (\overline{G}_{m}^{X}/q^{Z}, 1 ↦ X) is an object of \overline{M}(Γ_{1}(n))′ extending (\overline{G}_{m}^{q}/q^{Z}, α). It remains to prove that the induced morphism Spec(Z[ζ_{m}][[X]]) → \overline{M}(Γ_{1}(n))′ is formally etale.

Let (E/Spec(A), α) be an object of \overline{M}(Γ_{1}(n))′, with A local artinian. We let k denote the residue field of A and we suppose that after the base change to k the object (E/A, α) becomes isomorphic to the standard d-gon with a Γ_{1}(n)-structure such that α(1) lies on P^{1} × {1}. For N > 0 we define a scheme Z_{N} by:

Z_{N} = Isom_{A⊗_{Z}Z[ζ_{m}][[X]]/(X^{N})} ((E/A, α), (\overline{G}_{m}^{X}/q^{Z}, 1 ↦ X))

We want to show that for N large enough Z_{N} → Spec(A) is etale surjective. The surjectivity is obvious since Z_{N} ≠ ∅. The rest of the proof is done in two steps. Step1: Z_{N} → Spec(A) is unramified. We may suppose that A = k. Then we have to compare two objects of \overline{M}(P)′ over k[ζ_{m}][[X]]/(X^{N}). The first one is (E, α), with E the standard d-gon. The second one is (\overline{G}_{m}^{X}/q^{Z}, 1 ↦ X). Let us first consider the image of the non-smooth locus for both of them. Locally at a non-smooth point the first one is given by a local equation uv = 0, hence the image of the non-smooth locus is all of Spec(k[ζ_{m}][[X]]/(X^{N})). A local equation for the second one at a non-smooth point is uv = X, hence the image of the non-smooth locus is given by the equation X = 0. It follows that Z_{N} lies over Spec(k[ζ_{m}]), over which both generalized elliptic curves become isomorphic to the standard d-gon. Let us now use the Γ_{1}(n)- structures. In both cases we can interpret d · α(1) as an element of μ_{m}(k[ζ_{m}]). In the first case z := d · α(1) lies in μ_{m}(k) and in the second case d · α(1) = ζ_{m}. We find that Z_{N} lies over Spec(k[ζ_{m}]/(ζ_{m} − z)), i.e., over Spec(k). It follows that Z_{N} = Aut_{k}(E/k, α), which is easily seen to be either Spec(k) or (Z/2Z)_{k}. Step2: Z_{N} → Spec(A) is etale for N large enough. Let Z′ = Spec(B) be a connected (i.e., irreducible) component of Z_{N}. We have to show that Z′ → Spec(A) is etale. After replacing A by a finite etale extension we may suppose that B has residue field k. The canonical map k → k⊗_{A} B is then an isomorphism since Z′ → Spec(A) is unramified. By Nakayama’s Lemma, the map A → B is surjective. It remains to show that A → B is injective for N large enough. In order to do that it is sufficient to show that there exists a faithful flat extension A → A′ such that Spec(A′) → Spec(A) factors through Z′ → Spec(A), i.e., such that Z′ → Spec(A) acquires a section. That we will do now.

After a faithful finite flat base change (obtained by extracting a d-th root of some element of A^{∗}, to be precise) there exists a Γ_{1}(d)-structure β on E/Spec(A). By definition, β is an embedding of (Z/dZ)_{A} into E^{reg}. Let E′ be the quotient of E by the action of Z/dZ via β. Then E′/Spec(A) is a deformation of the standard 1-gon. By [11] III, §1.4.2 and Lemme 1.4.3 E′/Spec(A) arises by pullback from \overline{G}_{m}^{q}/q^{Z} over Z[[q]]. By [11] II, Prop. 1.17 (E/Spec(A), β) arises by pullback from (\overline{G}_{m}^{q}/q^{dZ}, q) over Z[[q]]. One now checks easily that over A[a]/(Φ_{m}(a^{d})) the pair (E, α) arises from (\overline{G}_{m}^{q}/q^{dZ}, a · q). But this last one is isomorphic to (\overline{G}_{m}^{X}/q^{Z}, X) via

Z[ζ_{m}][[X]] ⟶ Z[[q]][a]/Φ_{m}(a^{d}) ζ_{m} ↦ a^{−d} X ↦ qa

This finishes the proof of Prop. 2.2.1. □

2.3 Modular forms and q-expansions. Let N and k be positive integers. Let R be a ring in which N is invertible. We define the R-module of modular forms over R of level N and weight k as:

M(N, k)_{R} := H^{0}(\overline{M}(Γ_{1}(N))_{R}, \underline{ω}^{⊗k})  (2.3.1)

where \underline{ω} is the invertible sheaf 0^{∗}Ω_{E/\overline{M}}^{1}, or, more precisely, its pullback to \overline{M}(Γ_{1}(N))_{R}. In more down-to-earth terms this means that a modular form f of type (N, k) over R is a rule, compatible with cartesian squares, that associates to each (E/S/R, α) in \overline{M}(Γ_{1}(N))_{R} a section f(E/S, α) of \underline{ω}_{E/S}^{⊗k}. Of course we could have defined M(N, k)_{R} without N being invertible in R, but for example for R the ring of integers in a number field, \overline{M}(Γ_{1}(N))_{R} is not always the best possible model.

If N > 4 then the stack \overline{M}(Γ_{1}(N))_{R} is represented by the curve X_{1}(N)_{R} and modular forms of type (N, k) over R are just global sections of the invertible sheaf \underline{ω}^{⊗k} on X_{1}(N)_{R}. For general N we have the following description of M(N, k)_{R}. Let n ≥ 3, then \overline{M}(Γ(n), Γ_{1}(N))_{R[1/n]} is represented by a (possibly non-connected) smooth proper curve over R[1/n], say X, on which GL_{2}(Z/nZ) acts. Then M(N, k)_{R} ⊗_{R} R[1/n] can be identified with the submodule of GL_{2}(Z/nZ)-invariant sections of \underline{ω}^{⊗k} over X.

When R contains the Nth roots of unity one obtains the q-expansions of f at the various cusps of \overline{M}(Γ_{1}(N))_{R} by evaluating f on pairs (Tate(q^{d}), α), where Tate(q^{d}) is the Tate curve G_{m}/q^{dZ} over R[[q]](q^{−1}) and d|N. Explicitly: the q-expansion f_{d,α}(q) of f at the cusp (Tate(q^{d}), α) is the power series f(Tate(q^{d}), α)/(dt/t)^{⊗k} in R[[q]]. A form f is called a cusp form if all its q-expansions have constant term equal to zero.

The group (Z/NZ)^{∗} acts on \overline{M}(Γ_{1}(N)) by:

⟨a⟩ : (E/S, α) ↦ (E/S, aα),  (2.3.2)

for a ∈ (Z/NZ)^{∗}. This gives an action by (Z/NZ)^{∗} on modular forms:

(⟨a⟩^{∗}f)(E/S, α) = f(E/S, aα).  (2.3.3)

Let ε: (Z/NZ)^{∗} → R^{∗} be a character. A modular form f of type (N, k) is said to be of type (N, k, ε) if ⟨a⟩^{∗}f = ε(a)f for all a ∈ (Z/NZ)^{∗}. Of course, we always have ε(−1)f = (−1)^{k}f. We denote the R-module of forms of type (N, k, ε) by M(N, k, ε)_{R}, and its submodule of cusp forms by M^{o}(N, k, ε)_{R}.

For more details concerning modular forms in this setting see [18], Chapter 1 or [11], VII, §3.

2.4 Hecke operators. The R-modules M(N, k)_{R} with k ≥ 2 are equipped with certain endomorphisms, called Hecke operators. For each n ≥ 1 we have an operator T_{n}^{∗}. The algebra generated by all T_{n}^{∗} and ⟨a⟩^{∗} acting on M(N, k)_{R} is commutative and generated by the T_{l}^{∗} for l prime and the ⟨a⟩^{∗}. It follows that the T_{n}^{∗} also act on the submodules M(N, k, ε)_{R}. For a construction of the T_{n}^{∗} we refer to [18], §1.11; the hypothesis “k ≥ 2” is needed to use base change: M(N, k)_{R} = M(N, k)_{Z[1/N]} ⊗ R. There is no problem to construct T_{n}^{∗} on forms of weight one if n is invertible in R. The action of T_{n}^{∗} on M(N, k)_{Q} (k ≥ 1) is given by:

(T_{n}^{∗}f)(E/\overline{Q}, α) = n^{−1} ∑_{ϕ} ϕ^{∗}(f(E′/\overline{Q}, ϕα))  (2.4.1)

where ϕ: E → E′ ranges over the isogenies of degree n with source E such that ϕ is injective on the image of α. If f in M(N, k, ε)_{R} has q-expansion ∑ a_{n}q^{n} at (Tate(q), 1↦ζ_{N}) then we have for p prime:

(T_{p}^{∗}f)(Tate(q), 1↦ζ_{N}) = ∑_{n} (a_{np} + ε(p)p^{k−1}a_{n/p}) q^{n} (dt/t)^{⊗k}  (2.4.2)

with the conventions that a_{n/p} = 0 if p |̸n and ε(p) = 0 if p|N.

2.5 Galois representations. Let f be a cusp form over \overline{Q} of some type (N, k, ε) with k ≥ 2 and suppose that f is a common eigenvector for all the T_{p}^{∗}, p |̸N, with eigenvalues a_{p} ∈ \overline{Q}. Let K ⊂ \overline{Q} be the field generated by the a_{p}, p |̸N, and the ε(a), a ∈ (Z/NZ)^{∗}; then [K : Q] < ∞. Let λ be a finite place of K and let l be its residue characteristic. Let G_{Q} = Gal(\overline{Q}/Q) denote the absolute Galois group of Q. According to a theorem of Deligne ([12], Thm. 6.1; see also [5]) there exists a unique (up to isomorphism) continuous semi-simple representation ρ_{λ}: G_{Q} → GL_{2}(K_{λ}) which is unramified outside Nl and has the property that trace(ρ_{λ}(Frob_{p})) = a_{p} and det(ρ_{λ}(Frob_{p})) = ε(p)p^{k−1} for all p |̸Nl. In these equalities ρ_{λ}(Frob_{p}) denotes the image of a Frobenius element at p (unique up to conjugation). We will briefly describe a construction of ρ_{λ}; detailed constructions can be found in [5], [20] and [24].

First of all we may replace f by the unique new form with eigenvalues a_{p} for p |̸N (see [8], §2.4 and [2]); the field K remains the same, the level N is replaced by a divisor of it. Now f is an eigenform for all T_{n}^{∗}, n ≥ 1, with eigenvalues a_{n} ∈ K, of type (N, k, ε). Note that the a_{n} (n ≥ 1) generate K since f is defined over the field generated by the a_{n} and is an eigenvector for the ⟨a⟩^{∗}.

Let Y denote M(Γ_{1}(N))_{Q} and let π: E → Y be the universal elliptic curve. We have the sheaf F^{k} := Sym^{k−2}(R^{1}π_{∗}Q) on Y (C) and the sheaf F_{l}^{k} := Sym^{k−2}(R^{1}π_{∗}Q_{l}) on Y_{et}. There is the Shimura isomorphism ([7], Thm. 2.10):

H_{par}^{1}(Y (C), F^{k}) ⊗_{Q} C ˜ ⟶ M^{o}(N, k)_{C} ⊕ \overline{M^{o}(N, k)_{C}}  (2.5.1)

where H_{par} denotes “parabolic” cohomology: the image of the cohomology with compact support in the ordinary cohomology. This isomorphism is compatible with the T_{n}^{∗} and ⟨a⟩^{∗} acting on both sides in the usual way (see [7], Prop. 3.18).

Next let H be the opposite of the Q-algebra in End_{Q}(M^{o}(N, k)_{Q}) generated by the T_{n}^{∗} and the ⟨a⟩^{∗}. By definition, M^{o}(N, k)_{Q} is a right H-module and we denote by h^{∗} the endomorphism induced by h ∈ H. The algebra H is commutative and of finite dimension over Q. Let M^{o}(N, k)_{Q}^{∨} be the left H-module Hom_{Q}(M^{o}(N, k)_{Q}, Q); say that h ∈ H induces h_{∗} := h^{∗∨}. We claim that M^{o}(N, k)_{Q}^{∨} is free of rank 1 as H-module. Namely, it is faithful and for each maximal ideal m of H, M^{o}(N, k)_{Q}^{∨}/mM^{o}(N, k)_{Q}^{∨} = (M^{0}(N, k)[m])^{∨} is a 1-dimensional H/m-vector space because of q-expansions. It follows that H_{par}^{1}(Y (C), F^{k})^{∨} ⊗_{Q} C is a free H ⊗_{Q} C-module of rank 2. Hence H_{par}^{1}(Y (C), F^{k})^{∨} is a free H-module of rank 2 and H_{par}^{1}(Y\overline{Q}, F_{l}^{k})^{∨} is a free H ⊗_{Q} Q_{l}-module of rank 2. Let ϕ: H → \overline{Q} be the ring homomorphism such that for all h ∈ H: h^{∗}(f) = ϕ(h)f. Let m := ker(ϕ); then K = im(ϕ) = H/m and we have a surjection H⊗_{Q}Q_{l} → K_{λ}. Let V_{λ} := H_{par}^{1}(Y\overline{Q}, F_{l}^{k})^{∨}⊗_{H⊗Q_{l}}K_{λ}. Then V_{λ} is a 2-dimensional K_{λ}-vector space with a continuous action of G_{Q}. To be precise: we let σ ∈ G_{Q} act on H_{par}^{1}(Y ×_{SpecQ}Spec\overline{Q}, F_{l}^{k})^{∨} by (id × Spec(σ^{−1}))^{∗∨} (note that this is covariant in σ).

The action of G_{Q} on V_{λ} is unramified at p |̸Nl because π: E → Y has good reduction at such p. Congruence formulas as in [7], §4 show that Frob_{p} (p |̸Nl) satisfies the polynomial X^{2} − a_{p}X + ε(p)p^{k−1}. In order to show that this polynomial is the characteristic polynomial of Frob_{p} it suffices to prove that Frob_{p} has determinant ε(p)p^{k−1}. This one can do by computing the action of Frob_{p} on some non-degenerate alternating K_{λ}-bilinear form on V_{λ}. Such a form is found as follows.

The Weil pairing on π: E → Y gives a non-degenerate alternating pairing on R^{1}π_{∗}Q_{l} with values in the sheaf Q_{l}(−1) on Y. This induces a non-degenerate pairing on F_{l}^{k} with values in Q_{l}(2−k); this pairing is symmetric if k is even and alternating if k is odd. As explained in [7], §3.20, Poincaré duality then gives a non-degenerate Q_{l}-bilinear form (·|·) on H_{par}^{1}(Y\overline{Q}, F_{l}^{k})^{∨} with values in Q_{l}(k−1); one has (y|x) = (−1)^{k−1}(x|y) for all x and y. Let w_{ζ_{N}} be the automorphism of Y_{Q(ζ_{N})} = M(Γ_{1}(N))_{Q(ζ_{N})} defined by:

w_{ζ_{N}}: (E/S/Q(ζ_{N}), α) ↦ ((E′/S/Q(ζ_{N}), α′)  (2.5.2)

where β: E → E′ is the N-isogeny with ker(β) = im(α), im(α′) = ker(β^{t}) and ⟨α(1), α′(1)⟩_{β} = ζ_{N}. Here β^{t}: E′ → E denotes the dual of β and ⟨·, ·⟩_{β} denotes the pairing of [19], §2.8. The use of w_{ζ_{N}} is that if T_{n}^{t} denotes the correspondence (on Y_{Q(ζ_{N})}) dual to T_{n}, we have T_{n}^{t} = w_{ζ_{N}} T_{n}w_{ζ_{N}}^{−1} for all n ≥ 1 and ⟨a⟩^{−1} = w_{ζ_{N}} ⟨a⟩w_{ζ_{N}}^{−1} for all a ∈ (Z/NZ)^{∗}. We let w_{ζ_{N}} act on H_{par}^{1}(Y\overline{Q}, F_{l}^{k}) by:

w_{ζ_{N}}^{∗}: H_{par}^{1}(Y\overline{Q}, F_{l}^{k}) ⟶^{w_{ζ_{N}}^{∗}} H_{par}^{1}(Y\overline{Q}, w_{ζ_{N}}^{∗} F_{l}^{k}) ⟶^{Sym^{k−2}β^{∗}} H_{par}^{1}(Y\overline{Q}, F_{l}^{k})  (2.5.3)

where β: E → w_{ζ_{N}}^{∗} E; we let w_{ζ_{N} ∗} := w_{ζ_{N}}^{∗∨}. For h ∈ H let h_{∗}^{t} denote the endomorphism of H_{par}^{1}(Y\overline{Q}, F_{l}^{k})^{∨} which is dual to h_{∗} with respect to (·|·). Then we have h_{∗}^{t} = w_{ζ_{N} ∗}h_{∗}w_{ζ_{N} ∗}^{−1} for all h ∈ H and w_{ζ_{N} ∗}^{t} = (−1)^{k}w_{ζ_{N} ∗}. In other words, all h_{∗} are self-adjoint with respect to the non-degenerate alternating bilinear form ⟨x|y⟩ := (x|w_{ζ_{N} ∗}y) on H_{par}^{1}(Y\overline{Q}, F_{l}^{k})^{∨}.

The K_{λ}-vector space V_{λ} is canonically isomorphic to (H_{par}^{1}(Y\overline{Q}, F_{l}^{k})^{∨} ⊗_{Q_{l}} K_{λ}) ⊗_{H⊗_{Q}K_{λ}} K_{λ}, where H ⊗_{Q} K_{λ} → K_{λ}: a ⊗ b ↦ ϕ(a)b. We extend ⟨·|·⟩ to a K_{λ}-bilinear form, still denoted by ⟨·|·⟩, on H_{par}^{1}(Y\overline{Q}, F_{l}^{k})^{∨} ⊗_{Q_{l}} K_{λ}. Since f is a new form the quotient K = H/mH of H is actually a direct factor: H = K ×H′ as rings, for some (unique) H′. It follows that H ⊗_{Q} K_{λ} = K_{λ} ×H′′ and that H_{par}^{1}(Y\overline{Q}, F_{l}^{k})^{∨} ⊗_{Q_{l}} K_{λ} = V_{λ} ⊕ V′_{λ}. This direct sum decomposition is orthogonal with respect to ⟨·|·⟩ because the corresponding idempotents, lying in H ⊗_{Q} K_{λ}, are self-adjoint. It follows that the restriction of ⟨·|·⟩ to V_{λ} has all the properties we want, i.e., it is K_{λ}-bilinear, non-degenerate and alternating. One easily checks that Frob_{p} acts on it as desired.

3 Fourier expansions of new forms at various cusps.

3.1 The general problem. Let M be a positive integer. Let f be a new form on Γ_{1}(M) of some weight k, with character ε and with coefficients in \overline{Q}. Then we ask the following question: how are the q-expansions of f at the various cusps related? For M square free this problem has been solved by Asai in [1]. His method uses the Atkin-Lehner operators W_{d} and the diamond operators ⟨a⟩. It works because for M square free these operators permute the cusps transitively, but for general M this is not necessarily true. We will solve the problem in a situation described below. It might well be that the general case can be elegantly dealt with using some explicit information concerning the Kirillov models for the local factors of the automorphic representation attached to f.

3.2 Our problem. Let N be a positive integer, p a prime not dividing N and f a new form on X_{0} := \overline{M}(Γ_{0}(p^{2}), Γ_{1}(N))\overline{Q} of some weight k and character ε: (Z/NZ)^{∗} → \overline{Q}^{∗}. We want to relate the q-expansions of f at the p+1 cusps of X_{0} lying over the cusp (Tate(q), 1↦ζ_{N}) of \overline{M}(Γ_{1}(N))\overline{Q}, where ζ_{N} is a fixed Nth root of unity. The most natural way to do this seems to consist of first passing to a cover X of X_{0} on which a group G acts, permuting the relevant cusps transitively, and then applying the representation theory of G. In our case we take X = \overline{M}(Γ(p), Γ_{1}(N))\overline{Q}. Note that by definition 2.1.1 a Γ(p)-structure is not required to be symplectic. The group G will then be GL_{2}(F_{p}) and not its subgroup SL_{2}(F_{p}). We prefer to work with GL_{2}(F_{p}) because its representation theory is less complicated.

Let π: X → X_{0} be the morphism mapping (E/S, ϕ, α) to (E_{1}/S, ker(ϕ_{2}^{∘}ϕ_{1}^{t}), ϕ_{1}α) where ϕ_{1}: E → E_{1} and ϕ_{2}: E → E_{2} are the p-isogenies with ker(ϕ_{1}) = ⟨ϕ(1, 0)⟩ and ker(ϕ_{2}) = ⟨ϕ(0, 1)⟩ (see [19] 11.3.5). We let g in G = Aut(F_{p}^{2}) act from the right on X by sending (E/S, ϕ, α) to (E/S, ϕ^{∘}g, α). Then π is the quotient map for the action of the diagonal subgroup T of G. For f_{0} in H^{0}(X_{0}, \underline{ω}^{⊗k}) we define π^{∗}f_{0} in H^{0}(X, \underline{ω}^{⊗k}) by (π^{∗}f_{0})(E/S, ϕ, α) = ϕ_{1}^{∗}(f_{0}(E_{1}/S, ker(ϕ_{2}^{∘}ϕ_{1}^{t}), ϕ_{1}α)). We define operators T_{l}^{∗} for l ≠ p and ⟨a⟩^{∗} for a in (Z/NZ)^{∗} on H^{0}(X, \underline{ω}^{⊗k}) in the obvious way. The T_{l}^{∗} and the ⟨a⟩^{∗} commute with the action (from the left) of G. The injection π^{∗} of H^{0}(X_{0}, \underline{ω}^{⊗k}) into H^{0}(X, \underline{ω}^{⊗k}) intertwines the T_{l}^{∗} and the ⟨a⟩^{∗} on both sides. Now π^{∗}f generates a representation of G, namely the linear span V of the gπ^{∗}f, g ∈ G.

3.2.1 Lemma. The subspace of T -invariants V^{T} of V has basis π^{∗}f and V is irreducible.

Proof. The first statement follows from the multiplicity one principle on X_{0} applied to the T_{l}^{∗}, l ≠ p, and the ⟨a⟩^{∗} (recall that f is a new form so that we don’t need T_{p}^{∗}). Write V = ⊕_{1}^{r}V_{i} with all V_{i} irreducible subrepresentations. Then π^{∗}f = ∑ v_{i} with v_{i} ∈ V_{i}. Since π^{∗}f generates V we have v_{i} ≠ 0 for all i, hence dim V^{T} ≥ r. □

Let us now consider q-expansions. The Tate curve Tate(q) = G_{m}/q^{Z} over \overline{Q}((q)) has its ptorsion rational over \overline{Q}((q^{1/p})): the p-torsion points are the ζ_{p}^{a}q^{b/p} where ζ_{p} is a fixed pth root of unity and a, b are in F_{p}. It follows that the set of Γ(p)-structures ϕ on Tate(q) over \overline{Q}((q^{1/p})) is a G-torsor where g ∈ G acts by ϕ ↦ ϕ^{∘}g. We have an evaluation map:

H^{0}(X, \underline{ω}^{⊗k}) ⟶ ⊕_{ϕ}\overline{Q}[[q^{1/p}]], f ↦ (f(Tate(q), ϕ, 1↦ζ_{N})/(dt/t)^{⊗k})_{ϕ}  (3.2.2)

We let G act on ⊕_{ϕ}\overline{Q}[[q^{1/p}]] by permuting the factors: an element g in G sends (∑ a_{n,ϕ}q^{n/p})_{ϕ} to (∑ a_{n,ϕ}q^{n/p})_{ϕ^{∘}g^{−1}}; this action is compatible with the map 3.2.2. We identify the Galois group of \overline{Q}((q^{1/p})) over \overline{Q}((q)) with F_{p} by letting λ ∈ F_{p} send q^{1/p} to ζ_{p}^{λ}q^{1/p}. Then λ ∈ F_{p} induces the automorphism σ(λ) of the p-torsion of Tate(q) given by: ζ_{p}^{a}q^{b/p} ↦ ζ_{p}^{a+λb}q^{b/p}. Suppose that f(Tate(q), ϕ, 1↦ζ_{N}) = ∑ a_{n,ϕ}q^{n/p}(dt/t)^{⊗k} and let λ ∈ F_{p}. Base changing by \overline{Q}((q^{1/p})) → \overline{Q}((q^{1/p})), q^{1/p} ↦ ζ_{p}^{λ}q^{1/p} then gives: f(Tate(q), (σλ)^{∘}ϕ, 1↦ζ_{N}) = ∑ a_{n,ϕ}ζ_{p}^{λn}q^{n/p}(dt/t)^{⊗k}. In other words, the image of the map 3.2.2 is contained in the subspace consisting of (∑ a_{n,ϕ}q^{n/p})_{ϕ} such that a_{n,(σλ)ϕ} = ζ_{p}^{λn}a_{n,ϕ} for all n, λ and ϕ. Note that both the condition on the image of f we just found and the action of G do not mix terms of different degrees in q^{1/p}, so that it makes sense to project on the degree n parts.

For n ∈ Z let W_{n} be the space of functions h: G → \overline{Q} such that h(({1 \atop 0} {a \atop 1})x) = ζ_{p}^{na}h(x) for all a ∈ F_{p}. We let G act on W_{n} by right translations. Also let ϕ_{0} be the Γ(p)-structure on Tate(q) with ϕ_{0}(a, b) = ζ_{p}^{a}q^{b/p}. Then we can restate the results above as the following proposition.

3.2.3 Proposition. For f ∈ H^{0}(X, \underline{ω}^{⊗k}) let us define the a(f, n, ϕ) by

f(Tate(q), ϕ, 1↦ζ_{N}) = (∑ a(f, n, ϕ)q^{n/p}) (dt/t)^{⊗k}.

Then, for all n, mapping f to the function x ↦ a(f, n, ϕ_{0}^{∘}x) on G gives a G-equivariant map H^{0}(X, \underline{ω}^{⊗k}) → W_{n}. □

The usefulness of this result clearly depends on the multiplicity of the irreducible representation V in W_{n}.

3.3 Applying the representation theory of GL_{2}(F_{p}). The notation is as in §3.2. We have G = GL_{2}(F_{p}), B ⊂ G is the upper triangular Borel subgroup, T is the diagonal subgroup and U is the subgroup of order p of B. First of all we need a character table for G. The one given below is copied from [6].

3.3.1 Table. The character table of GL_{2}(F_{p}).

conjugacy class of  π(α, β), α ≠ β  π(Λ), Λ^{p} ≠ Λ α^{∘} det  π^{−}(α)
({x \atop 0} {0 \atop x}) x ∈ F_{p}^{∗}  (p + 1)α(x)β(x)  (p − 1)Λ(x)  α(x)^{2}  pα(x)^{2}
({x \atop 0} {0 \atop y}) x, y ∈ F_{p}^{∗} x ≠ y α(x)β(y) + α(y)β(x)  0  α(x)α(y) α(x)α(y)
({x \atop 0} {1 \atop x}) x ∈ F_{p}^{∗}  α(x)β(x)  −Λ(x)  α(x)^{2}  0
({z \atop 0} {0 \atop z^{p}}) z ∈ F_{p^{2}}^{∗} z^{p} ≠ z  0  −Λ(z) − Λ(z^{p}) α(z^{p+1}) −α(z^{p+1})

In this table α and β denote characters F_{p}^{∗} → \overline{Q}^{∗} and Λ denotes a character F_{p^{2}}^{∗} → \overline{Q}^{∗}. As is well known, the representation W_{n} is isomorphic to the induced representation Ind_{U}^{G}ρ_{n}, where ρ_{n} is the one-dimensional representation ρ_{n}({1 \atop 0} {a \atop 1}) = ζ_{p}^{na} of U. It is now a trivial matter to compute the multiplicities dim Hom_{G}(V, W_{n}) = dim Hom_{U}(Res_{U}^{G}V, ρ_{n}) and the dimensions dim V^{T}. One finds the following table.

3.3.2 Table.

V  π(α, β), α ≠ β π(Λ), Λ^{p} ≠ Λ α^{∘} det  π^{−}(α)
dim Hom_{G}(V, W_{n})  1 if p |̸n  1 if p |̸n  0 if p |̸n  1
2 if p|n  0 if p|n  1 if p|n
dim V^{T}  δ(αβ, 1)  δ(Λ|_{F_{p}}^{∗}, 1)  δ(α, 1) δ(α, 1) + δ(α^{2}, 1)

In this table the symbol δ denotes the function that takes the value 1 or 0 according to when its two arguments are equal or not.

Let us go back to the situation of §3.2. So f is a new form on X_{0}, V is the irreducible subrepresentation of H^{0}(X, \underline{ω}^{⊗k}) generated by π^{∗}f and π^{∗}f is a basis for V^{T}. Looking at the last row of Table 3.3.2 one sees that V ≅ π(α, β) implies β = α^{−1} and α ≠ 1, that V ≅ π(Λ) implies Λ|_{F_{p}}^{∗} = 1, that V ≅ α^{∘} det is impossible and that V ≅ π^{−}(α) implies that α has order two. Let n be a positive integer and let h_{n} ∈ W_{n} be given by h_{n}(x) = a(π^{∗}f, n, ϕ_{0}^{∘}x) as in Prop. 3.2.3. Looking at the second row in Table 3.3.2 one sees that h_{n} is determined up to scalar multiple by the conditions that it is T -invariant and that it generates a quotient representation of V, except in the case where V ≅ π(α, β) and p|n. To get such a function h′_{n} we can use the central idempotents in the group rings of G, T and U corresponding to the representations we want h′_{n} to be in. The central idempotent e_{χ} associated to an irreducible character χ of a finite group G is given by e_{χ} = χ(1)/(#G) ∑_{y} χ(y^{−1})δ_{y}, where the sum ranges over the elements of G (we view the group ring of G as the ring \overline{Q}[G] of functions on G with convolution as multiplication). From now on let χ be the character of G on V. We can start with an arbitrary element of \overline{Q}[G], project it into W_{n}, then into its χ-part and finally into the T -invariants. Starting with δ_{e}, the delta function at the unit element of G, gives us the function

h′_{n}(x) = ∑_{t∈T,u∈U} ρ_{n}(u^{−1})χ(uxt)  (3.3.3)

By construction we have that h′_{n} ≠ 0 if dim Hom_{G}(V, W_{n}) ≠ 0.

3.3.4 Proposition. In the situation of §3.2 we define the a(n, x) by:

π^{∗}f(Tate(q), ϕ_{0}^{∘}x, 1↦ζ_{N}) = (∑ a(n, x)q^{n/p}) (dt/t)^{⊗k}.

Suppose that p |̸n if χ = π(α, α^{−1}) for some α. Then the vector (a(n, x))_{x∈G} is a multiple of the vector (h′_{n}(x))_{x∈G} as defined in 3.3.3. □

Using this proposition we can almost solve the problem posed in the beginning of §3.2. Let us write:

f(Tate(q), μ_{p^{2}}, 1↦ζ_{N}) = (∑ a_{n}q^{n})(dt/t)^{⊗k}
f(Tate(ζ_{p}q), ⟨q^{1/p}⟩, 1↦ζ_{N}) = (∑ a′_{n}q^{n})(dt/t)^{⊗k}
f(Tate(q^{p^{2}}), ⟨q⟩, 1↦ζ_{N}) = (∑ a′′_{n}q^{n})(dt/t)^{⊗k}

(3.3.5)

Applying the definitions of π^{∗}f and ϕ_{0} one finds:

\begin{aligned}a_{n} & = & a(n, ({0 \atop 1} {1 \atop 0})) \\ a′_{n} & = & a(n, ({1 \atop 1} {0 \atop 1})) \\ a′′_{n} & = & ε(p)^{−1}p^{−k}a(n, ({1 \atop 0} {0 \atop 1})) \end{aligned}  (3.3.6)

A direct calculation shows that for p |̸n we have

h′_{n}({1 \atop 0} {0 \atop 1}) = p(p−1),  h′_{n}({0 \atop 1} {1 \atop 0}) = ±p(p−1)  (3.3.7)

where ± = α(−1) if χ = π(α, α^{−1}) or χ = π^{−}(α), and ± = −Λ(z_{0}) if χ = π(Λ) and z_{0}^{p−1} = −1. It follows that for p |̸n we have

a′_{n} = ±p^{−1}(p − 1)^{−1}h′_{n}({1 \atop 1} {0 \atop 1}) · a_{n},  a′′_{n} = ±ε(p)^{−1}p^{−k}a_{n}  (3.3.8)

where the sign ± is as in 3.3.7. It remains to see what the a′_{n} and a′′_{n} are for p|n. For the a′′_{n} this can be solved using the Atkin-Lehner automorphism w_{p^{2}} of X_{0} defined as follows:

w_{p^{2}}: (E/S, ker(ϕ), α) ↦ (E′/S, ker(ϕ^{t}), ϕα)  (3.3.9)

where ϕ: E → E′ is cyclic of degree p^{2}. On H^{0}(X_{0}, \underline{ω}^{⊗k}) we define an operator w_{p^{2}}^{∗} by:

(w_{p^{2}}^{∗}f_{0})(E/S, ker(ϕ), α) = ϕ^{∗}(f_{0}(E′, ker(ϕ^{t}), ϕα))  (3.3.10)

One easily checks that w_{p^{2}}^{2} = ⟨p^{2}⟩_{N}, that w_{p^{2}}^{∗2} = p^{2k}⟨p^{2}⟩_{N}^{∗} and that w_{p^{2}}^{∗} commutes with all T_{l}^{∗} (l ≠ p) and all ⟨a⟩_{N}^{∗} (a ∈ (Z/NZ)^{∗}). It follows that w_{p^{2}}^{∗}f = ±ε(p)p^{k}f; evaluating this on (Tate(q), μ_{p^{2}}, 1↦ζ_{N}) gives

∑ a′′_{n}q^{n} = ±ε(p)^{−1}p^{−k} ∑ a_{n}q^{n}  (3.3.11)

So we see that the equality in 3.3.8 concerning a′′_{n} is in fact true for all n.

It remains to determine the a′_{n} for p|n. Prop. 3.3.4 is of no help when one wants to express the a′_{pn} and a′′_{pn} in terms of a_{pn}, since a_{pn} = 0 for all n. For χ = π(Λ) it follows from 3.3.2 that a(pn, x) = 0 for all n and x, hence that a_{pn} = 0 = a′_{pn} = a′′_{pn}. The other two cases, χ = π(α, α^{−1}) or χ = π^{−}(α), will be dealt with in the next section.

3.4 Twisting. Suppose that χ = π(α, α^{−1}). We will show that f is a twist of a new form on X_{1} := \overline{M}(Γ_{1}(p), Γ_{1}(N))\overline{Q} and use this to solve our problem. For general information about twisting of modular forms one can consult [26], page 91.

Let β: F_{p}^{∗} → \overline{Q}^{∗} be a character. We will define an automorphism Θ_{β} of the \overline{Q}-vector space H^{0}(X, \underline{ω}^{⊗k}) such that:

\begin{aligned}T_{l}^{∗}^{∘}Θ_{β} & = & Θ_{β}^{∘}β(l)T_{l}^{∗} & ∀l ≠ p \\ g^{∘}Θ_{β} & = & Θ_{β}^{∘}β(det(g))g & ∀g ∈ G \end{aligned}  (3.4.1)

Let e_{p} denote the Weil pairing on the p-torsion of any elliptic curve (see e.g. [19], §2.8), let z be the locally constant μ_{p}(\overline{Q})-valued function e_{p}(ϕ(1, 0), ϕ(0, 1)) in H^{0}(X, O_{X}) and let b := ∑_{i} β(i)^{−1}z^{i}, where the sum ranges over i ∈ F_{p}^{∗}. Then one easily checks that the operator Θ_{β}: f ↦ bf on H^{0}(X, \underline{ω}^{⊗k}) satisfies 3.4.1 and that b is a unit if β is non-trivial.

Let V_{1} be the image of V under Θ_{α}. It follows from 3.4.1 and Table 3.3.1 that V_{1} ≅ π(α^{2}, 1) and that dim V_{1}^{H} = 1, where H ⊂ G is the subgroup \{({1 \atop 0} {∗ \atop ∗})\}. Let π_{1}: X → X_{1} be the morphism sending (E/S, ϕ, ψ) to (E/S, 1↦ϕ(1, 0), ψ). Then π_{1} identifies X_{1} with X/H, π_{1} is compatible with the T_{l}^{∗} for l ≠ p and the ⟨a⟩_{N}^{∗} and π_{1}^{∘}({x \atop 0} {a \atop y}) = ⟨x⟩_{p}^{∘}π_{1}. This implies that there exists a unique normalized new form f_{1} of weight k and character α^{2}ε on X_{1} such that π_{1}^{∗}f_{1} ∈ V_{1}. If we write:

f_{1}(Tate(q), 1↦ζ_{p}, 1↦ζ_{N}) = (∑ b_{n}q^{n})(dt/t)^{⊗k}  (3.4.2)

then we have a_{n} = α^{−1}(n)b_{n} for all n (we will always extend characters F_{p}^{∗}→\overline{Q}^{∗} to F_{p} by 0). The q-expansions of f_{1} at other cusps can be computed in various ways, see for example [1]??? and [16]??? ; one just evaluates the identity T_{p}^{∗}f_{1} = b_{p}f_{1} at such a cusp. First of all we have b_{p} ≠ 0. We define the b′_{n} by:

f_{1}(Tate(q)/\overline{Q}((q^{1/p})), 1↦q^{1/p}, 1↦ζ_{N}) = (∑ b′_{n}q^{n/p})(dt/t)^{⊗k}  (3.4.3)

For β: F_{p}^{∗} → \overline{Q}^{∗} a character and n ∈ Z let g(β, n) denote the Gauss sum ∑_{x} β(x)ζ_{p}^{nx}. Then:

b′_{n} = b_{p}^{−1}ε(p)p^{k−1}b′_{n/p} + p^{−1}α^{−2}(n)g(α^{2}, 1)b_{p}^{−1}b_{n}  (3.4.4)

A convenient way to express this result is the following Euler product expansion:

∑ b′_{n}n^{−s} = p^{−1}b_{p}^{−1}g(α^{2}, 1)(1 − λ′_{p}p^{−s})^{−1} ∏_{l≠p}(1 − α^{−1}(l)a_{l}l^{−s} + α^{−2}(l)ε(l)l^{k−1−2s})^{−1} (3.4.5)

where λ′_{p} is defined by: λ′_{p}b_{p} = ε(p)p^{k−1}. The q-expansions of f at various cusps can now be computed by expressing π^{∗}f in terms of π_{1}^{∗}f_{1}. By 3.4.1 we have Θ_{α^{−1}}π_{1}^{∗}f_{1} ∈ V. Recall that π^{∗}f is a basis for V^{T}. Unfortunately the projection of Θ_{α^{−1}}π_{1}^{∗}f_{1} in V^{T} is zero because ({x \atop 0} {0 \atop y})Θ_{α^{−1}}π_{1}^{∗}f_{1} = α(xy^{−1})Θ_{α^{−1}}π_{1}^{∗}f_{1}, but one can check that the projection of ({1 \atop 1} {0 \atop 1})Θ_{α^{−1}}π_{1}^{∗}f_{1} is non-zero. It follows that:

π^{∗}f = c ∑_{x} ({x \atop 0} {0 \atop 1})({1 \atop 1} {0 \atop 1})Θ_{α^{−1}}π_{1}^{∗}f_{1}  (3.4.6)

for some c ∈ \overline{Q}^{∗} (note that ∑({x \atop 0} {0 \atop 1}) projects into V^{T} because the central character of V is trivial). Evaluating 3.4.6 on say (Tate(q), ϕ_{0}^{∘}({0 \atop 1} {1 \atop 0}), 1↦ζ_{N}) one finds that c = g(α^{2}, 1)^{−1}b_{p}. Let the a′_{n} be as defined in 3.3.5. By evaluating 3.4.6 on (Tate(q), ϕ_{0}^{∘}({1 \atop 1} {0 \atop 1}), 1↦ζ_{N}) one computes that:

a′_{n} = α(−1)g(α^{2}, 1)^{−1}b_{p}g(α, 1)(j(α^{−1}, n)b′_{n} + b_{n/p})  (3.4.7)

where for any character β: F_{p}^{∗} → \overline{Q}^{∗} and n ∈ Z:

j(β, n) = ∑_{x} β(x)β(x − 1)ζ_{p}^{nx}  (3.4.8)

We summarize these results in the following proposition.

3.4.9 Proposition. Let f be a new form as in §3.2. Suppose that π^{∗}f generates the representation π(α, α^{−1}) of G for some α: F_{p}^{∗} → \overline{Q}^{∗}. Let f_{1} be the unique normalized new form on \overline{M}(Γ_{1}(p), Γ_{1}(N))\overline{Q} such that f is the twist of f_{1} by α^{−1}. Then the q-expansions of f and f_{1} at the various cusps are related by formulas 3.3.5, 3.4.2, 3.4.3, 3.4.4, 3.4.5 and 3.4.7.

Suppose now that χ = π^{−}(α). Using techniques as above one can show that there exists a unique normalized new form g of weight k and character ε, either on \overline{M}(Γ_{1}(N))\overline{Q} or on \overline{M}(Γ_{1}(p), Γ_{1}(N))\overline{Q}, such that a_{n} = α(n)b_{n} for all n, where T_{n}^{∗}g = b_{n}g. In the first case one finds:

a′_{n} = ε(p)^{−1}g(α, 1)^{−1} (p^{k}ε(p)b_{n/p^{2}} + j(α, n)b_{n})  (3.4.10)

In the second case we define the b′_{n} by:

g(Tate(q), 1↦q^{1/p}, 1↦ζ_{N}) = (∑ b′_{n}q^{n/p})(dt/t)^{⊗k}  (3.4.11)

Evaluating T_{p}^{∗}g = b_{p}g one obtains b_{p}^{2} = ε(p)p^{k−2} and b′_{n} = −p^{−1}b_{p}^{−1}b_{n} for all n. By the way, it follows that this case cannot occur if k = 1 since b_{p} has to be integral. For the a′_{n} one finds the following expression:

a′_{n} = ε(p)^{−1}g(α, 1)^{−1} (−pb_{p}b_{n/p} + j(α, n)b_{n})  (3.4.12)

3.4.13 Proposition. Let f be a new form as in §3.2. Suppose that π^{∗}f generates the representation π^{−}(α) of G, hence with α: F_{p}^{∗} → \overline{Q}^{∗} of order 2. Then there is a unique normalized new form g of weight k and character ε, either on \overline{M}(Γ_{1}(N))\overline{Q} or on \overline{M}(Γ_{1}(p), Γ_{1}(N))\overline{Q}, such that a_{n} = α(n)b_{n} for all n, where T_{n}^{∗}g = b_{n}g. In the first case the a′_{n} are given by 3.4.10 and in the second case by 3.4.12.

3.5 p-adic valuations. Let f be as in §3.2. In §6 we will need to know the p-adic valuation (in some sense) of the q-expansions of f at certain cusps. More precisely: we fix an embedding of \overline{Q} into \overline{Q}_{p} and we normalize the p-adic valuation v_{p} on \overline{Q}_{p} by setting v_{p}(p) = 1. Then we want to know the valuations of the power series ∑ a′_{n}q^{n} and ∑ a′′_{n}q^{n} in 3.3.5; by definition we have v_{p}(∑ a′_{n}q^{n}) = min\{v_{p}(a′_{n}) | n ≥ 0\}. Let \overline{Z}_{p} be the ring of integers in \overline{Q}_{p} and let \overline{F}_{p} be the residue field of \overline{Z}_{p}. Let F be a finite extension of F_{p}. Clearly any character α: F^{∗} → \overline{Q}_{p}^{∗} has its image in \overline{Z}_{p}^{∗}. We call α fundamental if α: F → \overline{Z}_{p} → \overline{F}_{p} is an embedding of fields. The Teichmüller character τ is the unique fundamental character τ: F_{p}^{∗} → \overline{Q}_{p}^{∗}. We denote the two fundamental characters F_{p^{2}}^{∗} → \overline{Q}_{p}^{∗} by τ_{2} and τ′_{2} = τ_{2}^{p}.

3.5.1 Lemma.

1. v_{p}(g(τ^{−m}, 1)) = \frac{m}{p−1} for 0 ≤ m < p − 1.

2. v_{p}(j(τ^{m}, 1)) = \{ \begin{matrix}1 − \frac{2m}{p−1} & if & 0 < m ≤ \frac{p−1}{2} \\ 0 & if & \frac{p−1}{2} ≤ m < p − 1 \end{matrix}

3. v_{p}(j(τ^{m}, 0)) = \{ \begin{matrix}1 & if & 0 < m < \frac{p−1}{2} \\ 0 & if & \frac{p−1}{2} ≤ m < p − 1 \end{matrix}

4. Let Λ = τ_{2}^{(p−1)m} with 0 < m < p + 1, let χ = π(Λ) and let h′_{n} be as in 3.3.3. Then h′_{n}(x) = 0 for all x if p|n. Assume that p |̸n and that p ≠ 2. Then:

(a) v_{p}(h′_{n}({1 \atop 0} {0 \atop 1})) = v_{p}(h′_{n}({0 \atop 1} {1 \atop 0})) = 1

(b) v_{p}(h′_{n}({1 \atop 1} {0 \atop 1})) = \{ \begin{matrix}\frac{p−m}{p−1} & if & m ≤ \frac{p+1}{2} \\ \frac{m−1}{p−1} & if & m ≥ \frac{p+1}{2} \end{matrix}

Proof. 1. Very well known, see for example …

2. We compute the image of j(τ^{m}, 1) in F_{p}[ζ_{p}] := Z_{p}[ζ_{p}]/pZ_{p}[ζ_{p}]:

j(τ^{m}, 1) mod p = ∑_{a} a^{m}(a − 1)^{m}ζ_{p}^{a} = ∑_{a} a^{m} ∑_{k=0}^{m} \binom{m}{k}(−1)^{m−k}a^{k}ζ_{p}^{a}
= (−1)^{m} ∑_{k=0}^{m}(−1)^{k}\binom{m}{k} ∑_{a} a^{m+k}ζ_{p}^{a} = (−1)^{m} ∑_{k=0}^{m}(−1)^{k}\binom{m}{k}g(τ^{m+k}, 1)

The statement now follows from part 1 of this lemma.

3. Well known because j(τ^{m}, 0) is τ(−1)^{m} times the usual Jacobi sum, see …

4. It follows from 3.3.2 that h′_{n}(x) = 0 if p|n. Assume that p |̸n. Part (a) follows from 3.3.7. The computation of v_{p}(h′_{n}({1 \atop 1} {0 \atop 1})) is more complicated. By definition 3.3.3 we have:

h′_{n}({1 \atop 1} {0 \atop 1}) = ∑_{b,x,y} ζ_{p}^{−nb}π(Λ)({x+xb \atop x} {by \atop y}), b ∈ F_{p}, x, y ∈ F_{p}^{∗}.

Write b = a−1. Note that ({ax \atop x} {(a−1)y \atop y}) is never scalar, hence the value of π(Λ) on it is determined by its trace (= ax + y) and its determinant (= xy). For any α ∈ F_{p} and β ∈ F_{p}^{∗} the number of solutions of

\{ \begin{matrix}ax + y & = & α \\ xy & = & β \end{matrix}

equals (\frac{α^{2}−4aβ}{p} ) +1, where (\frac{·}{p}) denotes the Legendre symbol. Using this fact, Table 3.3.1 and the non-triviality of Λ, we find:

h′_{n}({1 \atop 1} {0 \atop 1}) = −ζ_{p}^{n} ∑_{a∈F_{p}} ζ_{p}^{−na} ∑_{z∈F_{p^{2}}^{∗}} Λ(z) (\frac{(z + z^{p})^{2} − 4az^{p+1}}{p} ) .

The image of h′_{n}({1 \atop 1} {0 \atop 1}) in F_{p^{2}}[ζ_{p}] can be written as:

h′_{n}({1 \atop 1} {0 \atop 1}) mod p = −ζ_{p}^{n} ∑_{a∈F_{p}} ζ_{p}^{−na} ∑_{z∈F_{p^{2}}^{∗}} z^{(p−1)m} ((z + z^{p})^{2} − 4az^{p+1})^{(p−1)/2}

Expanding and rearranging terms gives:

h′_{n}({1 \atop 1} {0 \atop 1}) mod p = −ζ_{p}^{n} ∑_{k=1}^{(p−1)/2}(−4)^{k}\binom{\frac{p−1}{2}}{k} ∑_{z} z^{(p−1)(m+k+1)}(1 + z^{p−1})^{p−1−2k}g(τ^{k}, −n) (3.5.2)

Clearly the right hand side is of the form ∑ c_{k} g(τ^{k}, −n). It follows from part 1 of this lemma that v_{p}(g(τ^{k}, −n)) = 1 − k/(p−1), hence we have to find the largest k for which c_{k} ≠ 0. The exponent of z in each term of 3.5.2 is positive and less than 2(p^{2}−1). Because we sum over z, only terms with p^{2}−1 as exponent of z can give a non-zero contribution. It follows that we should look for the largest k for which there is a non-zero term depending on a and z as a^{k}z^{p^{2}−1}. This means that we are looking for the largest k such that there exists i with 0 ≤ i ≤ p−1−2k such that m+k + 1+i = p+1. Recall that 0 < m < p+1 and 0 < k ≤ (p−1)/2. It is then easy to see that one should take k = p−m and i = 0 if m ≥ (p+1)/2, and k = m−1 and i = p−1−2k if m ≤ (p+1)/2. □

3.5.3 Proposition. Let N be a positive integer, p > 2 a prime not dividing N and f a normalized new form on \overline{M}(Γ_{0}(p^{2}), Γ_{1}(N))\overline{Q} of some weight k ≥ 2 and character ε: (Z/NZ)^{∗} → \overline{Q}^{∗}. Let χ denote the character of the irreducible representation associated to f as in §3.2. Then we have χ = π(α, α^{−1}) for some α: F_{p}^{∗} → \overline{Q}^{∗} with α ≠ α^{−1}, or χ = π(Λ) for some Λ: F_{p^{2}}^{∗} → \overline{Q}^{∗} with Λ^{p} = Λ^{−1} ≠ Λ, or χ = π^{−}(α) with α: F_{p}^{∗} → \overline{Q}^{∗} of order two. Fix an embedding of \overline{Q} into \overline{Q}_{p} and normalize the p-adic valuation v_{p} on \overline{Q}_{p} by v_{p}(p) = 1. Let τ, τ_{2} and τ_{2}^{p} denote the fundamental characters of level 1 and 2 as in the beginning of §3.5. Let ζ_{p} and ζ_{N} be a pth and a Nth root of unity in \overline{Q}. Let the a_{n}, a′_{n} and a′′_{n} be as defined in 3.3.5. Then we have:

1. v_{p}(∑ a_{n}q^{n}) = 0,

2. v_{p}(∑ a′′_{n}q^{n}) = −k,

3. if χ = π(τ^{m}, τ^{−m}) with 0 < m < (p−1)/2 then v_{p}(∑ a′_{n}q^{n}) = −m/(p−1),

4. if χ = π(τ_{2}^{(p−1)m}) with 0 < m < (p+1)/2 then v_{p}(∑ a′_{n}q^{n}) = (1−m)/(p−1),

5. if χ = π^{−}(τ^{(p−1)/2}) then v_{p}(∑ a′_{n}q^{n}) = −1/2. Finally, the valuations of the q-expansions of f at the cusps (Tate(ζ_{p}^{i}q), ⟨q^{1/p}⟩, 1↦ζ_{N}), 1 ≤ i ≤ p−1, are all equal.

Proof. Part 1 is a consequence of the definition of “normalized”. Part 2 results from 3.3.11. Parts 3, 4 and 5 are an easy exercise using Lemma 3.5.1, Prop. 3.4.9, Prop. 3.3.4 and Prop. 3.4.13 (for 3 one has to use that in 3.4.5 v_{p}(λ′_{p}) ≥ 0). For the last statement one notes that Gal(\overline{Q}_{p}/Q_{p}) permutes those cusps transitively; that for any σ ∈ Gal(\overline{Q}_{p}/Q_{p}), σ(f) generates a representation with character σ(χ); that σ(χ) = χ (see 3.3.1), and that we have already proved that the valuation at one of those cusps depends only on that character. □

3.5.4 Remark. 1. Note that π(α, α^{−1}) = π(α^{−1}, α) and that π(Λ) = π(Λ^{p}).

2. The relation between χ and the Galois representation corresponding to f is given in Prop.4.8.

3. Note the curious case χ = π(τ_{2}^{p−1}) where v_{p}(∑ a′_{n}q^{n}) = 0.

4 A stable model for \overline{M}(Γ(p), Γ_{1}(N)) at p.

4.1 Construction of a stable model. In order to explain the construction of a stable model for \overline{M}(Γ(p), Γ_{1}(N)) at p, it is better to consider the following more general situation. Let S be the spectrum of a discrete valuation ring with perfect residue field, let s be the closed point of S and let η be the generic point. Let C → S be a curve: C → S is proper, flat and purely of relative dimension one. We suppose that C_{η} is smooth over η and that C is regular.

Under the hypotheses above, the irreducible components of C_{s} are Cartier divisors on C. After blowing up repeatedly in closed points of C_{s} we may assume that C_{s} is a Cartier divisor on C with normal crossings. Let n be the least common multiple of the multiplicities of the irreducible components of C_{s}, and let π_{0} be a uniformizer on S. Let T be S[π], with π^{n} = π_{0}, thus we have T → S totally ramified of degree n. Now we consider C̃_{T}, the normalization of the pullback of C to T. Let t be the closed point of T.

4.1.1 Proposition. If n is invertible on S, then the geometric fibre C̃_{T}_{,t̄} of C̃_{T}/T is a reduced curve whose singularities are ordinary double points (i.e., C̃_{T} → T is a semi-stable model).

Proof. By replacing S by its strict henselization we may suppose that k(s) is algebraically closed. We will check that C̃_{T}_{,t} is reduced and that its singularities are ordinary double points. In order to do this we compute C̃_{T} locally in the etale topology. Let x ∈ C(s). Since C_{s} is a normal crossings divisor on C there is an etale neighborhood U → C of x and a system X, Y of parameters in O_{U,x} such that X^{a}Y^{b} = π_{0}u with u ∈ O_{U,x} a unit and a, b non-negative integers with a ≠ 0. The numbers a and b are the multiplicities of the one or two irreducible components of U_{s} passing through x; b equals zero if and only if x lies on exactly one irreducible component of U_{s}. By assumption a is invertible on U, hence U[u^{1/a}] is finite etale over U. Changing X by an ath root of u we have: X^{a}Y^{b} = π_{0}. It follows that C_{T} is locally isomorphic to a subscheme of A_{T}^{2} defined by the equation X^{a}Y^{b} = π^{n}. The normalizations of the rings O_{T} [X, Y]/(X^{a}Y^{b}−π^{n}) can be easily computed, see for example [13], §2.2. One finds the following.

4.1.2 The normalization of Spec(O_{T} [X, Y]/(X^{a} − π^{n})) is the disjoint union, indexed by the ζ_{a} ∈ μ_{a}, of copies of Spec(O_{T} [Y]); on the copy labeled by ζ_{a} we have X = ζ_{a}π^{n/a}.

4.1.3 Suppose that b ≠ 0. Let c = gcd(a, b), write a = a′c, b = b′c, n = ca′b′m. Write 1 = a′e+b′d for some integers d and e. Then the normalization of Spec(O_{T} [X, Y]/(X^{a}Y^{b}−π^{n})) is the disjoint union, indexed be the ζ_{c} ∈ μ_{c}, of copies of Spec(O_{T} [V, Z]/(V Z − π^{m})). On the copy labeled by ζ_{c} we have: V = π^{a′em}X^{d}Y^{−e}, Z = π^{b′md}X^{−d}Y^{e}, X = ζ_{c}^{e}V^{b′} and

Y = ζ_{c}^{d}Z^{a′}.

From these results one concludes that indeed C̃_{T} → T is a semi-stable model. □

4.1.4 Remark. 1. If C_{η} is geometrically irreducible and has genus at least 2, one obtains the stable model (in the sense of [10]) of C_{η} over T by contracting the projective lines in C̃_{T}_{,t̄} intersecting the rest of C̃_{T}_{,t̄} in fewer than 3 points. By contracting and blowing up one also obtains the minimal model of C over T (n.b. C̃_{T} itself may not be regular).

2. The computations of the normalizations describe the morphism C̃_{T} → C_{T} locally in the etale topology. In particular, one knows the ramification structure of C̃_{T}_{,t} → C_{s,red}.

3. The choice of the uniformizing element π_{0} on S is unimportant, since all totally ramified extensions of degree n of S are isomorphic over the strict henselization of S.

4. On the contrary, if n is not invertible on S, there are lots of non-isomorphic (wildly ramified) extensions of the same degree. If one knows the action of the inertia subgroup of Gal(η̄/η) on H^{1}(Cη̄, Q_{ℓ}) one can pick the right extension. In the case of modular curves (of arbitrary level) this action is known ([5]). The problem is the description of the rings Ô_{C,x}.

4.2 Construction of a stable model for \overline{M}(Γ(p), Γ_{1}(N)). Let p be a prime, let N be an integer that is not divisible by p and let X := \overline{M}(Γ(p), Γ_{1}(N)) be the compactification (obtained by normalization, as in §2.1) of the algebraic stack classifying triples (E/S, ϕ, α) with E/S an elliptic curve, ϕ: F_{p} × F_{p} → E[p](S) a Drinfeld basis and α: Z/NZ → E[N](S) a Drinfeld Z/NZ-structure. For any elliptic curve E/S let e_{p}: E[p]×_{S} E[p] → μ_{p,S} denote the Weil pairing (see [19], §2.8). The morphism X → Spec(Z) factors through Spec(Z[ζ_{p}]) by sending (E/S, ϕ, α) to e_{p}(ϕ(1, 0), ϕ(0, 1)) ∈ Spec(Z[ζ_{p}])(S). Let X ˜⊗_{Z}Z[ζ_{p}] denote the normalization of X ⊗_{Z} Z[ζ_{p}]. For i ∈ F_{p}^{∗} let X_{i} denote the algebraic stack \overline{M}(Γ(p)^{ζ_{p}^{i}−can}, Γ_{1}(N)) over Z[ζ_{p}] (see [19], §9.4). By definition, for a scheme S over Z[ζ_{p}] an object in M(Γ(p)^{ζ_{p}^{i}−can}, Γ_{1}(N))(S) is a triple (E/S/Z[ζ_{p}], ϕ, α) with e_{p}(ϕ(1, 0), ϕ(0, 1)) = ζ_{p}^{i}; X_{i} is obtained from M(Γ(p)^{ζ_{p}^{i}−can}, Γ_{1}(N)) by normalization at infinity. The morphism

∐_{i∈F_{p}^{∗}} X_{i} ⟶ X ⊗ Z[ζ_{p}], (E/S/Z[ζ_{p}], ϕ, α) ↦ ((E/S, ϕ, α), e_{p}(ϕ(1, 0), ϕ(0, 1)))

induces an isomorphism

∐_{i∈F_{p}^{∗}} X_{i} ˜ ⟶ X ˜⊗Z[ζ_{p}]  (4.2.1)

because both sides are normal (in fact, both sides are regular).

We will now apply the construction of §4.1 to X ˜⊗Z[ζ_{p}] over Z[ζ_{p}]. The fact that we are dealing with an algebraic stack that is not necessarily a scheme is no problem because blowing up and normalization are compatible with etale descent. We have already remarked that X ˜⊗Z[ζ_{p}] is regular. The X_{i} ⊗_{Z[ζ_{p}]} F_{p} are described in [19], Thm. 13.7.6 and Thm. 13.8.4: X_{i} ⊗_{Z[ζ_{p}]} F_{p} has p+1 irreducible components, all isomorphic to \overline{M}(ExIg(p, 1), Γ_{1}(N)); these irreducible components are smooth over F_{p}, geometrically irreducible, have multiplicity one and intersect each other exactly at the supersingular points; the intersections are pairwise transversal.

Let Y → X ˜⊗Z[ζ_{p}] be the blow up in the supersingular points in characteristic p. Then Y ⊗_{Z[ζ_{p}]} F_{p} is a divisor on Y with normal crossings and over each supersingular point lies an exceptional curve having multiplicity p+1 in Y ⊗_{Z[ζ_{p}]} F_{p}. Let π_{0} be the uniformizer 1−ζ_{p} of Z[ζ_{p}] at p. According to Prop. 4.1.1 X^{st} := Y ˜⊗_{Z[ζ_{p}]}Z[ζ_{p}, π], where π^{p+1} = π_{0}, is a model for X over Z[ζ_{p}, π] that is semi-stable at p. The computations in the proof of Prop. 4.1.1 show that X^{st} is regular and that the fibre X_{p}^{st} := X^{st} ⊗_{Z[ζ_{p},π]} \overline{F}_{p} over \overline{F}_{p} has two kinds of irreducible components. First of all one has the irreducible components that map isomorphically onto an irreducible component of (X ˜⊗Z[ζ_{p}])\overline{F}_{p}; these components are isomorphic to \overline{M}(ExIg(p, 1), Γ_{1}(N))\overline{F}_{p}; we call them components of Igusa type. Secondly, there are the irreducible components that map to a supersingular point in X ˜⊗Z[ζ_{p}]; over each supersingular point in characteristic p there is exactly one such component; we call them components of Drinfeld type; they are all isomorphic. Each component of Drinfeld type maps with degree p+1 to some exceptional curve in Y ⊗_{Z[ζ_{p}]} \overline{F}_{p}. This shows a component of Drinfeld type is a cover of degree p+1 of P1 ; \overline{F}_{p} ramified exactly and totally over the p+1 F_{p}-rational points.

4.3 Components of Drinfeld type. In this section we want to get equations for the components of Drinfeld type in X_{p}^{st}. To do this we follow the construction of §4.2. Let s = (E/\overline{F}_{p}, ϕ, α) be a supersingular point in X_{i}(\overline{F}_{p}) for some i ∈ F_{p}^{∗}. We denote the corresponding component of Drinfeld type by D_{i,s}. Let W be the ring of Witt vectors of \overline{F}_{p}. By [19], Thm. 13.8.4, the complete local ring of X ˜⊗Z[ζ_{p}] at s is isomorphic to W [ζ_{p}][[x, y]]/(f), with f = f_{0} + π_{0}f_{1}, f_{0} = x^{p}y − xy^{p} + g, g ∈ (x, y)^{p+2} and f_{1} a unit. As explained in [13], §1.3.1, the completion of Y along the exceptional curve lying over s can be covered by two open affines isomorphic to Spf(A_{1}) and Spf(A_{2}) with A_{1} = W [ζ_{p}][v][[x]]/(f(x, vx)) and A_{2} = W [ζ_{p}][u][[y]]/(f(uy, y)) respectively.

It follows that the completion of X^{st} along D_{i,s} is covered by two open affines isomorphic to Spf(A_{j} ˜⊗_{W [ζ_{p}]}W [ζ_{p}, π]), j in {1, 2}. The normalization A_{j} ˜⊗_{W [ζ_{p}]}W [ζ_{p}, π] of A_{j} ⊗_{W [ζ_{p}]} W [ζ_{p}, π] is obtained by blowing up in the ideals (x, π) and (y, π) respectively (this can be seen from the computations in the proof of Prop. 4.1.1). Let us write out what happens for j = 1. Then:

A_{1} ⊗_{W [ζ_{p}]} W [ζ_{p}, π] = W [ζ_{p}, π][v][[x]]/ (x^{p+1}(v − v^{p}) + x^{p+2}h + π^{p+1}f_{1}(x, vx))

where we have written g(x, vx) = x^{p+2}h. Blowing up in (x, π) means setting π = xw. We find:

A_{1} ˜⊗_{W [ζ_{p}]}W [ζ_{p}, π] = W [ζ_{p}, π][v][[x]][w]/ (π − xw, v − v^{p} + xh + w^{p+1}f_{1}(x, vx))

The corresponding affine part of D_{i,s} is given by the equation x = 0. Substituting this we find that D_{i,s} is the smooth complete model of the affine curve in A2 ; \overline{F}_{p} given by the equation

aw^{p+1} = v^{p} − v  (4.3.1)

where a ∈ \overline{F}_{p}^{∗} is the image under W [ζ_{p}] → \overline{F}_{p} of f_{1}(0, 0). If we put α := w^{−1} and β := vw^{−1} then we find the equation:

−a = α^{p}β − αβ^{p}  (4.3.2)

for an other affine part of D_{i,s}.

In order to describe various actions on the D_{i,s} it is necessary to recall the construction in [19], §13.8 of the parameters x and y of the complete local ring Ô_{X_{i},s} of X_{i} at s. By definition we have a triple (E/Ô_{X_{i},s}, ϕ, α). Let Z be a parameter of the formal group of E. Then we can take x = Z(ϕ(1, 0)) and y = Z(ϕ(0, 1)). Putting things together we have:

x = Z(ϕ(1, 0)), y = Z(ϕ(0, 1)), y = vx, π = xw, α = π^{−1}x, β = π^{−1}y.  (4.3.3)

4.4 Components of Igusa type. In this section we want to give a precise description of the components of Igusa type in X_{p}^{st}. We have already seen in §4.2 that these are precisely the irreducible components of (X ˜⊗Z[ζ_{p}]) ⊗ \overline{F}_{p}. Because we have the isomorphism 4.2.1 between X ˜⊗Z[ζ_{p}] and the disjoint union of the X_{i}, i ∈ F_{p}^{∗}, it is sufficient to describe the irreducible components of the X_{i} ⊗_{Z[ζ_{p}]} \overline{F}_{p}. In fact, all X_{i} ⊗_{Z[ζ_{p}]} \overline{F}_{p} are the same: they classify triples

(E/S/\overline{F}_{p}, ϕ, α) with e_{p}(ϕ(1, 0), ϕ(0, 1)) = 1.

In [19], §13.7and §13.8 it is shown that the irreducible components of X_{i} ⊗_{Z[ζ_{p}]} \overline{F}_{p} are the compactifications of the p+1 stacks, labeled by the P ∈ P^{1}(F_{p}), classifying triples (E/S/\overline{F}_{p}, ϕ, α) with P ⊂ ker(ϕ), where we view P as a line in F_{p}^{2}. We denote the corresponding components in X_{p}^{st} by Ig_{i,P}, where i ∈ F_{p}^{∗} and P ∈ P^{1}(F_{p}). For every P ∈ P^{1}(F_{p}) we choose a surjective linear map L_{P}: F_{p}^{2} → F_{p} such that P = ker(L_{P}) as in [19], §13.8: for P = F_{p} · ({x \atop 1}) we take L_{P}: ({a \atop b}) ↦ a−bx and for P = F_{p} ·({1 \atop 0}) we take L_{P}: ({a \atop b}) ↦ b. Using these L_{P} we get isomorphisms:

Ig_{i,P} ˜→ \overline{M}(ExIg(p, 1), Γ_{1}(N))\overline{F}_{p}, (E/S/\overline{F}_{p}, ϕ, α) ↦ (E/S/\overline{F}_{p}, ϕ̄, α)  (4.4.1)

where ϕ̄: F_{p} → E(S) is the unique map such that ϕ = ϕ̄^{∘}L_{P}. Since these isomorphisms depend on the choice of the L_{P} we will avoid using them as much as we can.

4.5 The action of GL_{2}(F_{p}). The group G := GL_{2}(F_{p}) = Aut(F_{p}^{2}) acts from the right on X by automorphisms r(g):

r(g): (E/S, ϕ, α) ↦ (E/S, ϕ^{∘}g, α)

We let G act on X ⊗ Q(ζ_{p}, π) via its action on X; by construction this action extends uniquely to an action on X^{st}. The aim of this section is to describe the action that we get on X_{p}^{st}. First of all, since e_{p}(·, ·) is alternating we have:

e_{p}(ϕ^{∘}g(1, 0), ϕ^{∘}g(0, 1)) = e_{p}(ϕ(1, 0), ϕ(0, 1))^{det(g)}

Hence the action of g ∈ G on ∐_{i} X_{i} ⊗_{Z[ζ_{p}]} \overline{F}_{p} is given by:

((E/S/\overline{F}_{p}, ϕ, α), i) ↦ ((E/S/\overline{F}_{p}, ϕ^{∘}g, α), i det(g))

From this we can easily see the action of g ∈ G on the Ig_{i,P}. Each g induces an isomorphism r(g): Ig_{i,P} ˜→ Ig_{i det(g),g^{−1}P}. The stabilizer of Ig_{i,P} is the Borel subgroup B_{P} of SL_{2}(F_{p}) that fixes P. Let χ_{P}: B_{P} → F_{p}^{∗} be the character giving the action on the line P, then for g ∈ B_{P} we have L_{P}^{∘}g = χ_{P} (g)^{−1}L_{P} since det(g) = 1. It follows that the action of g ∈ B_{P} on \overline{M}(ExIg(p, 1), Γ_{1}(N))\overline{F}_{p} that we get via the isomorphism 4.4.1 is the so-called diamond operator:

⟨χ_{P} (g)^{−1}⟩_{p} : (E/S, ϕ̄, α) ↦ (E/S, χ_{P} (g)^{−1}ϕ̄, α)  (4.5.1)

Note that any other choice of L_{P} would lead to the same ⟨χ_{P} (g)^{−1}⟩_{p}.

Let us now consider the D_{i,s}. Each g ∈ G induces an automorphism r(g): D_{i,s} ˜→ D_{i det(g),s}. The stabilizer of D_{i,s} is SL_{2}(F_{p}). Let g = ({a \atop c} {b \atop d}) ∈ SL_{2}(F_{p}). Then g acts (from the left) on W [ζ_{p}][[x, y]]/(f) by:

r(g)^{#}x = r(g)^{#}Z(ϕ(1, 0)) = Z(ϕ^{∘}g(1, 0)) = Z(ϕ(a, c)) = Z(aϕ(1, 0) + cϕ(0, 1)) =
= ax + cy (mod (x, y)^{2})
r(g)^{#}y = bx + dy (mod (x, y)^{2})

It follows that g acts on D_{i,s} by:

r(g)^{#} : α ↦ aα + cβ, β ↦ bα + dβ  (4.5.2)

where α and β are the coordinates from 4.3.2.

4.6 The action of inertia. Let G_{Q} be the Galois group of \overline{Q} over Q. Let \overline{Z} be the integral closure of Z in \overline{Q}. We fix a surjection \overline{Z} → \overline{F}_{p} (these are permuted transitively by G_{Q}). Then we have a sequence of subgroups:

I_{p} ⊲ I ⊲ G_{p} ⊂ G_{Q},

where G_{p} is the decomposition group, I the inertia (or ramification) subgroup, and I_{p} the wild inertia subgroup. We can identify G_{p} with Gal(\overline{Q_{p}}/Q_{p}), G_{p}/I with Gal(\overline{F}_{p}/F_{p}), and I_{t} := I/I_{p} (the tame inertia group) with Gal(Q_{p}^{tr}/Q_{p}^{unr}) and with lim_{⟵} F_{p^{n}}^{∗} = lim_{⟵} μ_{n}(\overline{F}_{p}) by σ ↦ (σ(p^{1/n})p^{−1/n})_{p |̸n}.

For σ ∈ G_{Q} we define γ(σ) to be the automorphism id × Spec(σ) of X × Spec(\overline{Q}) = X^{st} ×_{Spec(Q(ζ_{p},π))} Spec(\overline{Q}). Note that γ is a right-action: γ(σ_{1}σ_{2}) = γ(σ_{2})^{∘}γ(σ_{1}). By construction, the automorphisms γ(σ) extend uniquely to automorphisms γ(σ) of X^{st} ×_{Spec(Z[ζ_{p},π])} Spec(\overline{Z}); the diagrams:

X^{st} ×_{Spec(Z[ζ_{p},π])} Spec(\overline{Z}) ⟶^{γ(σ)} X^{st} ×_{Spec(Z[ζ_{p},π])} Spec(\overline{Z})
↓  ↓
Spec(\overline{Z})  ⟶^{Spec(σ)}  Spec(\overline{Z})

are commutative. For each σ ∈ G_{p} we get an automorphism, still denoted γ(σ), of X_{p}^{st}, compatible with the automorphism Spec(σ) of Spec(\overline{F}_{p}). Finally, for σ ∈ I we get an \overline{F}_{p}- automorphism γ(σ) of X_{p}^{st}. This right-action of I on X_{p}^{st} is what we call the action of inertia.

Since X^{st} lives over Z[ζ_{p}, π] which is of degree p^{2}−1 over Z and totally ramified at p, the inertia action of I factors through an antihomomorphism γ̄: F_{p^{2}}^{∗} → Aut\overline{F}_{p}(X_{p}^{st}). The composition of I → F_{p^{2}}^{∗} and the norm F_{p^{2}}^{∗} → F_{p}^{∗} is the character giving the action of I on the pth roots of unity. It follows that the action of u ∈ F_{p^{2}}^{∗} on the left hand side of 4.2.1 is given by sending (E/S →^{f} Spec(Z[ζ_{p}]), ϕ, α)_{i} to (E/S →^{f′} Spec(Z[ζ_{p}]), ϕ, α)_{iu^{−p−1}}, where f′ = Spec(u^{p+1})^{∘}f. We conclude that u ∈ F_{p^{2}}^{∗} induces isomorphisms:

γ̄(u) : \{ \begin{matrix}Ig_{i,P} & ˜ ⟶ & Ig_{iu^{−p−1},P} \\ D_{i,s} & ˜ ⟶ & D_{iu^{−p−1},s} \end{matrix}  (4.6.1)

The stabilizers in F_{p^{2}}^{∗} of the Ig_{i,P} and the D_{i,s} are all equal to μ_{p+1}(F_{p^{2}}). Since the Ig_{i,P} already live in X ˜⊗Z[ζ_{p}] the γ̄(u) with u ∈ μ_{p+1}(F_{p^{2}}) act trivially on them. The action of the γ̄(u) with u ∈ μ_{p+1}(F_{p^{2}}) on the D_{i,s} can be read off from equations 4.3.2 and 4.3.3 (just note that γ(u)^{#}π = uπ mod π^{2}). One finds that for u ∈ μ_{p+1}(F_{p^{2}}) the automorphism γ̄(u) of D_{i,s} is given in the coordinates from 4.3.2 by:

γ̄(u)^{#}: α ↦ u^{−1}α, β ↦ u^{−1}β.  (4.6.2)

4.7 Proposition. The actions of G and F_{p^{2}}^{∗} from §4.5 and §4.6 on X_{p}^{st} commute. The stabilizer of Ig_{i,P} in F_{p^{2}}^{∗} × G is {(u, g) | u^{p+1} = det(g), g^{−1}P = P }; it acts on Ig_{i,P} by ⟨u^{p+1}χ_{P} (g^{−1})⟩_{p} (cf. 4.5.1). The stabilizer of D_{i,s} is \{(u, g) | u^{p+1} = det(g)\}; (u, ({a \atop c} {b \atop d})) in it acts by α ↦ u^{−1}(aα+cβ), β ↦ u^{−1}(bα + dβ) in the coordinates of 4.3.2.

Proof. The two actions commute by construction. The remaining statements follow from computations as in §4.5 and §4.6. □

4.8 Proposition. Let f and χ be as in Prop. 3.5.3. Let K ⊂ \overline{Q} be the field generated by the eigenvalues of f for the T_{n}^{∗} (n ≥ 1); then χ has values in K. Let λ |̸p be a finite place of K; let ρ_{λ}: G_{Q} → GL_{2}(K_{λ}) be associated to f as in §2.5. Then ρ_{λ}|_{I_{p}} (notation as in §4.6) is trivial and:

1. if χ = π(α, α^{−1}) with α ≠ α^{−1}: F_{p}^{∗} → \overline{K}^{∗} then (ρ_{λ}|_{I}) ⊗ \overline{K}_{λ} ≅ α ⊕ α^{−1},

2. if χ = π^{−}(α) with α: F_{p}^{∗} → K^{∗} of order 2 then ρ_{λ}|_{I} is an extension of α by α,

3. if χ = π(Λ) with Λ^{p} = Λ^{−1} ≠ Λ: F_{p^{2}}^{∗} → \overline{K}^{∗} then (ρ_{λ}|_{I}) ⊗ \overline{K}_{λ} ≅ Λ ⊕ Λ^{−1}.

Proof. Since f is defined over K (use a q-expansion) the G-representation V that it generates is defined over K hence its character χ has values in K. Let χ_{λ} := χ ⊗_{K} K_{λ}.

Let X^{o} denote the complement of the cusps in X and j: X^{o} ↪ X denote the inclusion. As in §2.5 one shows that the K_{λ}-representation ρ_{λ}^{∨}⊗χ_{λ} of G_{Q}×G occurs in H_{par}^{1}(Xo ; \overline{Q}, F_{l}^{k})⊗_{Q_{l}}K_{λ}; after replacing N by some multiple we can assume that N > 4. The cokernel of the inclusion j_{!}F_{l}^{k} ↪ j_{∗}F_{l}^{k} is supported on the cusps hence the map H_{c}^{1}(Xo ; \overline{Q}, F_{l}^{k}) → H^{1}(X\overline{Q}, j_{∗}F_{l}^{k}) is surjective. The Leray spectral sequence for j_{∗} shows that H^{1}(X\overline{Q}, j_{∗}F_{l}^{k}) → H^{1}(Xo ; \overline{Q}, F_{l}^{k}) is injective. It follows that H_{par}^{1}(Xo ; \overline{Q}, F_{l}^{k}) is equal to H^{1}(X\overline{Q}, j_{∗}F_{l}^{k}). The exact sequence of vanishing cycles (see [17], VII, Exp. XIII, Thm. 3.4(ii)) reads:

0 → H^{1}(X_{p}^{st}, j_{∗}F_{l}^{k}) → H^{1}(X^{st}_{\overline{Q}_{p}}, j_{∗}F_{l}^{k}) → ⨁_{x} F_{l}^{k}(−1)_{x}  (4.8.1)

where x ranges over the double points of X_{p}^{st}. If π denotes the normalization map of X_{p}^{st} then from the short exact sequence 0 → j_{∗}F_{l}^{k} → π_{∗}π^{∗}j_{∗}F_{l}^{k} → ⨁_{x} F_{l,x}^{k} → 0 (depending on the choice of an orientation of the graph associated to X_{p}^{st}) we get an exact sequence:

⨁_{x} F_{l,x}^{k} ⟶ H^{1}(X_{p}^{st}, j_{∗}F_{l}^{k}) ⟶ ⨁_{i,P} H^{1}(Ig_{i,P} , j_{∗}F_{l}^{k}) ⊕ ⨁_{i,s} H^{1}(D_{i,s}, F_{l,s}^{k}) ⟶ 0  (4.8.2)

The exact sequences 4.8.1 and 4.8.2 are compatible with the actions of G_{p} × G; the inertia group I ⊲ G_{p} acts via its tame quotient I_{t} = lim_{⟵} F_{p^{n}}^{∗}. The following lemma describes the terms of 4.8.1 and 4.8.2 as I_{t} × G-modules.

4.8.3 Lemma. (i). ⊕_{x}\overline{Q}_{l} ≅ ⊕_{s} ⊕_{α} α^{−1} ⊗ (π^{−}(α) ⊕ α^{∘} det), where s ranges over the super singular points in M(Γ_{1}(N))\overline{F}_{p} and α over all characters F_{p}^{∗} → \overline{Q}_{l}^{∗}.

(ii). ⊕_{i,P} H^{1}(Ig_{i,P} , j_{∗}F_{l}^{k}) ⊗ \overline{Q}_{l} is a direct sum of copies of α^{−1} ⊗ π(α, β), α^{−1} ⊗ π^{−}(α) and α^{−1} ⊗ α^{∘} det.

(iii). ⊕_{i,s}H^{1}(D_{i,s}, \overline{Q}_{l}) ≅ ⊕_{s} ⊕_{Λ} Λ^{−1} ⊗ π(Λ), with Λ^{p} ≠ Λ: F_{p^{2}}^{∗} → \overline{Q}_{l}^{∗}.

Proof. (i). Fix an s and let V := ⊕_{x↦s}\overline{Q}_{l}. As a representation of G, V ≅ ⊕_{α}(π^{−}(α) ⊕ α^{∘} det) since V is the induced of the trivial representation of some Borel subgroup of SL_{2}(F_{p}). Note that I_{t} acts on V via F_{p}^{∗} so that as a representation of F_{p}^{∗} × G, V is the direct sum over the α of β_{α} ⊗ π^{−}(α) and β′_{α} ⊗ α^{∘} det. Since the stabilizer subgroup of some x is the subgroup H := {(det(b), b) | b ∈ B} of F_{p}^{∗} × B all β_{α} ⊗ π^{−}(α) and β′_{α} ⊗ α^{∘} det must have non-zero H-invariants. From the formulas Res_{B}^{G}π^{−}(α) = Res_{B}^{T}(α, α) ⊕ (irr. of dim. p−1) and Res_{B}^{G}(α^{∘} det) = Res_{B}^{T}(α, α) it follows that β_{α} = β′_{α} = α^{−1}.

(ii). Let V := ⊕_{i,P} H^{1}(Ig_{i,P} , j_{∗}F_{l}^{k}). Note that the action of I_{t} on V factors through F_{p}^{∗}. As a representation of G, V is a direct sum of non-cuspidal representations since it is the induction from B to G of ⊕_{i}H^{1}(Ig_{i,(1:0)}, j_{∗}F_{l}^{k}) and B acts on ∐_{i} Ig_{i,(1:0)} through T. Suppose that W ≅ γ ⊗ π(α, β) is a subrepresentation of V ⊗ \overline{Q}_{l}. Let H be the subgroup \{(u, ({u \atop 0} {a \atop 1})) | u ∈ F_{p}^{∗}, a ∈ F_{p}} of F_{p}^{∗} × G. Then we must have W^{H} ≠ 0 since H acts trivially on the Ig_{i,(1:0)}. One computes that Res_{H}^{G}W is the direct sum of the characters γα and γβ of F_{p}^{∗} and an irreducible representation of dimension p−1. It follows that γ = α^{−1} and γ = β^{−1}. The cases π^{−}(α) and α^{∘} det are treated in the same way.

(iii). Fix an s. Let V := ⊕_{i}H^{1}(D_{i,s}, \overline{Q}_{l}). Note that I_{t} acts on V via F_{p^{2}}^{∗}. Using the Lefschetz trace formula one computes that, as a F_{p^{2}}^{∗}-representation, V ≅ ⊕_{Λ≠Λ^{p}}(p − 1)Λ. Let V_{Λ} denote the Λ-eigenspace of V. The trace of an element g in G on V_{Λ} is equal to the trace of gPr_{Λ} on V, where Pr_{Λ} = (p^{2} − 1)^{−1} ∑_{u} Λ(u^{−1})u. Using the Lefschetz trace formula one finds that V_{Λ} ≅ π(Λ^{−1}) as G-module. □

Note that det^{∘}ρ_{λ}|_{I} is trivial. Prop. 4.8 now follows from the exact sequences 4.8.1 and 4.8.2 and Lemma 4.8.3. □

5 A stable model for \overline{M}(Γ_{0}(p^{2}), Γ_{1}(N))a at p.

5.1 Construction of a stable model. We keep the notation from §4: X = \overline{M}(Γ(p), Γ_{1}(N)) and X^{st} is the model of X over Z[ζ_{p}, π] which is stable at p, with π^{p+1} = 1−ζ_{p}, as constructed in §4.2; in particular, the fibre X_{p}^{st} of X^{st} over \overline{F}_{p} is reduced, its irreducible components are smooth and intersect each other transversally. The right-action of the group G = GL_{2}(F_{p}) on X induces a right-action of G on X^{st}. Let X_{0} := \overline{M}(Γ_{0}(p^{2}), Γ_{1}(N)). The morphism X\overline{Q} → X_{0,\overline{Q}} described in §3.2 comes from a morphism X → X_{0}, which identifies X_{0} as the quotient of X by the diagonal subgroup T of G (away from the cusps this is [19] 11.3.5, then use that both X and X_{0} are regular). We define X_{0}^{st} := X^{st}/T; from general results about quotients (see §5.3) it follows that X_{0}^{st} is a model of X_{0} which is stable at p. Note that the order of T is prime to p, hence the fibre X_{0,p}^{st} of X_{0}^{st} over \overline{F}_{p} is equal to X_{p}^{st}/T.

Another way to construct X_{0}^{st} is to apply the construction of §4.1 to the blow up of \overline{M}(Γ_{0}(p^{2}), Γ_{1}(N)) in its supersingular points in characteristic p. Such a construction is given in [13].

5.2 Description of the stable model. From now on we assume that p > 2. Let us first compute the images of the irreducible components of Igusa type of X_{p}^{st} in X_{0,p}^{st}. In §4.5 we have seen how the elements of T act on the disjoint union of the Ig_{i,P}, where i ∈ F_{p}^{∗} and P ∈ P^{1}(F_{p}). The action of T on F_{p}^{∗} × P^{1}(F_{p}) has four orbits; representatives for them are: (1, (1 : 0)), (1, (0 : 1)), (1, (1 : 1)), and (1, (1 : d)) where d is some non-square in F_{p}. The stabilizers of these representatives are T ∩ SL_{2}(F_{p}), T ∩ SL_{2}(F_{p}), {±1} and {±1} respectively. It follows that the image in X_{0,p}^{st} of the disjoint union of the Ig_{i,P} consists of exactly four irreducible components: Ig_{(2,0)} := Ig_{1,(1:0)}/T ∩ SL_{2}(F_{p}), Ig_{(0,2)} := Ig_{1,(0:1)}/T ∩ SL_{2}(F_{p}), Ig_{+} := Ig_{1,(1:1)}/{±1}, and Ig_{−} := Ig_{1,(1:d)}/{±1}; we call them the components of Igusa type of X_{0,p}^{st}. We have isomorphisms: Ig_{(2,0)} ≅ Ig_{(0,2)} ≅ \overline{M}(Γ_{1}(N))\overline{F}_{p} and Ig_{+} ≅ Ig_{−} ≅ \overline{M}(Ig(p)/\{±1\}, Γ_{1}(N))\overline{F}_{p}, where M(Ig(p)) is the F_{p}-stack that classifies pairs (E/S, P) with P ∈ E^{(p)}(S) a generator of ker(V: E^{(p)} → E) (see [19], §12.3.1). Under the morphism X_{0,p}^{st} → \overline{M}(Γ_{0}(p^{2}), Γ_{1}(N))\overline{F}_{p} the irreducible components Ig_{(2,0)}, Ig_{(0,2)}, Ig_{+} and Ig_{−} are mapped to the irreducible components called of type (2, 0), (0, 2), (1, 1) and (1, 1) in [19] §13.5.6. As stacks over \overline{M}(Γ_{1}(N))\overline{F}_{p} we have isomorphisms Ig_{(2,0)} ≅ \overline{M}(Γ_{1}(N))\overline{F}_{p}, Ig_{+} ≅ Ig_{−} ≅ \overline{M}(Ig(p)/\{±1\}, Γ_{1}(N))\overline{F}_{p} and Ig_{(2,0)} ≅ \overline{M}(Γ_{1}(N))(p^{2}) ; \overline{F}_{p}, where \overline{M}(Γ_{1}(N))(p^{2}) ; \overline{F}_{p} denotes the pullback of \overline{M}(Γ_{1}(N))\overline{F}_{p} via the square of the \overline{F}_{p}-linear Frobenius on \overline{M}\overline{F}_{p}.

The stabilizer in T of an irreducible component D_{i,s} of Drinfeld type is T ∩ SL_{2}(F_{p}). It follows that the image in X_{0,p}^{st} of the disjoint union of the D_{i,s} consists of irreducible components D_{s} := D_{1,s}/T ∩ SL_{2}(F_{p}), labeled by the super singular s in M(Γ_{1}(N))(\overline{F}_{p}); we call them the components of Drinfeld type of X_{0,p}^{st}. By construction, the D_{s} are smooth irreducible curves over \overline{F}_{p}; they are all isomorphic and one can easily verify that their genus is p−1. An element ({t \atop 0} {0 \atop t^{−1}} ) acts on the coordinates of D_{i,s} in 4.3.2 by: α ↦ tα, β ↦ t^{−1}β. It follows that v_{1} := αβ and u_{1} := α^{p−1} are coordinates of an affine open of D_{s}; equation 4.3.2 gives: u_{1}^{2}v_{1}+au_{1}−v_{1}^{p} = 0.

Figure 1: A picture of X_{0,p}^{st} → \overline{M}(Γ_{1}(N))\overline{F}_{p}.

[figure: ∞; Ig^{2,0}; Ig^{0,2}; Ig^{+}; Ig^{−}; D_{s}; D_{s′}; D_{s′′}; s; s′; s′′; \overline{M}(Γ_{1}(N))\overline{F}_{p}]

In other words, D_{s} is the smooth complete model of the hyperelliptic curve given by:

u_{2}^{2} = v_{1}^{p+1} + \frac{1}{4}a^{2}  (5.2.1)

where u_{2} = u_{1}v_{1} + \frac{1}{2}a.

The double points of X_{0,p}^{st} can be described as follows: the components of Igusa type are all disjoint and the D_{s} are all disjoint; each D_{s} meets all four components of Igusa type at the unique supersingular point lying over s under the morphism to \overline{M}(Γ_{1}(N))\overline{F}_{p}. Figure 1 gives a picture of the situation. We will use the following notation for the double points in

X_{0,p}^{st}(\overline{F}_{p}): x_{(2,0),s} := Ig_{(2,0)} ∩ D_{s}, x_{(0,2),s} := Ig_{(0,2)} ∩ D_{s}, x_{+,s} := Ig_{+} ∩ D_{s} and x_{−,s} := Ig_{−} ∩ D_{s}.

The complete local rings of X_{0}^{st} at the double points can be easily computed using either 4.1.2 and 4.1.3 or Prop. 5.3.3. The complete local rings at x_{(2,0),s} and x_{(0,2),s} are isomorphic to W [π][[x, y]]/(xy − π^{p−1}) where W denotes the ring of Witt vectors of \overline{F}_{p}. The complete local rings at x_{+,s} and x_{−,s} are isomorphic to W [π][[x, y]]/(xy − π^{2}).

As in §4.6 let I be an inertia subgroup of G_{Q} and let γ̄: F_{p^{2}}^{∗} → Aut\overline{F}_{p}(X_{0,p}^{st}) be the antihomomorphism giving the inertia action. The inertia action preserves the irreducible components D_{s}, Ig_{(2,0)} and Ig_{(0,2)}; it interchanges Ig_{+} and Ig_{−}. Let u ∈ F_{p^{2}}^{∗}. Then γ̄(u) is the identity on Ig_{(2,0)} and on Ig_{(0,2)}; on D_{s} it is the automorphism given by:

γ̄(u)^{#}: u_{2} ↦ u_{2}, v_{1} ↦ u^{p−1}v_{1}  (5.2.2)

where u_{2} and v_{1} are the coordinates of 5.2.1. If u ∈ F_{p^{2}}^{∗} is a square then it induces the automorphism ⟨u^{(p+1)/2}⟩_{p} on Ig_{+} and on Ig_{−}. It will be useful to have a list giving the inertia action on the cotangent spaces of irreducible components of X_{0,p}^{st} at the double points.

5.2.3 Table. Inertia action on cotangent spaces at double points. The bottom row in this table gives the action of γ̄(u)^{∗} on the cotangent space given in the top row for u ∈ F_{p^{2}}^{∗} fixing the corresponding double point (for x_{(2,0),s} and x_{(0,2),s} this is no condition on u; for x_{±,s} it means that u is a square).

x_{(2,0),s}^{∗}Ω_{Ig_{(2,0)}/\overline{F}_{p}}^{1} x_{(2,0),s}^{∗}Ω_{D_{s}/\overline{F}_{p}}^{1} x_{±,s}^{∗}Ω_{Ig_{±}/\overline{F}_{p}}^{1} x_{±,s}^{∗}Ω_{D_{s}/\overline{F}_{p}}^{1} x_{(0,2),s}^{∗}Ω_{Ig_{(0,2)}/\overline{F}_{p}}^{1} x_{(0,2),s}^{∗}Ω_{D_{s}/\overline{F}_{p}}^{1}
u^{0}  u^{p−1}  u^{p+1}  u^{1−p}  u^{0}  u^{p−1}

5.3 Some generalities on quotients of stable curves. The following is based on [19], pp. 508–510. There it is proved that the quotient of a smooth curve over a noetherian regular scheme by a finite group acting trivially on the base is again a smooth curve. We do the same kind of computations for stable curves over discrete valuation rings.

5.3.1 Lemma. Let V be a complete discrete valuation ring with uniformizer π. Let A := V [[x, y]]/(xy −π^{e}) and let G be a finite subgroup of Aut_{V} (A) that does not interchange the two branches modulo π. Let A^{G} be the subring of G-invariants in A. Let B := V [[u, v]]/(uv − π^{eg}), where g := #G. Define B ↪ A by u ↦ N(x), v ↦ N(y), where N: A → A^{G} is the map a → ∏_{σ∈G} σ(a). Then the image of B in A is A^{G}.

Proof. Clearly we have that B ⊂ A^{G}. It is a well known fact that both A and B are normal. Therefore it suffices to show that A is finitely generated as a B-module (then the same holds for A^{G}) and that the field of fractions Frac(B) of B is equal to Frac(A^{G}).

We will show that 1, x, y,…,x^{g−1}, y^{g−1} generate A as B-module. For each σ ∈ G we have σ(x) = xt_{σ} and σ(y) = yt_{σ}^{−1} with t_{σ} ∈ A^{∗}. Hence N(x) = x^{g}t and N(y) = y^{g}t^{−1} for some t ∈ A^{∗}. Now let a ∈ A. Using the relation xy = π^{e} we can write a = ∑ a_{i}x^{i} + ∑ b_{i}y^{i} with a_{i} and b_{i} in V. It follows that we can write a = ∑_{i=0}^{g−1} a_{i}x^{i} + ∑_{i=0}^{g−1} b_{i}y^{i} + ua′ + va′′ with a′ and a′′ in A. Iteration of this process shows that a is a linear combination over B of the x^{i} and y^{i} with 0 ≤ i < g.

Let B′ be the local ring of Spec(B) at the generic point of branch given by u = 0 and let A′ be the local ring of Spec(A) at the generic point of x = 0. Then B′ ⊂ A′ is a finite extension of discrete valuation rings, both with uniformizer π. Hence the degree of the extension of fraction fields is the degree of the extension of residue fields. Let k := V/πV. The extension of the residue fields can be written as k((v)) ⊂ k((y)) and 1, y,…,y^{g−1} is a basis of k((y)) over k((v)) because v = y^{g}t^{−1}. It follows that Frac(A^{G}) = Frac(B) since Frac(A) has dimension g over both of them. □

5.3.2 Lemma. Let V and A be as in Lemma 5.3.1 and let G be a finite subgroup of Aut_{V} (A) that interchanges the two branches mod π. Let H be the subgroup of G that stabilizes the two branches. Then A^{G} is regular and as parameters one can take π and N_{H}(x) + σN_{H}(y) where σ ∉ H and N_{H}: A → A^{H} is the map a ↦ ∏_{σ∈H} σ(a).

Proof. Let us first take the quotient by H and let h := #H. By Lemma 5.3.1 we are reduced to the situation where Z/2Z acts on V [[x, y]]/(xy − π^{eh}). Now we have σ(x) = yu with u a unit. We replace the coordinate y by σ(x). Then the situation is: A = V [[x, y]]/(xy − π^{eh}u) and σ(x) = y. Any a ∈ A can be written uniquely as a = a_{0} + ∑_{i>0} a_{i}x^{i} + ∑_{i>0} b_{i}y^{i} with a_{0}, a_{i} and b_{i} in V. We see that a = σ(a) if and only if a_{i} = b_{i} for all i. In other words, we have A^{σ} = {a_{0} + ∑_{i>0} a_{i}(x^{i} + y^{i})}. All the x^{i} + y^{i} can be expressed (with coefficients in Z) in t := x + y and xy = π^{eh}u. Iteration shows that A^{σ} = V [[t]]. □

5.3.3 Proposition. Let V be a discrete valuation ring with uniformizer π and residue field k. Let X → Spec(V) a stable curve whose generic fibre is smooth. Let G be a finite subgroup of Aut_{V} (X). Then we have:

(i) The special fibre of G\X is reduced and its singularities are ordinary double points. In other words: G\X is a semi-stable curve over V.

(ii) The image in (G\X)_{k} of a smooth point P of X_{k} is smooth and if π and x are parameters at P then π and N_{Stab}(x) are parameters at the image of P.

(iii) Let P ∈ X_{k}(k) be singular, say with complete local ring isomorphic over V to the ring V [[x_{P} , y_{P}]]/(x_{P} y_{P} = π^{e_{P}}). Let Q be the image of P in (G\X)_{k}(k). Then Q is singular if and only if Stab_{P} fixes the two branches at P. If Q is singular then the complete local ring of Q is isomorphic to V [[x_{Q}, y_{Q}]]/(x_{Q}y_{Q} − π^{e_{Q}}) with e_{Q} = e_{P} · #Stab_{P}, x_{Q} = N_{Stab_{P} (x_{P})} and y_{Q} = N_{Stab_{P} (y_{P})}. If Q is smooth then π and N_{H}(x_{P}) + σN_{H}(x_{P}) are parameters at Q, where H ⊂ Stab_{P} is the subgroup stabilizing the two branches at P and σ ∈ Stab_{P}, σ ∉ H.

(iv) Let Y be an irreducible component of X_{k} and Z the image of Y in (G\X)_{k}. Let H := Stab_{Y} and K := ker(H → Aut_{k}(Y)). Then the restriction to Y of the quotient morphism X → G\X factorizes as Y → (H/K)\Y → Z, where the first morphism is separable and the second is purely inseparable of degree #K.

Proof.

6 Divisors and leading terms of new forms.

6.1 Definitions. Let f be a normalized new form of weight k ≥ 2 on \overline{M}(Γ_{0}(p^{2}), Γ_{1}(N))\overline{Q} as in §3.2. We fix an embedding \overline{Q} → \overline{Q}_{p}. Let X_{0}^{st} denote the model of \overline{M}(Γ_{0}(p^{2}), Γ_{1}(N)) over Z[π] as constructed in §5.1; recall that π^{p+1} = 1−ζ_{p}. We fix an embedding Z[π] → \overline{Z}_{p}. Then we can view f as a rational section of the invertible sheaf \underline{ω}^{⊗k} on X_{0,\overline{Z}_{p}}^{st}. As such, f has a divisor. This is not completely “standard” since \overline{Z}_{p} is not noetherian, so we must explain what we mean. Let v_{p}: \overline{Q}_{p}^{∗} → Q denote the normalized valuation. Then v_{p} extends uniquely to the local rings of X_{0,\overline{Z}_{p}}^{st} at the generic points of the irreducible components of the fibre X_{0,p}^{st} over \overline{F}_{p}. We can now define:

div(f) = H + a_{(2,0)}Ig_{(2,0)} + a_{(0,2)}Ig_{(0,2)} + a_{+}Ig_{+} + a_{−}Ig_{−} + ∑_{s} a_{s}D_{s}  (6.1.1)

with H an effective “horizontal” divisor (as in the theory of arithmetic surfaces, i.e., it is the closure of the scheme of zeros of f over \overline{Q}_{p}), and with the a_{∗} ∈ Q the valuations of f at the corresponding generic points. If f has valuation a_{∗} along an irreducible component C of X_{0,p}^{st} then (p^{−a_{∗}}f)|_{C} is a rational section of \underline{ω}^{⊗k}|_{C}, called the leading term of f along C. The numbers a_{(2,0)}, a_{(0,2)}, a_{+} and a_{−} can be read off from the q-expansions of f. More precisely, we have formal charts at infinity:

1: (Tate(q), μ_{p^{2}}, 1↦ζ_{N})  over Z_{p}[π, ζ_{N}][[q]]
2: (Tate(ζ_{p}^{i}q), ⟨q^{1/p}⟩, 1↦ζ_{N}) over Z_{p}[π, ζ_{N}][[q]]  (i ∈ F_{p}^{∗})
3: (Tate(q), ⟨q^{1/p^{2}}⟩, 1↦ζ_{N}) over Z_{p}[π, ζ_{N}][[q^{1/p^{2}}]]

(6.1.2)

The closed fibre of chart 1 is a chart for Ig_{(2,0)}; chart 3 gives a chart for Ig_{(0,2)} and one can check that chart 2 gives a chart for Ig_{+} (resp. Ig_{−}) if −i is (resp. is not) a square in F_{p}. Prop. 3.5.3 can now be interpreted as follows.

6.2 Proposition. Let f be as in §6.1, χ as in §3.2 and a_{n}, a′_{n} and a′′_{n} as in 3.3.6. 1. a_{(2,0)} = v_{p}(∑ a_{n}q^{n}) = 0.

2. a_{(0,2)} = v_{p}(∑ a′′_{n}q^{n}) = −k.

3. if χ = π(τ^{m}, τ^{−m}) with 0 < m < (p−1)/2 then a_{+} = a_{−} = v_{p}(∑ a′_{n}q^{n}) = −m/(p−1),

4. if χ = π(τ_{2}^{(p−1)m}) with 0 < m < (p+1)/2 then a_{+} = a_{−} = v_{p}(∑ a′_{n}q^{n}) = (1−m)/(p−1),

5. if χ = π^{−}(τ^{(p−1)/2}) then a_{+} = a_{−} = v_{p}(∑ a′_{n}q^{n}) = −1/2. □

Let a := a_{+} = a_{−}; then we have −\frac{1}{2} ≤ a ≤ 0. Let b := min_{s} a_{s}. Our main interest in this section will be to get more information on b and to determine the leading terms of f along the components of X_{0,p}^{st} of Igusa type.

6.3 Lemma. Let V be a complete discrete valuation ring. Let n ≥ 1, t ≠ 0 in the maximal ideal of V and let X := Spec(V [[x, y]]/(xy − t)). Suppose that f ∈ O_{X}(X), considered as a function on the subscheme defined by the ideal (y, π), has a zero of order k at x = 0. Then the valuation of f at the generic point of the subscheme defined by (x, π) is at most k times the valuation of t.

Proof. Let π ∈ V be a uniformizer and let f̃ = ∑_{i,j≥0} a_{i,j}x^{i}y^{j} ∈ V [[x, y]] be a lift of f. Using the relation xy = t we can write f̃ = a_{0} + ∑_{i>0} a_{i}x^{i} + ∑_{i>0} b_{i}y^{i}. Modulo (y, π) we have: f̃ = \overline{a_{0}} + ∑_{i>0} \overline{a_{i}}x^{i}. Hence \overline{a_{i}} = 0 for i < k and \overline{a_{k}} ≠ 0. On the open subscheme D(y) of X where y is invertible the subscheme defined by (x, π) is defined by the equation π = 0. It follows that the valuation we are after is equal to the valuation of a_{0} + ∑_{i>0} a_{i}t^{i}y^{−i} + ∑_{i>0} b_{i}y^{i} in O_{X}(D(y)) = V [[y, ty^{−1}]][y^{−1}]. The valuation of the coefficient of y^{−k} is k times the valuation of t. □

6.4 Proposition. We have:

b ≤ a + ⌊\frac{k − 1}{2}⌋\frac{2}{p^{2} − 1}

where for any real number x, ⌊x⌋ is the largest integer less than or equal to x.

Proof. We may suppose that b > a. Then div(p^{−a}f) = H−aIg_{(2,0)}−(a+k)Ig_{(0,2)}+∑_{s}(a_{s}−a)D_{s}. Since X_{0,\overline{Z}_{p}}^{st} is normal, p^{−a}f is a section of \underline{ω}^{⊗k}(−cusps) in an open neighborhood of Ig_{+}. Recall that in [19] §12.8 a construction is given of a global section A^{1/(p−1)} of \underline{ω} on \overline{M}(Ig(p)) which has a first order zero at every supersingular point, and no other zeros. Because Ig_{+} is isomorphic over \overline{M}(Γ_{1}(N))\overline{F}_{p} to \overline{M}(Ig(p)/\{±1\}, Γ_{1}(N)), it follows that there is at least one s where (p^{−a}f)|_{Ig_{+}} has a zero of order at most ⌊\frac{k−1}{2}⌋. The complete local ring of X_{0}^{st}_{,Z[π]} at x_{+,s} is isomorphic to W [π][[x, y]]/(xy − π^{2}) (see §5.2). We can apply Lemma 6.3 with V a finite extension of W [π] over which f is defined. □

In the same way one can easily show that b > −k. However, we need a better lower bound for b. To get such a bound we will use the Kodaira-Spencer morphism.

6.5 The Kodaira-Spencer morphism. Recall ([19], §10.13.10, and [15], III, §9) that for any morphisms of schemes f: X → S and g: S → T, with f proper and smooth, one has the Kodaira-Spencer class:

KS ∈ Γ (S, (R^{1}f_{∗}T_{X/S}) ⊗_{O_{S}} Ω_{S/T}^{1} )  (6.5.1)

inducing a morphism of O_{S}-modules:

KS: (R^{1}f_{∗}T_{X/S})^{∨} ⟶ Ω_{S/T}^{1}  (6.5.2)

where T_{X/S} = \underline{Hom}_{O_{X}}(Ω_{X/S}^{1}, O_{X}) and (−)^{∨} denotes the O_{S}-dual. For f′: X′ → S′ obtained from f by a base change S′ → S over T, there is an obvious compatibility between the elements in 6.5.1 for f and f′. If R^{1}f_{∗}T_{X/S} is locally free, then it is of course equivalent to give 6.5.1 or 6.5.2.

For E → S an elliptic curve we will interpret 6.5.2, via Serre duality, as a morphism KS: \underline{ω}_{E/S}^{⊗2} → Ω_{S/T}^{1} . It is known ([19], Thm. 10.13.11) that for the universal elliptic curve E/M the Kodaira-Spencer morphism is an isomorphism with a first order pole at infinity:

KS: \underline{ω}^{⊗2} ˜ ⟶ Ω_{\overline{M}/Z}^{1}(cusps)  (6.5.3)

Equivalently, viewing KS as a section of the line bundle \underline{ω}^{⊗−2} ⊗_{O_{M}} Ω_{M/Z}^{1}, we can say that the divisor of KS on \overline{M} is “−cusps”. We want to study the divisor of KS for X_{0,\overline{Z}_{p}}^{st} → Spec(\overline{Z}_{p}), but since Ω_{X_{0}^{st}/\overline{Z}_{p}}^{1} is not a line bundle, we will consider the composition:

KS: \underline{ω}^{⊗2} ⟶ Ω_{X_{0}^{st}/\overline{Z}_{p}}^{1} ↪ ω_{X_{0}^{st}/\overline{Z}_{p}}  (6.5.4)

where ω_{X_{0}^{st}/\overline{Z}_{p}} is the dualizing sheaf ([11], I, §2.3). As in §6.1 multiplicities of components of X_{0,p}^{st} are measured with the normalized p-adic valuation.

6.5.5 Proposition. 1. The divisor of the Kodaira-Spencer morphism in 6.5.4 is:

−cusps + 2Ig_{(0,2)} + \frac{1}{p + 1} ∑_{s} D_{s}

2. On X_{\overline{Z}_{p}}^{st}: div(KS: \underline{ω}^{⊗2} → ω_{X^{st}/\overline{Z}_{p}}) = −cusps + ∑_{i,P} Ig_{i,P} + \frac{p}{p+1} ∑_{i,s} D_{i,s}.

Proof. We will only prove (1); the proof of (2) is analogous. The multiplicities of the cusps and the components of Igusa type can be seen in the formal charts 6.1.2 at infinity. One knows ([19], Thm. 10.13.11) that KS for Tate(q)/Z[[q]]/Z maps (dt/t)^{⊗2} to dq/q. In all three charts, (dt/t)^{⊗2} is a local generator of \underline{ω}^{⊗2}; local generators for ω_{X_{0}^{st}/\overline{Z}_{p}} are dq/q, dq/q and d(q^{1/p^{2}})/q^{1/p^{2}} = p^{−2}dq/q. It remains to compute the multiplicities of the D_{s}. To do this we use the computations of [13], §2.2.4 alluded to at the end of §5.1.

As explained in [19], Thm. 13.4.7, the complete local ring of M(Γ_{0}(p^{2}), Γ_{1}(N)) at a supersingular point s ∈ M(Γ_{1}(N))(\overline{F}_{p}) is isomorphic to W [[x, y]]/(f_{0} + pf_{1}), with f_{0} = (x^{p^{2}} − y)(x − y)^{p−1}(x − y^{p^{2}}) and f_{1} a unit. Moreover, the coordinates x and y are the “local moduli of source and target”. Let f be a local generator of \underline{ω}^{⊗2} at s. Then, since M(Γ_{1}(N)) → M is etale at s, KS(f) = unit · dx. In [13], §2.2.4 it is shown that u and v, defined by y = vx and x = uπ^{p−1}, induce non-zero rational functions on D_{s} and that du induces a non-zero rational differential form on D_{s}. It follows that the valuation of dx along D_{s} is the valuation of π^{p−1}, i.e., 1/(p + 1). □

6.6 Proposition. Let f be as in §6.1 and a and b as in Prop. 6.4. Then we have:

a − \frac{k}{p + 1} ≤ b

Proof. Let us first suppose that k is even. Then ω(f) := KS^{⊗k/2}(f) is a section of ω_{X_{0}^{st}/\overline{Z}_{p}}^{⊗k/2} with divisor:

div(ω(f)) = H − \frac{k}{2}cusps + a(Ig_{+} + Ig_{−}) + ∑_{s} a′_{s}D_{s}  (6.6.1)

with a′_{s} = a_{s} + \frac{k}{2(p+1)}. We have to prove that for all s: a − a′_{s} ≤ \frac{k}{2(p+1)}. Assume that for some s we have a − a′_{s} > \frac{k}{2(p+1)}. Then p^{−a′_{s}}ω(f)|_{D_{s}} is a global section of ω_{X_{0}^{st}/\overline{Z}_{p}}^{⊗k/2}|_{D_{s}}; according to Lemma 6.3 is has zeros of order at least k/2 at x_{(2,0),s} and x_{(0,2),s}. On the other hand, deg_{D_{s}}(ω_{X_{0}^{st}/\overline{Z}_{p}}) = p+1. One concludes that at x_{+,s} or x_{−,s}, p^{−a′_{s}}ω(f)|_{D_{s}} has a zero of order at most k(p − 1)/4. Lemma 6.3 now gives a − a′_{s} ≤ k/(2(p+1)). This finishes the proof for k even.

Let us now deal with the odd k. In that case we can simply repeat the proof for k even but now applied to f^{⊗2}; the fact that f^{⊗2} is not an eigenform does not matter at all. □

6.7 q-expansions of leading terms. Let f, a and b be as in Prop. 6.6. We are interested in the q-expansions of the leading terms of f along the components of X_{0,p}^{st} that are of Igusa type. More precisely, we want to evaluate f on the formal charts 6.1.2 and take the leading terms (in the p-adic sense as in §6.1) of the power series obtained in that way. We have already computed those power series in §3.3 and §3.4; their p-adic valuations are given in Prop. 3.5.3. It seems convenient to formulate the result one gets not on the Ig_{∗} but on \overline{M}(Γ_{1}(N))\overline{F}_{p} and on \overline{M}(Ig(p), Γ_{1}(N))\overline{F}_{p}. Recall from §5.2 that we have isomorphisms:

ϕ_{(2,0)}: \overline{M}(Γ_{1}(N))\overline{F}_{p} ˜ ⟶ Ig_{(2,0)}, ϕ_{(0,2)}: \overline{M}(Γ_{1}(N))\overline{F}_{p} ˜ ⟶ Ig_{(0,2)}  (6.7.1)

and two morphisms:

ϕ_{±}: \overline{M}(Ig(p), Γ_{1}(N))\overline{F}_{p} ⟶ Ig_{±}  (6.7.2)

that are the quotient for the action of {1, −1} on Ig(p). Note that ϕ_{(0,2)} is “exotic” ([19], §11) in the sense that it is not compatible with the maps to \overline{M}; the other three are not exotic. Via these morphisms we can interpret the leading terms (p^{−a_{∗}}f)|_{Ig_{∗}} as modular forms on \overline{M}(Γ_{1}(N))\overline{F}_{p} or on \overline{M}(Ig(p), Γ_{1}(N))\overline{F}_{p} with poles (possibly) in the super singular points. Using the description of the inertia action on Ig_{±} given in §5.2 one finds that ϕ_{±}^{∗}(p^{−a}f|_{Ig_{±}}) has eigenvalue x^{−2(p−1)a} for ⟨x⟩_{p}^{∗}, x ∈ F_{p}^{∗}. Let (again) A^{1/(p−1)} ∈ \overline{M}(Ig(p), \underline{ω}) denote the (p−1)th root of the Hasse invariant A as constructed in [19], §12.8; then A^{1/(p−1)} has non-zero constant q-expansion at all cusps and ⟨x⟩_{p}^{∗}A^{1/(p−1)} = xA^{1/(p−1)} for all x in F_{p}^{∗}. It follows that f′_{±} := (A^{1/(p−1)})^{2(p−1)a}ϕ_{±}^{∗}(p^{−a}f|_{Ig_{±}}) is a modular form on \overline{M}(Γ_{1}(N))\overline{F}_{p}, possibly with poles in the supersingular points, of weight k + 2(p−1)a, with character ε, whose q-expansion at (Tate(q), 1↦ζ_{N}) is up to a constant factor and some (dt/t)’s equal to that of ϕ_{±}^{∗}(p^{−a}f|_{Ig_{±}}) at (Tate(q), q, 1↦ζ_{N}). Using the results of §3 it is now straightforward (but somewhat long) to prove the following proposition.

6.7.3 Proposition. The four forms ϕ_{(2,0)}^{∗}(f|_{Ig_{(2,0)}}), ϕ_{(0,2)}^{∗}(p^{k}f|_{Ig_{(0,2)}}) and f′_{±} are eigenforms for all T_{l}^{∗} (including l=p) with character ε and weights k, p^{2}k and k + 2(p−1)a respectively. The first two have eigenvalues \overline{a_{l}}, where \overline{a_{l}} is the image of a_{l} as in 3.3.5 under \overline{Z}_{p} → \overline{F}_{p}. The eigenvalues of f′_{±} depend on the character χ associated to f as in §3.2. For χ = π(τ^{m}, τ^{−m}) with 0 < m < (p−1)/2 one has T_{l}^{∗}f′_{±} = l^{−m}\overline{a_{l}}f′_{±} (l ≠ p) and T_{p}^{∗}f′_{±} = \overline{λ′_{p}}f′_{±} (λ′_{p} as in 3.4.4). For χ = π(τ_{2}^{(p−1)m}), with 0 < m < (p+1)/2, one has T_{l}^{∗}f′_{±} = l^{1−m}\overline{a_{l}}f′_{±} (l ≠ p) and T_{p}^{∗}f′_{±} = 0. For χ = π^{−}(τ^{(p−1)/2}) one has T_{l}^{∗}f′_{±} = l^{(p−1)/2}\overline{a_{l}}f′_{±} (l ≠ p) and T_{p}^{∗}f′_{±} = \overline{b_{p}}f′_{±} (b_{p} as in Prop. 3.4.13). □

6.8 Forms of weight 2 defined over unramified extensions. From now on we suppose that f is a normalized new form on \overline{M}(Γ_{0}(p^{2}), Γ_{1}(N))\overline{Q} of weight 2, character ε and that f is defined over an unramified subextension of Q_{p} → \overline{Q}_{p} (recall that we have a fixed \overline{Q} → \overline{Q}_{p}). In this case we have a_{(2,0)}∈Z, a_{(0,2)}∈Z, a∈(p−1)^{−1}Z and a_{s}∈(p+1)^{−1}Z for all s (cf. 6.1.1). Let ω(f) := KS(f) ∈ H^{0}(X_{0,\overline{Z}_{p}}^{st}, ω_{X_{0}^{st}/\overline{Z}_{p}})⊗Q (note that since f is a cusp form, KS(f) has no pole at ∞); then div(ω(f)) is given in 6.6.1. Let b′ be the minimum of the a′_{s}, hence b′ = b + 1/(p+1). According to Prop.’s 6.4 and 6.6 we have:

a − \frac{1}{p + 1} ≤ b′ ≤ a + \frac{1}{p + 1}  (6.8.1)

Recall that we already know a in terms of χ (cf. Prop. 6.2), hence in terms of the ρ_{λ}|_{I} with λ |̸p (cf. Prop. 4.8). Now we want to find the exact value of b′ in terms of the representation \overline{ρ} associated to f. By definition, \overline{ρ}: G_{Q} → GL_{2}(\overline{F}_{p}) is the unique (up to isomorphism) semisimple continuous representation unramified outside pN such that trace(\overline{ρ}(Frob_{l})) = \overline{a_{l}} and det(\overline{ρ}(Frob_{l})) = ε(l)l^{k−1} for all l |̸pN.

Recall that the filtration w(g) of a non-zero mod p modular form g ∈ H^{0}(\overline{M}(Γ_{1}(N))\overline{F}_{p}, \underline{ω}^{⊗k}) of weight k is defined to be the smallest integer k′ for which there exists a form g′ ∈ H^{0}(\overline{M}(Γ_{1}(N))\overline{F}_{p}, \underline{ω}^{⊗k′}) which, at some cusp, has q-expansion equal to that of g (see for example [14], §3 and references given there). Equivalently, (p−1)^{−1}(k − w(g)) is the minimum of the orders of the zeros of g at the super singular points. We extend this definition to forms g that are allowed to have poles at the super singular points (the form g′ is of course not allowed to have poles).

6.8.2 Proposition. Under the assumptions made at the beginning of §6.8 we have:

1. w(ϕ_{(2,0)}^{∗}(f|_{Ig_{(2,0)}})) = 2 − (p^{2}−1)b

2. w(f′_{±}) = 2 + (p^{2}−1)(a−b) = p+1 + (p^{2}−1)(a−b′) ≤ 2p

3. there exist super singular s∈\overline{M}(Γ_{1}(N))\overline{F}_{p} with a_{s} = b and x_{(2,0),s}, x_{(0,2),s}, x_{+,s} and x_{−,s} not in the support of H.

Proof. Let us write a = −α/(p−1); then 0 ≤ α ≤ (p−1)/2. Recall that b∈(p+1)^{−1}Z and that b′ = b + 1/(p+1). According to 6.8.1 there are at most three possible values for b′. Let us first deal with the case 0 < α < (p−1)/2. Then there are exactly two possibilities: b′ = −(α+1)/(p+1) and b′ = −α/(p+1); they are equivalent to b′ < a and b′ > a, respectively. If b′ < a and a_{s} = b then p^{−b′}ω(f)|_{D_{s}} is a non-zero global section of ω_{X_{0}^{st}/\overline{Z}_{p}}|_{D_{s}} on which inertia acts via I → F_{p^{2}}^{∗} followed by F_{p^{2}}^{∗} → \overline{F}_{p}^{∗}: u ↦ u^{(p−1)(α+1)} (see §4.6 for conventions about inertia). The degree of ω_{X_{0}^{st}/\overline{Z}_{p}}|_{D_{s}} is p+1. From Table 5.2.3 it follows that p^{−b′}ω(f)|_{D_{s}} has zeros at x_{(2,0),s} and x_{(0,2),s} of order α+1 and zeros at x_{+,s} and x_{−,s} of order (p−1)/2 − α.

The complete local ring at x_{+,s} of X_{0}^{st} is isomorphic to W [π][[x, y]]/(xy−π^{2}) (see §5.2). It follows from the construction as in §4.1 of X_{0}^{st} that the section p^{−b′}ω(f) over this ring is already defined over W [π^{2}], i.e., to study its divisor we may view p^{−b′}ω(f) as an element of the regular local ring W [π^{2}][[x, y]]/(xy − π^{2}). Since the p-adic valuation of p^{−b′}ω(f) along Ig_{+} equals a−b′ and v_{p}(π^{2}) = (p^{2}−1)/2 we have:

(H · D_{s})_{x_{+,s}} + \frac{p^{2} − 1}{2}(a − b′) = \frac{p − 1}{2} − α

where (H · D_{s})_{x_{+,s}} denotes the local intersection number. It follows that (H · D_{s})_{x_{+,s}} = 0. Likewise: (H · D_{s})_{x_{−,s}} = 0. This shows that x_{+,s} and x_{−,s} are not in the support of H. Using this information one can verify that w(f′_{±}) = 2 + (p^{2}−1)(a−b).

To prove that x_{(2,0),s} is not in the support of H we do a similar computation. Now we view p^{−b′}ω(f) as an element of W [π^{p−1}][[x, y]]/(xy−π^{p−1}). The fact that −b′/v_{p}(π^{(p−1)}) is the order of vanishing of p^{−b′}ω(f)|_{D_{s}} at x_{(2,0),s} gives the result. One can now verify that w(ϕ_{(2,0)}^{∗}f|_{Ig_{(2,0)}}) = 2 − (p^{2}−1)b.

Suppose now that b′ > a, i.e., that b′ = −α/(p+1). Then p^{−a}ω(f)|_{Ig_{+}} is a global section of ω_{X_{0}^{st}/\overline{Z}_{p}}|_{Ig_{+}} on which inertia acts via I → F_{p^{2}}^{∗} followed by F_{p^{2}}^{∗} → \overline{F}_{p}^{∗}: u ↦ u^{(p+1)α}. It follows that ϕ_{+}^{∗}(p^{−a}ω(f)|_{Ig_{+}}) is a global section of Ω^{1}(s.s.) on \overline{M}(Γ_{1}(N), Ig(p)) on which ⟨x⟩^{∗} has eigenvalue x^{2α} for all x∈F_{p}^{∗}; therefore it has zeros of order 2α modulo p−1 at the supersingular points. A computation shows that deg_{\overline{M}(Γ_{1}(N),Ig(p))} Ω^{1}(s.s.) = (p+1) deg_{\overline{M}(Γ_{1}(N))}(s.s.)− (p−1) deg_{\overline{M}(Γ_{1}(N))}(cusps). This implies that there exist super singular s where ϕ_{+}^{∗}(p^{−a}ω(f)|_{Ig_{+}}) has a zero of order exactly 2α, or equivalently, where p^{−a}ω(f)|_{Ig_{+}} has a zero of order exactly α. Fix such a s. Since (p^{2}−1)(b′−a)/2 = α, a computation as above shows that a_{s} = b and that x_{+,s} is not in the support of H. Likewise: x_{−,s} is not in the support of H. It follows that p^{−b′}ω(f)|_{D_{s}} has poles of order exactly α at x_{±,s}. The action of inertia shows that p^{−b′}ω(f)|_{D_{s}} has zeros of order exactly α modulo p+1 at x_{(2,0),s} and x_{(0,2),s}. Moreover, the orders of vanishing at x_{(2,0),s} and x_{(0,2),s} are equal. Since deg_{D_{s}} ω_{X_{0}^{st}} = p+1 we see that p^{−b′}ω(f)|_{D_{s}} has zeros of order exactly α at x_{(2,0),s} and x_{(0,2),s}. It follows that x_{(2,0),s} and x_{(0,2),s} are not in the support of H and that w(f′_{±}) and w(ϕ_{(2,0)}^{∗}(f|_{Ig_{(2,0)}})) are as claimed.

It remains to study the cases a = 0 and a = −1/2. Suppose first that a = 0. Then χ = π(τ_{2}^{(p−1)}), hence T_{p}^{∗}f = 0 by Prop. 3.3.4 and Lemma 3.5.1(4). By Prop. 6.7.3 ϕ_{(2,0)}^{∗}(f|_{Ig_{(2,0)}}) and f′_{+} are eigenforms for all T_{l}^{∗} (including l = p) and have the same eigenvalues, so their q-expansions are equal up to a scalar factor. Consequently: w(f′_{±}) = w(f). According to 6.8.1 b′ ∈ {1/(p+1), 0, −1/(p+1)}. If b′ = 1/(p+1) then b = 0, f|_{Ig_{(2,0)}} has no poles at the super singular points and since it is a cusp form of weight two it cannot have zeros at all super singular s hence w(f) = 2. If b′ = −1/(p+1) and a_{s} = b then p^{1/(p+1)}ω(f)|_{D_{s}} is a non-zero global section of ω_{X_{0}^{st}/\overline{Z}_{p}}|_{D_{s}} on which inertia acts by u ↦ u^{p−1}; hence it has zeros of order exactly 1 and (p−1)/2 at x_{(2,0),s} and x_{±,s}, respectively; computations as above prove the required statements.

Suppose now that b′ = 0. Then ω(f)|_{Ig_{(2,0)}}, ω(f)|_{Ig_{(0,2)}} and ω(f)|_{Ig_{±}} are global sections of Ω^{1}(s.s.) with q-expansions equal up to scalar factors. If all residues of ω(f)|_{Ig_{(2,0)}} at the super singular points are zero then the same is true on the other components of Igusa type and the ω(f)|_{D_{s}} are global sections of Ω_{D_{s}}^{1} for all s on which inertia acts trivially; but no such non-zero differential forms exist. Hence not all residues are zero. But then g := Aϕ_{(2,0)}^{∗}f|_{Ig_{(2,0)}} is a cusp form on \overline{M}(Γ_{1}(N))\overline{F}_{p} of weight p+1, w(g) = p+1 and T_{p}^{∗}g = 0; by [14], Prop. 3.3 this is impossible. It follows that b′ = 0 is impossible.

Finally, suppose that a = −1/2. According to 6.8.1 at most three values for b′ are possible. If b′ = −1/2−1/(p+1) and a_{s} = b then p^{−b′}ω(f)|_{D_{s}} is a non-zero global section of Ω_{D_{s}}^{1} on which inertia acts by u ↦ u^{−(p−1)(p+3)/2}. But no such differential forms exist, hence b′ = −1/2 or b′ = −1/2+1/(p+1). In these two cases computations as above prove the required statements. □

6.8.3 Corollary. Let k(\overline{ρ}) ∈ Z be the “Serre weight” associated to \overline{ρ} as in [14], Def. 4.3. Suppose that \overline{ρ} is irreduible. If k(\overline{ρ}) > p+1 then b = (2 − k(\overline{ρ}))/(p^{2}−1). If k(\overline{ρ}) = 2 and \overline{ρ} is of level one (see [14], §2.4) then b = −2/(p+1). If k(\overline{ρ}) = 2 and \overline{ρ} is of level two then b = 0. The case k(\overline{ρ}) = p+1 does not occur.

Proof. Let us first remark that the difference between k(\overline{ρ}) and Serre’s original definition in [25], §1–2 of the “weight” of \overline{ρ} is irrelevant in our situation; see [14], Rem. 4.4 and note that we have p > 2 and k(\overline{ρ}) ≡ 2(p−1). Theorem 4.5 of [14] shows that there exists a mod p cuspidal eigenform g of weight k(\overline{ρ}), level N and character ε such that ϕ_{(2,0)}^{∗}f|_{Ig_{(2,0)}} and g have the same eigenvalues for all T_{l}^{∗} (l ≠ p). If k(\overline{ρ}) > p+1 then T_{p}^{∗}g = 0 by [16], Prop. 4.12, hence ϕ_{(2,0)}^{∗}f|_{Ig_{(2,0)}} and g have the same q-expansions at a fixed cusp, so k(\overline{ρ}) = w(g) = w(ϕ_{(2,0)}^{∗}f|_{Ig_{(2,0)}}). If k(\overline{ρ}) = 2 and \overline{ρ} is of level two then T_{p}^{∗}g = 0 by [14], Thms. 2.5 and 2.6, hence w(ϕ_{(2,0)}^{∗}f|_{Ig_{(2,0)}}) = 2, hence b = 0. If k(\overline{ρ}) = 2 and \overline{ρ} is of level one then T_{p}^{∗}g ≠ 0 hence w(ϕ_{(2,0)}^{∗}f|_{Ig_{(2,0)}}) = w(θ^{p−1}g = 2p. (see [14], §3). If k(\overline{ρ}) = p+1 then T_{p}^{∗}g ≠ 0 and w(ϕ_{(2,0)}^{∗}f|_{Ig_{(2,0)}}) = w(θ^{p−1}g = p^{2}+p hence −b = 1 + 1/(p+1) which is impossible. □

6.8.4 Remark. 1. The representation \overline{ρ} an only be reducible if \overline{ρ}|_{I} ∼ χ_{p}^{α}⊕χ_{p}^{β}, where χ_{p}: G_{Q} → F_{p}^{∗} is the p-cyclotomic character and (α, β) equals (0, 1) or ((p−1)/2, (p+1)/2).

2. We have already seen in Prop. 4.8 that a is determined by ρ_{λ}|_{I}. Cor. 6.8.3 shows that b is determined by \overline{ρ}|_{I}, at least if \overline{ρ} is irreducible. Eqn. 6.8.1 gives a strong restriction on the possible combinations (ρ_{λ}|_{I}, \overline{ρ}|_{I}).

7 Modular parametrizations.

7.1 Definitions. For M a positive integer, let X_{0}(M) denote coarse moduli scheme over Z associated to the stack \overline{M}(Γ_{0}(M)). Let J_{0}(M) be the Néron model over Z of the jacobian J_{0}(M)_{Q} of X_{0}(M). An elliptic curve E over Q is said to be modular if it is an isogeny factor of some J_{0}(M)_{Q}; the smallest M for which this happens is called the level of E. The Shimura-Taniyama conjecture states that every elliptic curve E over Q is modular, and that the level of E equals the conductor of (the system of l-adic representations of) E. One knows that for modular elliptic curves the level equals the conductor ([5], [20] and [9]). A modular elliptic curve E of level M is called strong if there exists a closed immersion E↪J_{0}(M)_{Q}. It follows from the multiplicity one principle that such an immersion is unique up to sign.

Let E be a strong modular elliptic curve of level M. The corresponding strong modular parametrization ϕ: X_{0}(M)_{Q} → E is obtained as follows:

\begin{matrix}J_{0}(M)_{Q} & → & E \\ ↑ & ↗_{ϕ} \\ X_{0}(M)_{Q} \end{matrix}  (7.1.1)

where J_{0}(M)_{Q} → E is the dual of one of the two closed immersions E↪J_{0}(M)_{Q} and where X_{0}(M)_{Q} → J_{0}(M)_{Q} is the standard immersion sending the cusp ∞ to 0. Let E be the Néron model over Z of E. Then the Z-module Γ(E, Ω^{1}) is free of rank one; let ω be one of the two generators. We get the differential form ϕ^{∗}ω on X_{0}(M)Q. There is also another differential form on X_{0}(M)_{Q} related to E: the normalized newform ∑ a_{n}q^{n}dq/q corresponding to E. The multiplicity one principle gives:

ϕ^{∗}ω = c ∑_{n≥1} a_{n}q^{n} \frac{dq}{q}  (7.1.2)

for some c ∈ Q^{∗}. This number c is called the Manin constant of E (see [21], [22], [23] and [4]).

8 Manin constants.

37

9 More on parametrizations.

38

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