The kernel of the Eisenstein ideal

by János Csirik

B.A. (University of Cambridge) 1994

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in Mathematics

in the GRADUATE DIVISION

of the

UNIVERSITY of CALIFORNIA at BERKELEY

Committee in charge: Professor Kenneth A. Ribet, Chair

Professor Robert F. Coleman Professor Steven N. Evans

Spring 1999

The dissertation of János Csirik is approved:

Chair Date \overline{Date} \overline{Date}

University of California at Berkeley Spring 1999 The kernel of the Eisenstein ideal

Copyright 1999 by János Csirik

Abstract

The kernel of the Eisenstein ideal

by

János Csirik Doctor of Philosophy in Mathematics

University of California at Berkeley

Professor Kenneth A. Ribet, Chair

Let N be a prime number, and let J_{0}(N) be the Jacobian of the modular curve X_{0}(N). Let 𝕋 denote the endomorphism ring of J_{0}(N). In a seminal 1977 article, B. Mazur introduced and studied an important ideal I ⊆ 𝕋, the Eisenstein ideal. In this dissertation we give an explicit construction of the kernel J_{0}(N)[I] of this ideal (the set of points in J_{0}(N) that are annihilated by all elements of I). We use this construction to determine the action of the group Gal(\overline{ℚ}/ℚ) on J_{0}(N)[I]. Then we apply our results to study the structure of the old subvariety of J_{0}(NM), where M is a prime number distinct from N. Our results were previously known in the special case where N − 1 is not divisible by 16.

Professor Kenneth A. Ribet Dissertation Committee Chair

This dissertation is dedicated to

my mother, Erzsébet Czachesz; my father, János Csirik; and my wife, Susan Harrington.

I thank them for their love and support.

Contents

1 Introduction 1 2 Notation and setup 4 3 The Units of X_{1}(N) 8 4 Some units on X_{0}^{#} (N) 13 5 The Galois structure of J_{0}(N)[I] 23 6 The old subvariety of J_{0}(NM) 29 7 A unit calculation on X_{0}^{#} (N, M) 40 Bibliography 48

Acknowledgements

First of all, I thank Ken Ribet for many helpful conversations and for suggesting

this problem to me. I also thank Arthur Ogus for telling me some useful facts about étale cohomology.

I thank Matt Baker and Kevin Buzzard for the many fun conversations and semi-

nars we participated in together. I also learned much from conversations with Hendrik W. Lenstra, Jr., and Bjorn Poonen.

I thank Donald Knuth for creating TEX, the typesetting system used for this disser-

tation, and for assigning it to the public domain. I also thank the creators of the computer program PARI–GP [1], which was used for some preliminary calculations for this dissertation.

Chapter 1

Introduction

Let N be a prime number and let J_{0}(N) denote the Jacobian of the modular

curve X_{0}(N). The variety J_{0}(N) possesses certain naturally defined endomorphisms T_{ℓ} (for all primes ℓ ≠ N) and w. These endomorphisms together with ℤ (the multiplications by integers) generate the Hecke ring 𝕋_{N} of endomorphisms of J_{0}(N). In his celebrated article Modular Curves and the Eisenstein Ideal [11], Mazur defined the Eisenstein ideal I in 𝕋_{N} as the ideal generated by 1 + w and the 1 + ℓ − T_{ℓ} and used it to identify the possible rational torsion subgroups of elliptic curves defined over the rational numbers. The Galois module J_{0}(N)(\overline{ℚ})[I] plays an important role in [11] and later studies of the arithmetic geometry of the curve X_{0}(N).

Mazur proved that

J_{0}(N)[I] ≅ ℤ/nℤ × ℤ/nℤ

as groups, for n = (N − 1)/ gcd(N − 1, 12). In this dissertation we will study the action of the group Gal(\overline{ℚ}/ℚ) on J_{0}(N)[I]. The group J_{0}(N)[I] has two noteworthy Galois-invariant subgroups. The cuspidal subgroup C is generated by the divisor c = 0 − (the formal difference of the two cusps of X_{0}(N)). The group C is cyclic of order n and is pointwise fixed by Gal(\overline{ℚ}/ℚ). The Shimura subgroup Σ is a finite flat subgroup scheme of J_{0}(N) such that

Σ(\overline{ℚ}) = ker(β^{∗} : J_{0}(N) ) J_{1}(N)),

where β^{∗} is induced by the usual degeneracy map β : X_{1}(N) ) X_{0}(N). The group Σ is also cyclic of order n, but is isomorphic to μ_{n} as a group scheme.

In this dissertation we shall give an explicit construction of J_{0}(N)[I], and apply

the construction in various ways. Mazur’s paper [11] contains an explicit construction of J_{0}(N)[I] only in the case N ≢ 1 (mod 16), although he remarks in a few places that a general description would be desirable. Our construction identifies the action of Gal(\overline{ℚ}/ℚ) on J_{0}(N)[I].

If n is odd (equivalently N ≢ 1 (mod 8)) then C ∩ Σ = 0, so J_{0}(N)[I] ≅ C ⊕ Σ and

therefore we know the Galois action on J_{0}(N)[I].

If n is even then C ∩ Σ ≠ 0 and more is needed to find the Galois action. In

this case C + Σ has index 2 in J_{0}(N)[I]. Therefore it suffices to find an “extra” point P in J_{0}(N)[I] that is not in C + Σ. The knowledge of the Galois action on P, Σ and C then gives a description of the Gal(\overline{ℚ}/ℚ)-action of J_{0}(N)[I]. For the case n ≡ 2 (mod 4) (or equivalently N ≡ 9 (mod 16)), Mazur finds P by considering the Nebentypus covering X_{0}^{#} (N) ) X_{0}(N) of degree 2. Using a function constructed by Ogg and Ligozat, he obtains a divisor d on X_{0}^{#} (N) which turns out to be the pullback of a certain divisor on X_{0}(N) that gives the extra point on J_{0}(N).

This dissertation uses other coverings X_{0}^{#} (N) ) X_{0}(N) to generalize Mazur’s con-

struction and find extra points of J_{0}(N)[I] for any N ≡ 1 (mod 8). To find suitable divisors on our modular curves X_{0}^{#} (N), we use the theory of modular units: rational functions on a modular curve whose divisors are concentrated at the cusps. Our coverings X_{0}^{#} (N) ) X_{0}(N) are all intermediate to X_{1}(N) ) X_{0}(N), enabling us to rely on the theory of modular units on X_{1}(N). The units of X(N) are treated in Kubert and Lang’s [8]. We recall some of their results in Chapter 2. We then use the results of Chapter 2 to develop some results about the units of X_{1}(N) in Chapter 3. (The references [6, 8] also treat this case but restrict their attention to units whose divisors are supported at the rational cusps, and don’t explicitly give the data necessary for the descent to X_{0}^{#} (N).) In Chapter 4, we construct a divisor on X_{0}^{#} (N) and establish properties of the divisor that make our later arguments work. In Chapter 5, we prove that the extra point we obtain is in J_{0}(N)[I] and use this fact to prove the following theorem, conjectured by Ribet. For any positive integer k, let χ_{k} denote the kth cyclotomic character Gal(\overline{ℚ}/ℚ) ) (ℤ/kℤ)^{×} obtained via the identification Gal(ℚ(μ_{k})/ℚ)) ≅ (ℤ/kℤ)^{×}.

Theorem 1.1 J_{0}(N)[I] has a basis e_{1}, e_{2} over ℤ/nℤ such that a) c = e_{1} + 2e_{2};

b) e_{1} generates Σ; c) σ ∈ Gal(\overline{ℚ}/ℚ) acts via left multiplication by

( \begin{matrix}χ_{n}(σ) & (1 − χ_{2n}(σ))/2 \\ 0 & 1 \end{matrix} )

with respect to the given basis e_{1} = \binom{1}{0}, e_{2} = \binom{0}{1}.

In Chapters 6 and 7, we give another application of our construction of J_{0}(N)[I]. Let M be a prime number distinct from N. The old part of J_{0}(NM) is the abelian

subvariety of J_{0}(NM) generated by

A = im(α : J_{0}(N)^{2} ) J_{0}(NM)) and B = im(β : J_{0}(M)^{2} ) J_{0}(NM)),

where α and β are certain naturally defined degeneracy morphisms (to be specified precisely in Chapter 6). Each of A and B was determined by Ribet in [16]. Therefore, to complete the description of the old part of J_{0}(NM), we need to determine A ∩ B. This project was carried out partially by Ribet in [18], where he determined the odd part of A ∩ B. In [9], Ling went on to obtain partial results about the even part of A ∩ B when neither N nor M is congruent to 1 modulo 16. In Chapter 6, we prove that A ∩ B is Eisenstein and we use Theorem 1.1 to completely determine A ∩ B to obtain

Theorem 1.2 Let N and M be distinct primes. Let m = (M − 1)/ gcd(M − 1, 12), and let D^{−,−} = P_{1} − P_{N} − P_{M} + P_{NM} ∈ J_{0}(NM). Then A ∩ B is the unique subgroup of order gcd(n, m) of the cyclic group generated by D^{−,−}.

The symbols P_{1}, P_{N}, P_{M} and P_{NM} are the conventional names for the cusps of

X_{0}(NM). Their meaning will be explained in Chapter 6. Theorem 1.2 answers questions in [18], [16] and [12].

Chapter 7 contains some modular unit calculations that are used in Chapter 6.

Chapter 2

Notation and setup

For any non-zero rational number x, let num(x) denote the numerator of x, that

is, the smallest positive integer n such that n/x is an integer.

We will now briefly summarize the relevant properties of the modular curves we

will be using. The reader can find a thorough treatment of these in [3], as well as in the references cited below.

Let N be a positive integer. We shall consider the usual modular curves X_{0}(N),

X_{1}(N) and X(N), and their Jacobians J_{0}(N), J_{1}(N) and J(N). These correspond to the moduli problems of classifying an elliptic curve with a cyclic subgroup of order N, an elliptic curve with a point of order N, and an elliptic curve with an embedding of μ_{N} × ℤ/Nℤ compatible with the Weil pairing, respectively. These curves are all defined over ℚ, as are the usual degeneracy maps (which are Galois coverings) α : X(N) ) X_{1}(N), β : X_{1}(N) ) X_{0}(N) and γ = β ∘ α.

The curves X_{0}(N)_{ℂ}, X_{1}(N)_{ℂ} and X(N)_{ℂ} can also be regarded as compactified quo-

tients of the complex upper half plane ℌ^{∗}/Γ_{0}(N), ℌ^{∗}/Γ_{1}(N) and ℌ^{∗}/Γ(N), respectively, where

Γ_{0}(N) = | ({a \atop c} {b \atop d}) ∈ SL_{2}ℤ : c ≡ 0 (mod N)∥ ,
Γ_{1}(N) = | ({a \atop c} {b \atop d}) ∈ SL_{2}ℤ : c ≡ 0, d ≡ 1 (mod N)∥ ,
Γ(N) = | ({a \atop c} {b \atop d}) ∈ SL_{2}ℤ : a ≡ 1, b ≡ 0, c ≡ 0, d ≡ 1 (mod N)∥ ,

and these subgroups of SL_{2}ℤ act on the complex upper half plane ℌ via fractional linear transformations. The points introduced during the compactification are called cusps.

Now let N be an odd prime number, and let

r = (N − 1)/2.

The curve X_{0}(N) has two cusps, denoted 0 and . They are both defined over

ℚ and are distinguished by the fact that under the natural map X_{0}(N)_{ℂ} = ℌ^{∗}/Γ_{0}(N) ) X(1)_{ℂ} = ℌ^{∗}/SL_{2}ℤ, the cusp 0 is ramified with index N and the cusp is unramified.

The curve X_{1}(N) has N − 1 cusps that come in two groups. We shall use Klimek’s

notation in [6] for them. The cusps P_{1}, P_{2}, …, P_{r} are defined over ℚ and are mapped to 0 under β : X_{1}(N) ) X_{0}(N). The cusps Q_{1}, Q_{2}, …, Q_{r} are defined over ℚ(μ_{N})^{+} (the maximal totally real subfield of the Nth cyclotomic field) and are mapped to under β. All the cusps of X_{1}(N) are unramified with respect to β.

The curve X(N) has (N^{2} − 1)/2 cusps and we use Shimura’s notation in [15] to

regard them as pairs ±\binom{x}{y} with x, y ∈ 𝔽_{N}, not both equal to 0. In this representation, Gal(X(N)_{ℂ}/X_{0}(N)_{ℂ}) ≅ PSL_{2}𝔽_{N} acts naturally from the left. For 1 ≤ i ≤ r, the cusps \binom{∗}{i} are all defined over ℚ(μ_{N}) and map unramifiedly to P_{i} under α : X(N) ) X_{1}(N). For 1 ≤ i ≤ r, the cusps \binom{i}{0} are all defined over ℚ(μ_{N})^{+} and map to Q_{i} under α with ramification index N.

Shimura’s notation can be used to label the cusps of any modular curve. We

shall now provide the translations to Shimura’s system of all the names we use. On the curve X_{0}(N), the cusps 0 and (respectively) are called \binom{0}{1} and \binom{1}{0} (respectively). On the curve X_{1}(N), for any 1 ≤ t ≤ r, our notation P_{t} corresponds to \binom{0}{t}, while Q_{t} corresponds to \binom{t}{0}.

Recall that a unit of a modular curve is a rational function on the curve that has

its divisor concentrated at the cusps. (It is a unit of the ring of rational maps from the noncuspidal points of the curve to the affine line.) In [8], Kubert and Lang determined all the units of X(N). We briefly recall their results here, using ℤ/Nℤ × ℤ/Nℤ as the indexing group instead of their \frac{1}{N} ℤ/ℤ × \frac{1}{N} ℤ/ℤ. Let e = (e_{1}, e_{2}) be a pair of integers such that not both of e_{1} and e_{2} are divisible by N. One can use the classical Weierstrass σ and Dedekind η functions to define the Klein form k_{e}(τ) on ℌ. This form enjoys the properties

∀α = ({a \atop c} {b \atop d}) ∈ SL_{2}ℤ,  k_{e}(ατ) = (cτ + d)^{−1}k_{eα} (τ)  (K1)

(where eα denotes usual matrix multiplication) and

∀f = (f_{1}, f_{2}) ∈ Nℤ × Nℤ,  k_{e+f} (τ) = ε(e, f)k_{e}(τ),  (K2)

where

ε(e, f) = (−1)^{\frac{f_{1} f_{2}}{N^{2}} + \frac{f_{1}}{N} + \frac{f_{2}}{N}} exp(\frac{πi}{N^{2}} (e_{1}f_{2} − e_{2}f_{1})) .

These Klein forms are then used to define for all e = (e_{1}, e_{2}) ∈ (ℤ × ℤ) \ (Nℤ × Nℤ) the Siegel function

g_{e}(τ) = k_{e}(τ)η^{2}(τ).

Recall that

∀α = ({a \atop c} {b \atop d}) ∈ SL_{2}ℤ,  η^{2}(ατ) = ψ(α)(cτ + d)η^{2}(τ),  (N)

where ψ is defined by its values on the two standard generators of SL_{2}ℤ as

ψ (({1 \atop 0} {1 \atop 1})) = exp(\frac{πi}{6} ), ψ (({0 \atop −1} {1 \atop 0})) = exp(\frac{πi}{2} ) = i.

Now [8, Chapter 4, Theorem 1.3] says

Theorem 2.1 The units of X(N)_{ℂ} are exactly the functions of the form

g = c ∏_{e∈E} g_{e}(τ)^{m(e)} ,

for some constant c and some finite set E ⊆ ℤ × ℤ, where the m(e) satisfy the conditions

∑_{e∈E} m(e) ≡ 0 (mod 12),  (U1)
∑_{e=(e_{1},e_{2})∈E} e_{1}^{2}m(e) ≡ 0 (mod N),  (U2)
∑_{e=(e_{1},e_{2})∈E} e_{1}e_{2}m(e) ≡ 0 (mod N),  (U3)
∑_{e=(e_{1},e_{2})∈E} e_{2}^{2}m(e) ≡ 0 (mod N).  (U4)

The order of such a g at a cusp P = \binom{x}{y} of X(N) is as follows. Pick some α ∈

PSL_{2}𝔽_{N} such that α\binom{1}{0} = P, and let α be some lift of α to SL_{2}ℤ. Let (c_{1}(e), c_{2}(e)) = eα be the components of the usual matrix product of e and α; thus, we may take c_{1}(e) = xe_{1}+ye_{2}. Then

ord_{P}(g) = ∑_{e∈E} m(e)\frac{N}{2}B_{2} (\frac{c_{1}(e) mod N}{N} ) ,  (2.1)

where B_{2}(X) = X^{2} − X + 1/6 is the second Bernoulli polynomial, and we used x mod N to denote the smallest non-negative residue of x modulo N. For details and a derivation using the q-expansion of g, see [8, Chapter 2, §3].

Let e ∈ E and α ∈ SL_{2}ℤ. From (K1) and (N), we conclude that

g_{e}(ατ) = ψ(α)g_{eα} (τ),  (2.2)

so using (U1) and the fact that ψ(α)^{12} = 1 for any α, for g(τ) = c ∏ _{e∈E} g_{e}(τ)^{m(e)} we have

g(ατ) = c ∏_{e∈E} g_{eα} (τ)^{m(e)} .  (2.3)

By (K2), if e ≡ e′ (mod N), then g_{e}/g_{e′} is a root of unity. By (K1) with α =

({−1 \atop 0} {0 \atop −1}), if e + e′ = (0, 0), then g_{e}/g_{e′} = −1 (here we also used the fact that with this α, the ψ and the cτ + d factors in (N) are both −1, and so they multiply to 1). Hence all units of X(N)_{ℂ} can be put into the form g = c′ ∏ _{e∈E′} g_{e}(τ)^{m(e)} , with

E′ =  { (0, 1), (0, 2), …, (0, r),
(1, 0), (1, 1), (1, 2),  …,  (1, N − 1),
(2, 0), (2, 1), (2, 2),  …,  (2, N − 1),
⋮
(r, 0), (r, 1), (r, 2),  …,  (r, N − 1) }.

Kubert and Lang [8, Chapter 5, Theorem 3.1] then prove that the degree zero divisors on X(N) concentrated at the cusps span a finite subgroup of the divisor class group (this was also proved in general for all modular curves by Manin and Drinfeld, see [10, footnote to Corollary 3.6]). The number of cusps on X(N) is (N^{2} − 1)/2, which is also the cardinality of the set E′. Since the divisor of a rational function has degree 0, this implies that the functions g_{e} with e ∈ E′ are independent except for a single relation. A simple calculation using (2.1) shows that ∏ _{e∈E′} g_{e}(τ) is a constant, providing the sought-after relation. To sum up, we have shown that

Fact 2.2 The function g = c ∏ _{e∈E′} g_{e}(τ)^{m(e)} is constant if and only if the m(e) are the same for all e ∈ E′.

Chapter 3

The Units of X_{1}(N)

We now determine units of X_{1}(N)_{ℂ}. They can be identified with the units of

X(N)_{ℂ} that are invariant under Gal(X(N)_{ℂ}/X_{1}(N)_{ℂ}), which is generated by T = ({1 \atop 0} {1 \atop 1}) ∈ Gal(X(N)_{ℂ}/X(1)_{ℂ}) ≅ PSL_{2}𝔽_{N}. Therefore the units of X_{1}(N) can be determined from the knowledge of the units of X(N) and their transformation properties under T.

Definition 3.1 For all 1 ≤ i ≤ r, let

g_{i}(τ) = g_{(0,i)}(τ)

and

s_{i}(τ) = ∏_{j=0}^{N−1} g_{(i,j)} (τ).

Theorem 3.2 The units of X_{1}(N)_{ℂ} are exactly the functions of the form

g(τ) = c ∏_{i=1}^{r} g_{i}(τ)^{c_{i}} s_{i}(τ)^{d_{i}} ,

where c is a constant and the c_{i} and d_{i} satisfy

∑_{i=1}^{r} c_{i} + N ∑_{i=1}^{r} d_{i} ≡ 0 (mod 12),  (V1)
∑_{i=1}^{r} i^{2}c_{i} ≡ 0 (mod N),  (V2)
∑_{i=1}^{r} i^{2}d_{i} ≡ 0 (mod N),  (V3)

Proof. Let g(τ) = c ∏ _{e∈E′} g_{e}(τ)^{m(e)} be a unit of X(N). By determining the con-

ditions the m(e) must satisfy to be invariant under T, we can find a criterion for g to be a unit of X_{1}(N). Assume then that g is invariant under the action of T. By (2.3) and Fact 2.2, m : E′ ) ℤ must be constant on the orbits of T (acting on E′ from the right). Since (e_{1}, e_{2})T = (e_{1}, e_{1} + e_{2}), this shows that g can be written as a product of g_{i} and s_{i} as above, with c_{i} = m((0, i)) and d_{i} = m((i, 0)) = m((i, 1)) = ⋯ = m((i, N − 1)).

Condition (U1) translates immediately to (V1).

The condition (U2) translates as

∑_{e∈E′} e_{1}^{2}m(e) = N ∑_{i=1}^{r} i^{2}d_{i} ≡ 0 (mod N),

so it is necessarily satisfied. The condition (U3) translates as

∑_{e∈E′} e_{1}e_{2}m(e) = ∑_{i=1}^{r} i ∑_{j=0}^{N−1} jd_{i} = ∑_{i=1}^{r} iNrd_{i} ≡ 0 (mod N),

so it is also necessarily satisfied. Lastly, (U4) in our case is

∑_{e∈E′} e_{2}^{2}m(e) = ∑_{i=0}^{r} i^{2}c_{i} + ∑_{j=0}^{r} ∑_{i=0}^{N−1} i^{2}d_{j} = ∑_{i=0}^{r} i^{2}c_{i} + N\frac{(2N − 1)r}{3} ∑_{j=0}^{r} d_{j} ≡ 0 (mod N).

Since N is not divisible by 3, r(2N − 1) must be, so (U4) translates to (V2).

It remains to check that our function g is actually invariant under the action of T,

as opposed to being just invariant up to multiplication by a constant. This is not automatic, despite the fact that (2.3) has no extra constant factors. Indeed, by restricting our indexing set to E′ we sometimes have to convert some g_{e} (with e ∉ E′) occurring in (2.3) to some g_{e′} with e′ ∈ E′, thereby introducing a factor of a root of 1.

Using (K2), we can see that when we are acting by T on our g, this factor will be

∏_{j=1}^{r} ∏_{k=0}^{j−1} ε((j, k), (0, N))^{d_{j}} = ∏_{j=1}^{r} ∏_{k=0}^{j−1}(−1)^{d_{j}} exp(\frac{πi}{N} jd_{j})
= ∏_{j=1}^{r}(−1)^{jd_{i}} exp(\frac{πi}{N} j^{2}d_{j}) .

Hence to ensure invariance, we need

N ∑_{j=1}^{r} jd_{j} + ∑_{j=1}^{r} j^{2}d_{j} ≡ 0 (mod 2N).

The condition mod 2 is automatic since N is odd and j ≡ j^{2} (mod 2). The condition mod N is just (V3).

It is also clear from the above calculations that any such g as given in the statement

of the theorem satisfies (U1–4) and hence is actually a T-invariant unit of X(N), so we have completed the proof. ■

Remark. For any i, the Atkin–Lehner involution (associated to the matrix ({0 \atop −N} {1 \atop 0}))

interchanges the functions g_{i} and s_{i} up to constants. (For more details, see the proof of Theorem 3.5.) Therefore, we expect the conditions (V1–3) to be invariant under exchanging the c_{i}s and the d_{i}s. This is clear for (V2) and (V3), but it is also true for (V1). Indeed, since (N, 6) = 1, we have N^{2} ≡ 1 (mod 12), and hence

12 | C + ND {} 12 | N^{2}C + ND {} 12 | NC + D,

where we used the notation C = ∑ _{i=1}^{r} c_{i} and D = ∑ _{i=1}^{r} d_{i}.

Theorem 3.3 The order of such a function g (mentioned in Theorem 3.2) at the cusps is

ord_{P_{t}}(g) = ∑_{i=1}^{r} (\frac{c_{i}N}{2} B_{2} (\frac{it mod N}{N} ) + \frac{d_{i}}{12})
ord_{Q_{t}} (g) = ∑_{i=1}^{r} (\frac{c_{i}}{12} + \frac{d_{i}N}{2} B_{2} (\frac{it mod N}{N} )).

Proof. A straightforward calculation using (2.1) and the fact that ∑ _{i=1}^{r} B_{2}(i/N) =

−r/(6N) gives the order of g at the cusps of X(N). Then, using the ramification indices of the cusps of X(N) over the cusps of X_{1}(N), we obtain our result. ■

Finally, we give the transformation properties of our functions under the Galois

group of X_{1}(N)_{ℂ} over X_{0}(N)_{ℂ}.

Definition 3.4 As in [15, §2], for a odd positive integer u, let

{·}_{u} : ℤ ) {0, 1, …, (u − 1)/2}

be the function defined by

{a}_{u} ≡ ±a (mod u).

For brevity, let {·} denote {·}_{N}.

Theorem 3.5 For any b ∈ \{1, …, r\} and α = ({s \atop Nh} {f \atop t}) ∈ Γ_{0}(N), we have

g_{b}(ατ) = ψ(α)κ(α; b)g_{{bt}}(τ),

with

κ(α; b) = (−1)^{bh} exp(πi (−\frac{b^{2}ht}{N} )) (−1)^{⌊bt/N⌋} .

Although this fact will not be needed later, for reference we state that for any a ∈ {1, …, r} and α as above, we have

s_{a}(ατ) = ψ(α)^{N}κ′(α; a)s_{{as}}(τ),

with

κ′(α; a) = (−1)^{af} exp(πi (\frac{a^{2}sf}{N} + r\frac{a − \{as\}}{N} )) (−1)^{⌊as/N⌋} .

Proof. Using (2.2) we obtain

g_{b}(ατ) = g_{(0,b)}(ατ) = ψ(α)g_{(0,b)α} (τ) = ψ(α)g_{(bhN,bt)} (τ)
= ψ(α)ε((0, bt), (bhN, 0))g_{(0,bt)} (τ)
= ψ(α)(−1)^{bh} exp(\frac{πi}{N} (−b^{2}ht)) g_{(0,bt)} (τ).

If bt ≡ {bt} (mod N) then

g_{(0,bt)} (τ) = ε((0, {bt}), (0, N⌊bt/N⌋))g_{(0,{bt})}(τ)
= (−1)^{⌊bt/N⌋} g_{{bt}}(τ).

If bt ≡ −{bt} (mod N) then

g_{(0,bt)} (τ) = ε((0, −{bt}), (0, N(⌊bt/N⌋ + 1)))g_{(0,−{bt})}(τ)
= (−1)^{⌊bt/N⌋} g_{{bt}}(τ).

In either case, this completes the proof of the formula for g_{b}(ατ).

For s_{a}(ατ), a similar but more involved calculation can be used. Alternatively,

one might use the Atkin–Lehner involution w_{N} = ({0 \atop −N} {1 \atop 0}) ∈ SL_{2}ℤ and the q-expansion of g_{(e_{1},e_{2})} in [8, Chapter 2, §1, K4] via the identity w_{N}(τ) = ({0 \atop −1} {1 \atop 0}) (Nτ) to conclude that g_{a}(wτ)/s_{a}(τ) = c exp(πira/N), for some constant c that does not depend on a, and thereby reduce the calculation to the one done above. ■

Remark. Theorems 3.2 and 3.3 allow us to determine the group of divisors

supported at the cusps for any particular N. For example, consider the conjecture by Klimek in [6, p. 3]. Let J_{1}^{}(N) denote the group of divisors on X_{1}(N) supported at the set {P_{1}, P_{2}, …, P_{r}}. Klimek proved that

#J_{1}^{}(N) = 4^{1−r}N ∏_{χ≠1} B_{2,χ},

(where χ runs over all non-trivial even characters of (ℤ/Nℤ)^{×}, and B_{2,χ} denotes the generalized Bernoulli numbers of Kubota-Leopoldt, see also [8, Chapter 6, Theorem 3.4] for another proof), and conjectured (presumably without the benefit of a computer) that the group J_{1}^{}(N) is always cyclic. He confirmed this conjecture for all N ≤ 23. A simple (computer-aided) calculation using Theorems 3.2 and 3.3 shows that the conjecture is false for N = 29. We obtain

\begin{matrix}N & J_{1}^{}(N) \\ 2, 3, 5, 7 & 0 \\ 11 & ℤ/5ℤ \\ 13 & ℤ/19ℤ \\ 17 & ℤ/584ℤ \\ 19 & ℤ/4383ℤ \\ 23 & ℤ/37181ℤ × ℤ/11ℤ \\ 29 & ℤ/64427244ℤ × ℤ/4ℤ × ℤ/4ℤ \\ 31 & ℤ/1772833370ℤ × ℤ/10 \\ ⋮ & ⋮ \end{matrix}

Chapter 4

Some units on X_{0}^{#} (N)

Assume from now on that

N ≡ 1 (mod 8),

(in particular N ≥ 17).

Definition 4.1 Let C_{N} denote the group (ℤ/Nℤ)^{×}/ ± 1.

Since β : X_{1}(N)_{ℂ} ) X_{0}(N)_{ℂ} is a cyclic Galois covering of degree r, it has a unique

intermediate covering of X_{0}(N)_{ℂ} of any degree dividing r. Because of its uniqueness, any such curve is defined over ℚ (for a thorough treatment of these intermediate curves, see [2, IV, §3]). Letting n = (N−1)/ gcd(N−1, 12), we know from [11, II, §2] that the intermediate curve X_{2}(N)_{ℂ} ) X_{0}(N)_{ℂ} of degree n (the Shimura covering) is the largest étale covering of X_{0}(N)_{ℂ} through which β factors. (As remarked before, the cusps of X_{0}(N) are not branch points for β; it is the points with j = 0 and j = 1728 that ramify in β.)

Definition 4.2 Write n as

n = 2^{k}v

where 2^{k} is the largest power of 2 that divides n (and v = n/2^{k} is an odd integer). Set z = 3 if N ≡ 1 (mod 3) and z = 1 otherwise. For future use, we also set q = 3/z here.

Let

ϕ : X_{0}^{#} (N) ) X_{0}(N)

be the unique covering of degree 2^{k} through which β factors. Let J_{0}^{#} (N) = Jac(X_{0}^{#} (N)).

Observe that the definitions of k, v, z imply that

r = 2^{k+1}zv.

Since 2^{k} divides n, the Shimura covering factors through ϕ. This implies that ϕ is étale and that Σ_{0} = ker(ϕ^{∗} : J_{0}(N) ) J_{0}^{#} (N)) is contained in Σ = ker(J_{0}(N) ) J_{2}(N)).

The Galois group X_{1}(N)_{ℂ} over X_{0}(N)_{ℂ} is isomorphic to C_{N}, with ({s \atop Nh} {f \atop t}) mapping

to {t}. Let ξ be a generator of C_{N}. We will abuse notation to lighten it, and let the same ξ denote the generator of Gal(X_{1}(N)_{ℂ}/X_{0}(N)_{ℂ}) ≅ C_{N} and the corresponding generator of the Galois group of the function field extension; so that (ξf)(τ) = f(ξτ) for all functions f on X_{1}(N). Let Ω denote the set of 2^{k}th powers in C_{N}. Then Ω is the Galois group of X_{1}(N) over X_{0}^{#} (N) and

#Ω = 2zv.

By its uniqueness property, X_{0}^{#} (N) is Galois over X_{0}(N).

The curve X_{0}^{#} (N) is the coarse moduli space for the problem of classifying elliptic

curves with a point of order N, where (E, P) and (E′, P′) are to be considered equivalent if there is an isomorphism δ : E ) E′ such that δ(P) = ±b · P′ for some b ∈ Ω.

We shall now construct some units of X_{0}^{#} (N). They will first be given as units

of X_{1}(N), and to check that they are actually units of X_{0}^{#} (N) we shall need the following lemmas. In these lemmas (and later), for any element b ∈ C_{N} we let b̃ denote the representative for b in the set {1, 2, …, r}. For example, ∑ _{b∈C_{N}} b̃ = ∑ _{i=1}^{r} i = r(r + 1)/2.

Lemma 4.3 For any coset Ω′ of Ω,

∑_{b∈Ω′} b̃^{2} ≡ 0 (mod N)

Proof. Let μ be a primitive root modulo N. Then a set of representatives for C_{N}

in ℤ are 1, μ, …, μ^{(N−3)/2} . The representatives of a coset Ω′ of Ω are μ^{j}, μ^{j+2^{k}} , μ^{j+2·2^{k}} , …, μ^{j+((N−1)/2^{k+1} −1)2^{k}} for some 0 ≤ j < 2^{k}. If

b̃ ≡ ±μ^{j+t2^{k}} (mod N),

then

b̃^{2} ≡ μ^{2j+t2^{k+1}} (mod N),

so

∑_{b∈Ω′} b̃^{2} ≡ μ^{2j} ∑_{t=0}^{(N−1)/2^{k+1} −1} μ^{2^{k+1} t} = μ^{2j} \frac{μ^{N−1} − 1}{μ^{2^{k+1}} − 1} ≡ 0 (mod N)

by Fermat’s Little Theorem. ■

In the proofs of the next two lemmas, we shall use the following convention.

Convention 4.4 For P a statement, let [P] be 1 if P is true, 0 if P is false. Lemma 4.5 Let t be an integer relatively prime to N. Then

S = ∑_{b∈C_{N}} ⌊b̃t/N⌋

is even if and only if t is a square modulo N.

Proof. Note that ∑ _{b∈C_{N}} b̃ = r(r + 1)/2 is even (since r is divisible by 4), so

S = ∑_{b∈C_{N}} ⌊b̃t/N⌋ ≡ ∑_{b∈C_{N}}^{r} b̃t − ∑_{b∈C_{N}} N⌊b̃t/N⌋ = ∑_{b∈C_{N}} (b̃t mod N) (mod 2).

Now b̃t mod N is either {b̃t} or N − {b̃t} depending on whether b̃t mod N ≤ r or not, respectively. Therefore, we can write S as follows

S ≡ ∑_{b∈C_{N}} {b̃t} + ∑_{b∈C_{N}} [b̃t mod N > r](−{b̃t} + N − {b̃t})
≡ ∑_{b∈C_{N}} b̃ + ∑_{b∈C_{N}} [b̃t mod N > r](−{b̃t} + N − {b̃t})
≡ ∑_{b∈C_{N}} [b̃t mod N > r](N − 2{b̃t}) ≡ ∑_{b∈C_{N}} [b̃t mod N > r] (mod 2)

Define m to equal ∑ _{b∈C_{N}} [b̃t mod N > r]. Then

(−1)^{m}r! = ∏_{b∈C_{N}} {b̃t} ≡ ∏_{b∈C_{N}} (b̃t) = t^{r}r! (mod N)

Since N does not divide r!, this implies that

(−1)^{m} ≡ t^{r} (mod N).

Since t^{r} ≡ 1 (mod N) exactly when t is a square modulo N, this proves our lemma. ■

Lemma 4.6 Let Ω′ be a coset of Ω and s, f, t, h integers with st − Nfh = 1 and {t} ∈ Ω and

S = h(1 − t) ∑_{b∈Ω′} b̃ + ∑_{b∈Ω′} ⌊b̃t/N⌋.

The parity of S does not depend on the choice of Ω′.

Proof. First observe that st − Nfh = 1 implies that if t is even then h must be

odd, so in any case h(1 − t) ≡ t + 1 (mod 2). Therefore,

S ≡ (t + 1) ∑_{b∈Ω′} b̃ − N ∑_{b∈Ω′} ⌊b̃t/N⌋ = ∑_{b∈Ω′} b̃ + ∑_{b∈Ω′} (b̃t mod N) (mod 2).

Since t ∈ Ω, for b ranging over Ω the reductions of {b̃t} to C_{N} just range over Ω′, so we can once again use the method of the proof of Lemma 4.5. Accordingly,

S ≡ 2 ∑_{b∈Ω′} b̃ + ∑_{b∈Ω′} [b̃t mod N > r](−{b̃t} + N − {b̃t})
≡ ∑_{b∈Ω′} [b̃t mod N > r] (mod 2).

Let m = ∑ _{b∈Ω′} [b̃t mod N > r]. Then

(−1)^{m} ∏_{b∈Ω′} b̃ = ∏_{b∈Ω′} {b̃t} ≡ ∏_{b∈Ω′} (b̃t) = t^{#Ω} ∏_{b∈Ω′} b̃ (mod N)

Since N does not divide ∏ _{b∈Ω′} b̃,

(−1)^{m} ≡ t^{#Ω} (mod N),

and it is plain that the parity of m (which is the same as the parity of S) depends only on the choice of t and not on the choice of Ω′. ■

Recall from Definition 4.2 that q = 3/z.

Theorem 4.7 Define the following three functions on X_{1}(N):

f(τ) = ( ∏_{b∈Ω} gb̃(τ)^{2^{k} q}) ( ∏_{b∈C_{N}} gb̃(τ)^{−q} ) ,
g(τ) = ( ∏_{b∈Ω} gb̃(τ)^{−q}) ( ∏_{b∈ξΩ} gb̃(τ)^{q} ) ,
h(τ) = ( ∏_{b∈Ω} gb̃(τ)^{2q}) .

Then the following are true:

(a) the group Gal(X_{1}(N)_{ℂ}/X_{0}^{#} (N)_{ℂ}) is the subgroup of Gal(X_{1}(N)_{ℂ}/X_{0}(N)_{ℂ}) that fixes the function f;

(b) the functions g and h are invariant under Gal(X_{1}(N)_{ℂ}/X_{0}^{#} (N)_{ℂ}) and can therefore be regarded as being defined on X_{0}^{#} (N)_{ℂ};

(c) ξf = (−1)g^{2^{k}} f;

(d) g(ξg)(ξ^{2}g) …(ξ^{2^{k} −1}g) = −1;

(e) the divisor div(f) is divisible by 2^{k} in Div^{0}(X_{0}^{#} (N)).

Remark. Henceforth we will regard f, g and h as functions on X_{0}^{#} (N). By (a)

above, f descends to no smaller cover of X_{0}(N).

Note. The function f above is the analogue in our situation of the Ogg–Ligozat

function f_{OL} that was used in [11, II, Proposition (12.2)]. (In that paper, f_{OL} is called f.) Note however that if we restrict to the case of N ≡ 9 (mod 16) (equivalently n ≡ 2 (mod 4)) considered in that paper, our function f does not equal the function f_{OL}. Instead, the “correct” function f (the one we are using above) is equal to f_{OL}^{q}. However, since q is always equal to 1 or 3, and Mazur was constructing a point in a group of exponent two, f and f_{OL} worked equally well.

Proof. First we need to check (using Theorem 3.2) that f is actually a function on

X_{1}(N). Conditions (V1) and (V3) are clearly satisfied, since, in the notation of Theorem 3.2, each d_{i} is zero, and ∑ c_{i} is also zero. For (V2), note that we need that N divide

q (∑_{b∈Ω} b̃^{2}2^{k} − ∑_{b∈C_{N}} b̃^{2} ) = q (2^{k} ∑_{b∈Ω} b̃^{2} − \frac{r(r + 1)}{6}N)

The first term is divisible by N by Lemma 4.3, the second is divisible by N since clearly r(r + 1)/6 is an integer.

We have now confirmed that f is defined on X_{1}(N). It remains to check that the

largest subgroup Θ of C_{N} that fixes f is in fact Ω. Since the coefficients in f for those gb̃ with b ∈ Ω are different from those for which b ∉ Ω, we must have Θ ⊆ Ω.

To check Θ = Ω then, it remains to show that for any α = ({s \atop Nh} {f \atop t}) ∈ Γ_{0}(N) with

{t} ∈ Ω, we have f(ατ) = f(τ). Using Theorem 3.5 (and (V1) to get rid of the ψ factors), we need to confirm that

C = q (∑_{b∈Ω} 2^{k}(Nb̃h − b̃^{2}ht + N⌊b̃t/N⌋)) −

q ( ∑_{b∈C_{N}} (Nb̃h − b̃^{2}ht + N⌊b̃t/N⌋)) ≡ 0 (mod 2N)

The expression C can be thought of as six separate sums, and it turns out that each of them is divisible by 2N. This is obvious for the first, third, fourth and fifth; follows by Lemma 4.3 for the second; and follows by Lemma 4.5 for the sixth. Hence we have proved (a).

Similarly, we can check that g is defined on X_{0}^{#} (N). Again using Theorem 3.5, we

need to confirm that

− q(∑_{b∈Ω} (Nb̃h − b̃^{2}ht + N⌊b̃t/N⌋))

+ q( ∑_{b∈ξΩ} (Nb̃h − b̃^{2}ht + N⌊b̃t/N⌋)) ≡ 0 (mod 2N)

The divisibility by N is immediate by Lemma 4.3, and for divisibility by 2 observe that for any coset Ω′ of Ω we have

S = ∑_{b∈Ω′} (Nb̃h − b̃^{2}ht + N⌊b̃t/N⌋)
≡ ∑_{b∈Ω′} (b̃h − b̃ht + ⌊b̃t/N⌋) ≡ h(1 − t) ∑_{b∈Ω′} b̃ + ∑_{b∈Ω′} ⌊b̃t/N⌋ (mod 2).

Now an application of Lemma 4.6 completes the proof.

For h(τ), we need 12|2q(#Ω) to verify (V1). But 2q(#Ω) = 2q · 2zv = 12v so this

is clear. The rest of the proof is analogous to the proof for g(τ). This completes the proof of (b).

To prove (c), we calculate

(ξf)(τ) = f(ξτ) = ( ∏_{b∈Ω} gb̃(ξτ)^{2^{k} q}) ( ∏_{b∈C_{N}} gb̃(ξτ)^{−q} )
= ( ∏_{b∈Ω} gξ̃b(τ)^{2^{k} q}κ(ξ; b̃)^{2^{k} q}) ( ∏_{b∈C_{N}} gξ̃b(τ)^{−q}κ(ξ; b̃)^{−q} ) ,

so (since (−1)^{q} = −1) it suffices to show that

∏_{b∈Ω} κ(ξ; b̃)^{2^{k}} ∏_{b∈C_{N}} κ(ξ; b̃)^{−1} = −1.

Pick some ({s \atop Nh} {f \atop t}) ∈ Γ_{0}(N) that lifts ξ. Then t will necessarily generate (ℤ/Nℤ)^{×}, and in particular it will be a non-square modulo N. By Theorem 3.5, it suffices to show that

exp(\frac{πi}{N} (2^{k} ∑_{b∈Ω} (Nb̃h − b̃^{2}ht + N⌊b̃t/N⌋)) ) ×

exp (\frac{πi}{N} (− ∑_{b∈C_{N}} (Nb̃h − b̃^{2}ht + N⌊b̃t/N⌋)) ) = −1.

The first exponential is clearly 1 by Lemma 4.3 and because 2^{k} is even. Clearly

exp (\frac{πi}{N} (− ∑_{b∈C_{N}} (Nb̃h − b̃^{2}ht)) ) = 1,

and we are done with (c) by Lemma 4.5.

For (d), note that

g = ( ∏_{b∈Ω} g−q ; b̃ ∏_{b∈ξΩ} gq ; b̃)
ξg = ( ∏_{b∈ξΩ} g−q ; b̃ ∏_{b∈ξ^{2}Ω} gq ; b̃) ( ∏_{b∈Ω} κ(ξ; b̃)^{−q}) ( ∏_{b∈ξΩ} κ(ξ; b̃)^{q} )
ξ^{2}g = ( ∏_{b∈ξ^{2}Ω} g−q ; b̃ ∏_{b∈ξ^{3}Ω} gq ; b̃) ( ∏_{b∈Ω} κ(ξ; b̃)^{−q}) ( ∏_{b∈ξΩ} κ(ξ; b̃)^{q} ) ×

× ( ∏_{b∈ξΩ} κ(ξ; b̃)^{−q} ) ( ∏_{b∈ξ^{2}Ω} κ(ξ; b̃)^{q} )

= ( ∏_{b∈ξ^{2}Ω} g−q ; b̃ ∏_{b∈ξ^{3}Ω} gq ; b̃) ( ∏_{b∈Ω} κ(ξ; b̃)^{−q}) ( ∏_{b∈ξ^{2}Ω} κ(ξ; b̃)^{q} )

⋮

ξ^{2^{k} −1}g = ( ∏_{b∈ξ^{2^{k} −1} Ω} g−q ; b̃ ∏_{b∈Ω} gq ; b̃) ( ∏_{b∈Ω} κ(ξ; b̃)^{−q}) ( ∏_{b∈ξ^{2^{k} −1} Ω} κ(ξ; b̃)^{q} ) .

Therefore

g(ξg)(ξ^{2}g) …(ξ^{2^{k} −1}g) = ( ∏_{b∈Ω} κ(ξ; b̃)^{−q})^{2^{k}} ( ∏_{b∈C_{N}} κ(ξ; b̃)^{q} ) = \frac{fg^{2^{k}}}{ξf} = −1,

by (c). Thus the proof of (d) is complete.

We will use Theorem 3.3 to calculate div(f). It is immediately clear that ord_{Q_{t}} (f) =

0 for all the cusps Q_{t}. On the other hand, letting Ω′ denote the coset of Ω containing the reduction of b̃t, and using the fact that B_{2}(x) = B_{2}(1 − x), we can calculate

ord_{P_{t}}(f) = q (∑_{b∈Ω} 2^{k} \frac{N}{2}B_{2} (\frac{b̃t mod N}{N} ) − ∑_{b∈C_{N}} \frac{N}{2}B_{2} (\frac{b̃t mod N}{N} ))
= q (2^{k−1}N ∑_{b∈Ω′} (\frac{b̃^{2}}{N^{2}} − \frac{b̃}{N} + \frac{1}{6}) − \frac{N}{2} \frac{(−r)}{6N})
= q2^{k−1} ∑_{b∈Ω′} (\frac{b̃^{2}}{N} − b̃) + q (2^{k−1}N\frac{#Ω}{6} + \frac{r}{12}) .

By Lemma 4.3, the first term is an integer and clearly it is divisible by 2^{k}. Using the identities #Ω = 2zv, r = 2^{k+1}zv, qz = 3, the latter term in the above sum reduces to

2^{k−1}(N + 1)v,

which is divisible by 2^{k} since N + 1 is even. So we have completed the proof (e). ■

We will need the following lemma. For a curve X defined over ℚ and a field K

containing ℚ, denote the function field of X over K by K(X). It is well known that a finite abelian group equipped with a continuous action of Gal(\overline{ℚ}/ℚ) is the same as a finite étale commutative group scheme over ℚ. We will use this identification throughout the rest of this dissertation.

Lemma 4.8 Let ϕ : X ) Y be a finite étale map of projective curves, with X, Y and ϕ defined over ℚ. Assume that ϕ is Galois after some finite base extension F/ℚ. Let Γ = Gal(\overline{ℚ}(X)/\overline{ℚ}(Y)) ≅ Gal(F(X)/F(Y)) and assume that Γ is commutative. By the Picard functoriality of the Jacobians, we have an exact sequence

0 −) K −) Jac(Y)(\overline{ℚ}) −)^{ϕ^{∗}} Jac(X)(\overline{ℚ})^{Γ},

where K denotes the finite Gal(\overline{ℚ}/ℚ)-module ker(ϕ^{∗}). Then

(a) the group scheme K is isomorphic to Γ^{D}, the Cartier dual of Γ;

(b) if Γ is cyclic, then ϕ^{∗} surjects onto Jac(X)(\overline{ℚ})^{Γ}.

Remark. To make sense of Γ^{D}, we need to consider Γ as an étale group scheme

over ℚ. The group Γ is naturally acted upon by Gal(\overline{ℚ}/ℚ) as follows: for any σ ∈ Gal(\overline{ℚ}/ℚ) and τ ∈ Γ, let σ · τ = σ̃τσ̃^{−1} where σ̃ ∈ Gal(\overline{ℚ}(X)/ℚ(X)) ≅ Gal(\overline{ℚ}/ℚ) is any lift of σ.

Proof. Taking the exact sequence of low degree terms for the Hochschild–Serre

spectral sequence of the étale cohomology of 𝔾_{m} over the base \overline{ℚ} as in [14, III, Theorem 2.20], we obtain

\begin{aligned}0 & −) & H^{1}(Γ, H^{0}(X_{et}, 𝔾_{m})) & −) & H^{1}(Y_{et}, 𝔾_{m}) & −) \\ & −) & H^{0}(Γ, H^{1}(X_{et}, 𝔾_{m})) & −) & H^{2}(Γ, H^{0}(X_{et}, 𝔾_{m})) \end{aligned}

which is

0 −) H^{1}(Γ, \overline{ℚ}^{×}) −) Pic(Y) −)^{ϕ^{∗}} Pic(X)^{Γ} −) H^{2}(Γ, \overline{ℚ}^{×}).

Now since Γ acts trivially on \overline{ℚ}^{×}, H^{1}(Γ, \overline{ℚ}^{×}) ≅ Hom(Γ, \overline{ℚ}^{×}) ≅ Γ^{D}. The kernel of ϕ^{∗} : Pic(Y) ) Pic(X) is contained in Jac(Y), so we have proved (a).

If Γ is cyclic, then by [21, VIII §4], H^{2}(Γ, \overline{ℚ}^{×}) ≅ (\overline{ℚ}^{×})^{Γ}/(\overline{ℚ}^{×})^{N} = 0. But if ϕ^{∗} :

Pic(Y) ) Pic(X)^{Γ} is surjective, then so is ϕ^{∗} : Jac(Y) ) Jac(X)^{Γ}, which proves (b). ■

Now we are almost ready to find extra points in J_{0}(N)[I]. Recall that c denotes the

divisor 0 − on X_{0}(N).

Theorem 4.9 Let d = \frac{1}{2^{k}} div(f), considered as a point on J_{0}^{#} (N). Then

(a) the divisor d is rational over ℚ;

(b) the divisor d is in the image of ϕ^{∗} : J_{0}(N) ) J_{0}^{#} (N);

(c) 2d = ϕ^{∗}(v · c).

Remark. In essence, we are trying to find “one half of c” in the group J_{0}(N)[I]/Σ.

Assertion (c) in the above theorem shows that d is “one half of v · c”. Recall from Definition 4.2 that v is the odd part of n, so this is as good as finding half of c, but some calculations work out simpler this way. Assertion (b) will be used to show that our point pulls back to J_{0}(N), and assertion (a) will be used to show that we are actually finding points in J_{0}(N)[I].

Proof. As can be seen from the proof of Theorem 4.7(e), div(f) is concentrated at

the cusps of X_{0}^{#} (N) that lie over the cusp 0 of X_{0}(N). All of these cusps are rational over ℚ, hence so is d, proving (a).

By Lemma 4.8(b), it suffices to check that d is fixed by ξ, the generator of the

group Gal(X_{0}^{#} (N)/X_{0}(N)). By Theorem 4.7(c), div(f) − div(ξf) = −2^{k} div(g), so

d − ξd = \frac{1}{2^{k}} div(f) − \frac{1}{2^{k}} div(ξf) = div(1/g),

which is a principal divisor, so d = ξd in J_{0}^{#} (N), concluding our proof of (b).

Let d′ = div(f)/2^{k−1} − div(h) be a divisor on X_{0}^{#} (N). Using Theorem 3.3, for

any 1 ≤ t ≤ r,

ord_{Q_{t}} (d′) = \frac{1}{2^{k−1}} (∑_{b∈Ω} \frac{2^{k}q}{12} − ∑_{b∈C_{N}} \frac{q}{12}) − ∑_{b∈Ω} \frac{2q}{12} = 0 − 2zvq/6 = −v,

and

ord_{P_{t}}(d′) = ∑_{b∈Ω} \frac{2^{k}qN}{2^{k}} B_{2} (\frac{b̃t mod N}{N} ) − ∑_{b∈C_{N}} \frac{qN}{2^{k}} B_{2} (\frac{b̃t mod N}{N} )

− ∑_{b∈Ω} \frac{2qN}{2} B_{2} (\frac{b̃t mod N}{N} ) = −\frac{qN}{2^{k}} \frac{(−r)}{6N} = v.

Hence d′ = ϕ^{∗}(v · c), so

2d − div(h) = ϕ^{∗}(v · c)

as divisors. But h is a function defined on X_{0}^{#} (N) by Theorem 4.7(b), so this proves (c). ■

Chapter 5

The Galois structure of J_{0}(N)[I]

Theorem 5.1 Let D denote the group generated by d in J_{0}^{#} (N). Let A = (ϕ^{∗})^{−1}D. Then

(a) all the points of D are unramified at N;

(b) all the points of A are unramified at N;

(c) the group A is contained in J_{0}(N)[I].

Remark. Since # ker(ϕ) = #D = 2^{k}, the group A has cardinality 2^{2k}. Therefore

part (c) of the above theorem implies that A is the whole of the 2-primary component of J_{0}(N)[I]. Since the odd part of J_{0}(N)[I] is the direct sum of the odd parts of C and Σ, we have now completed the concrete description of J_{0}(N)[I] that we were aiming for.

Proof. Assertion (a) is immediate from Theorem 4.9(a), since the points of D are

rational. (Note that since the action of Gal(\overline{ℚ}/ℚ) on the cusps of X_{0}^{#} (N) factors through the cyclotomic character χ_{N}, the only way for a divisor supported at the cusps to be unramified at N is to be rational.)

Assertion (b) follows from [11, II, Lemma (16.5)]. Note that since the lemma just

cited applies only to points of prime power order, we have to apply it separately to each of the primary components of the point of A in question.

Multiplication by 2^{k} annihilates d. Therefore 2^{k}A ⊆ ker(ϕ^{∗}) ⊆ Σ, so certainly

all points in A are torsion points. By [20, Prop. 3.3], all torsion points of J_{0}(N) that are unramified at N are in J_{0}(N)[I], so we have proved A ⊆ J_{0}(N)[I]. ■

Remark. For the reader’s convenience we summarize another proof of part (c)

of Theorem 5.1 that avoids invoking [20]. This proof also does not need the results of parts (a) and (b) of Theorem 5.1. We shall use the terminology and notation of [11]. Fix an embedding \overline{ℚ}_{N} ) \overline{ℚ} and let J be the Néron model of J_{0}(N) over ℤ_{N}. Let J_{/𝔽_{N}} denote the special fiber of J, and let J_{/𝔽_{N}}^{0} denote the irreducible component of the identity in J_{/𝔽_{N}} . Let Σ_{/𝔽_{N}} denote the reduction of Σ to J_{/𝔽_{N}} . Note that Σ ≅ μ_{n}, and so Σ is unramified at N. Therefore, by [22, Lemma 2], Σ reduces injectively to Σ_{/𝔽_{N}} . Then, by [11, II, Proposition (11.9)],

Σ_{/𝔽_{N}} ∩ J_{/𝔽_{N}}^{0} = 0.

Thus, a point of Σ that reduces to a point in J_{/𝔽_{N}}^{0} must be zero. We shall now use this observation to show that A ⊆ J_{0}(N)[I].

It suffices to show that for arbitrary point x of A and any element T of I, we have

Tx = 0. The group of irreducible components of J_{/𝔽_{N}} is Eisenstein, as can be seen from the title (and contents) of [4] (see also [19]). Therefore, the operator T sends the reduction of x into the identity component. In other words, Tx reduces into J_{/𝔽_{N}}^{0} .

On the other hand, we can use the formulae in [24, Section 2] to define actions of

T_{l} (for l ≠ N) and w on (J_{1}(N) and therefore on) J_{0}^{#} (N) that are compatible with the actions defined on J_{0}(N) via the map ϕ^{∗}, and calculate (in the spirit of the proof of Theorem 4.9(c)) that D is annihilated by each 1 + l − T_{l} and by 1 + w. Let T′ be a lift of T to the ring ℤ[…, T_{l}, …, w] and let T′′ be the image of T′ in End(J_{0}^{#} (N)). Then we have a commutative diagram

J_{0}(N) −)^{ϕ^{∗}} J_{0}^{#} (N)
↓_{T} ↓_{T′′}
J_{0}(N) −)^{ϕ^{∗}} J_{0}^{#} (N).

Here x is mapped to ϕ^{∗}x ∈ D which is annihilated by T′′. By the commutativity of the diagram we must have Tx ∈ ker(ϕ^{∗}) = Σ. This completes our proof that Tx = 0. ■

Now that we established that A ⊆ J_{0}(N)[I], we will determine the action of

Gal(\overline{ℚ}/ℚ) on A and then assemble what we know to find the action of Gal(\overline{ℚ}/ℚ) on the whole of J_{0}(N)[I].

Definition 5.2 Let λ : X_{0}^{##} (N) ) X_{0}^{#} (N) be a minimal covering of X_{0}^{#} (N) on which f^{1/2^{k}} is defined.

By Theorem 4.7, parts (a) and (e), the degree of λ is 2^{k} and λ is étale. In fact, after

base extension to ℚ(μ_{2^{k+1}} ), λ becomes a Galois covering with Galois group Γ. The group Γ can be regarded as a finite étale group scheme over ℚ, and by Lemma 4.8(a), A will be its Cartier dual. This allows us to determine the action of Gal(\overline{ℚ}/ℚ) on A.

Convention 5.3 Choose once and for all a primitive 2^{k+1}st root of unity ζ ∈ \overline{ℚ}. Then ζ^{2} is the primitive 2^{k}th root of unity that we will use in explicit Cartier duality calculations.

Theorem 5.4 Let K denote the function field of X_{0}(N) over ℚ and L the function field of X_{0}^{#} (N) over ℚ, so that L(f^{1/2^{k}} ) is the function field of X_{0}^{##} (N) over ℚ.

(a) L(f^{1/2^{k}} , ζ)/K(ζ) is a Galois extension with

Γ = Gal(L(f^{1/2^{k}} , ζ)/K(ζ)) ≅ ℤ/2^{k}ℤ × ℤ/2^{k}ℤ.

In terms of the basis described in the proof, any element σ of Gal(\overline{ℚ}/ℚ) acts on Γ via the matrix

( \begin{matrix}1 & 0 \\ (χ_{2^{k+1}} (σ) − 1)/2 & χ_{2^{k}} (σ) \end{matrix} ) .

(b) The abelian group A is isomorphic to ℤ/2^{k}ℤ × ℤ/2^{k}ℤ, with σ ∈ Gal(\overline{ℚ}/ℚ) acting via

( \begin{matrix}χ_{2^{k}} (σ) & (1 − χ_{2^{k+1}} (σ))/2 \\ 0 & 1 \end{matrix} ) .

Proof. We know that the field extension L/K is Galois of degree 2^{k} with cyclic

Galois group generated by ξ, and this remains true for L(ζ)/K(ζ). Clearly L(f^{1/2^{k}} , ζ)/L(ζ) is also Galois (and cyclic) of degree 2^{k}. Since K(ζ) (and hence L(ζ)) contains all 2^{k}th roots of unity, and by Theorem 4.7(c), (ξf)/f = (ζg)^{2^{k}}, we can conclude that L(f^{1/2^{k}} , ζ) = L((ξf)^{1/2^{k}} , ζ). This way we obtain that L(f^{1/2^{k}} , ζ) contains all the 2^{k}th roots of f, ξf, …, and therefore that L(f^{1/2^{k}} , ζ)/K(ζ) is a Galois extension.

To determine the group Γ = Gal(L(f^{1/2^{k}} , ζ)/K(ζ)), observe that the field extension

L(f^{1/2^{k}} , ζ)/K(ζ) contains all the conjugates of its generator f^{1/2^{k}} . Therefore it is obtained as a splitting field of the polynomial F whose roots are all the 2^{k}th roots of all conjugates of f. Since ξf = (−1)g^{2^{k}} f, the 2^{k}th roots of ξf are ζgf^{1/2^{k}} , ζ^{3}gf^{1/2^{k}} , …, ζ^{2^{k+1} −1}gf^{1/2^{k}} . Then

ξ^{2}f = ξ((−1)g^{2^{k}} f) = (−1)(ξg)^{2^{k}} (ξf)
= (−1)(ξg)^{2^{k}} (−1)g^{2^{k}} f = (ξg)^{2^{k}} g^{2^{k}} f,

so the 2^{k}th roots of ξ^{2}f are (ξg)gf^{1/2^{k}} ,ζ^{2}(ξg)gf^{1/2^{k}} , …, ζ^{2^{k+1} −2}(ξg)gf^{1/2^{k}} . Hence it is clear that the roots of F are exactly the

δ_{i,j} = ζ^{2i+j} (∏_{k=0}^{j−1}(ξ^{k}g)) f^{1/2^{k}} ,

where i and j range over the interval [0, 2^{k} − 1].

To determine Γ, observe that it must act simply transitively on the set of all roots

of F. Let ρ ∈ Γ be such that

ρ : δ_{0,0} = f^{1/2^{k}} ) δ_{1,0} = ζ^{2}f^{1/2^{k}} .

Taking 2^{k}th powers, we see that ρ fixes f and hence all of L(f^{1/2^{k}} , ζ). So ρ sends δ_{i,j} to δ_{i+1,j} (with δ_{2^{k} ,j} to be interpreted as δ_{0,j}).

Now consider the element ξ̄ ∈ Γ for which

ξ̄ : δ_{0,0} = f^{1/2^{k}} ) δ_{0,1} = ζgf^{1/2^{k}} .

Taking 2^{k}th powers again, we see that ξ̄ sends f to ξf, so it acts as ξ on L(ζ) (thereby justifying our choice of name for it). Note that

ξ̄(δ_{0,1}) = ξ̄(ζgf^{1/2^{k}} ) = ζ(ξ̄g)(ξ̄f^{1/2^{k}} ) = ζ(ξg)ζgf^{1/2^{k}} = ζ^{2}(ξg)gf^{1/2^{k}} = δ_{0,2},

similarly

ξ̄(δ_{0,2}) = ξ̄(ζ^{2}(ξg)gf^{1/2^{k}} ) = ζ^{3}(ξ^{2}g)(ξg)gf^{1/2^{k}} = δ_{0,3},

and so on. Finally, using Theorem 4.7(d) we obtain

ξ̄(δ_{0,2^{k} −1}) = ξ̄(ζ^{2^{k} −1}(ξ^{2^{k} −2}g) …(ξg)gf^{1/2^{k}} ) = ζ^{2^{k}} (ξ^{2^{k} −1}g) …(ξg)gf^{1/2^{k}}
= (−1)(−1)f^{1/2^{k}} = f^{1/2^{k}} = δ_{0,0}.

Hence ξ̄ sends δ_{i,j} to δ_{i,j+1} (with δ_{i,2^{k}} to be interpreted as δ_{i,0}).

This shows that Γ is generated by two commuting elements of order 2^{k}. In other

words, we have Γ ≅ ℤ/2^{k}ℤ × ℤ/2^{k}ℤ, and we can represent elements of Γ as column vectors over ℤ/2^{k}ℤ, with ξ̄ corresponding to \binom{1}{0} and ρ corresponding to \binom{0}{1}.

As for the action of Gal(\overline{ℚ}/ℚ) on Γ, take some σ ∈ Gal(\overline{ℚ}/ℚ) and consider its

natural action on L(f^{1/2^{k}} , ζ) that leaves L(f^{1/2^{k}} ) fixed. Both σξ̄σ^{−1} and ρ^{(χ_{2^{k+1}} (σ)−1)/2} ξ̄ fix ζ and

σξ̄σ^{−1} : f^{1/2^{k}} ) f^{1/2^{k}} ) ζgf^{1/2^{k}} ) ζ^{χ_{2^{k+1}} (σ)} gf^{1/2^{k}}
ρ^{(χ_{2^{k+1}} (σ)−1)/2} ξ̄ : f^{1/2^{k}} ) ζgf^{1/2^{k}} ) ζ^{χ_{2^{k+1}} (σ)} gf^{1/2^{k}} .

Therefore σξ̄σ^{−1} = ρ^{(χ_{2^{k+1}} (σ)−1)/2} ξ̄. Similarly both σρσ^{−1} and ρ^{χ_{2^{k}} (σ)} fix ζ and

σρσ^{−1} : f^{1/2^{k}} ) f^{1/2^{k}} ) ζ^{2}f^{1/2^{k}} ) ζ^{2χ_{2^{k+1}} (σ)} f^{1/2^{k}}
ρ^{χ_{2^{k}} (σ)} : f^{1/2^{k}} ) ζ^{2χ_{2^{k}} (σ)} f^{1/2^{k}} = ζ^{2χ_{2^{k+1}} (σ)} f^{1/2^{k}} .

Therefore σρσ^{−1} = ρ^{χ_{2^{k}} (σ)} . Hence σ ∈ Gal(\overline{ℚ}/ℚ) does act on the elements of Γ (represented by column vectors) as required. With this the proof of (a) is complete.

For (b), a simple calculation shows that if G is an étale group scheme over ℚ that

is isomorphic to ℤ/mℤ × ℤ/mℤ with a Galois action described by

({a(σ) \atop c(σ)} {b(σ) \atop d(σ)} ) : Gal(\overline{ℚ}/ℚ) ) GL_{2}(ℤ/mℤ),

then its Cartier dual G^{D} is also isomorphic to ℤ/mℤ × ℤ/mℤ, but with a Galois action described in terms of the usual dual basis by

({χ_{m}(σ)a(σ^{−1}) \atop χ_{m}(σ)b(σ^{−1})} {χ_{m}(σ)c(σ^{−1}) \atop χ_{m}(σ)d(σ^{−1})} ) : Gal(\overline{ℚ}/ℚ) ) GL_{2}(ℤ/mℤ).

In our case this means that A ≅ Γ^{D} is isomorphic to ℤ/2^{k}ℤ×ℤ/2^{k}ℤ with the action

of σ ∈ Gal(\overline{ℚ}/ℚ) described by

({χ_{2^{k}} (σ) \atop 0} {χ_{2^{k}} (σ)(χ_{2^{k+1}} (σ^{−1}) − 1)/2 \atop 1} )

in terms of the basis ξ̄^{D} = \binom{1}{0}, ρ^{D} = \binom{0}{1}. But χ_{2^{k}} (σ)(χ_{2^{k+1}} (σ^{−1}) − 1)/2 ≡ (1 − χ_{2^{k+1}} (σ))/2 (mod 2^{k}) so we have completed the proof of this theorem. ■

Proof of Theorem 1.1. Since the quotient group Gal(L(ζ)/K(ζ)) of Γ is spanned

by ξ, the dual subgroup Σ_{0} = ker(J_{0}(N) ) J_{0}^{#} (N)) is spanned by ξ̄^{D} in A. One checks easily that d ∈ ker(J_{0}^{#} (N) ) J_{0}^{##} (N)) corresponds to the image of ρ in A/Σ_{0} under Cartier duality, so we can see by Theorem 4.9(c) that v · c ∈ A is represented by some vector \binom{∗}{2} in A. But since v · c ∈ A is Gal(\overline{ℚ}/ℚ)-invariant, we can use Theorem 5.4(b) to conclude that v · c = \binom{1}{2}.

The odd part of J_{0}(N)[I] is a direct product μ_{v} × ℤ/vℤ. The constant part is gener-

ated by 2^{k}c, so we can choose a basis g_{1}, 2^{k}c so that for any σ ∈ Gal(\overline{ℚ}/ℚ), σ(g_{1}) = χ_{v}(σ)g_{1} and σ(2^{k}c) = 2^{k}c. Taking the basis consisting of g_{1} and g_{2} = (2^{k}c − g_{1})(v + 1)/2 instead, we have 2^{k}c = g_{1} + 2g_{2} and σ acts via

({χ_{v}(σ) \atop 0} {(1 − χ_{2v}(σ))/2 \atop 1} ) .

Now pick integers a, b such that va + 2^{k}b = 1. Then

e_{1} = aξ̄^{D} + bg_{1}
e_{2} = aρ^{D} + bg_{2}

is a basis of J_{0}(N)[I] that clearly has all properties required in Theorem 1.1.

Finally, observe that if N ≢ 1 (mod 8), then n = v and the 2-primary part of

J_{0}(N)[I] is 0. So formally setting ξ̄^{D} = ρ^{D} = 0, we still have v · c = n · c = 0 = ξ̄^{D} + 2ρ^{D} and ρ^{D} ∈ Σ. The above argument about the prime-to-2 part works without a change, so we have proved Theorem 1.1 in this case too. ■

Remark. As described in [20], H. W. Lenstra and K. Ribet proved a version of The-

orem 1.1, where the expression (1 − χ_{2n}(σ))/2 in the statement of the theorem is replaced by a function

b : Gal(\overline{ℚ}/ℚ) ) ℤ/nℤ,

satisfying the properties that for each σ, τ ∈ Gal(\overline{ℚ}/ℚ),

b(στ) = b(σ) + χ_{n}(σ)b(τ),
2b(σ) = 1 − χ_{n}(σ),

and that the kernel of b cuts out the 2nth cyclotomic field. We shall show here that his result is strictly weaker than Theorem 1.1.

Indeed, let b_{0}(σ) = (1 − χ_{2n}(σ))/2, and let

ϵ : Gal(\overline{ℚ}/ℚ) ) ℤ/2ℤ

be a homomorphism that factors through Gal(ℚ(μ_{n})/ℚ). It is easy to check that for any such ϵ, the function

b(σ) = b_{0}(σ) + \frac{n}{2}ϵ(σ)

satisfies all of the above conditions. Since there is more than one choice for such a function ϵ when n divisible by 4, we have shown that the above result is weaker than Theorem 1.1.

Chapter 6

The old subvariety of J_{0}(NM)

Let N and M be distinct primes. The modular curve X_{0}(NM) classifies elliptic

curves equipped with a subgroup of order NM. (Recall from elementary group theory that all abelian groups of order NM are cyclic.) Forgetting all but the N-primary part of such a subgroup yields a subgroup of order N, thereby giving rise to a degeneracy map

π_{1} : X_{0}(NM) ) X_{0}(N),

which is defined over ℚ.

The Atkin–Lehner involution

w_{M} : X_{0}(NM) ) X_{0}(NM)

is another morphism defined over ℚ that can be associated to a moduli-theoretic operation. The operation is as follows. Let (E, C) denote a pair of an elliptic curve E and a subgroup C of order NM. Let C_{N} (respectively, C_{M}) be the N-primary (respectively, M-primary) subgroup of C, so that C = C_{N} × C_{M}. Let γ : E ) E′ be the degree M isogeny with kernel C_{M}. Then it is easy to see that the group γ(E[M]) is cyclic of order M, and that the group γ(C_{N}) is cyclic of order N. Then the operation we are looking for sends the pair (E, C) to the pair (E′, γ(E[M]) × γ(C_{N})).

On X_{0}(NM)_{ℂ} = ℌ^{∗}/Γ_{0}(NM), the map w_{M} is induced by the action of the matrix

({−aN \atop −NM} {b \atop M} ) acting on ℌ^{∗} via the corresponding Moebius transformation, where a and b are integers chosen so that bN − aM = 1.

The curve X_{0}(NM) has four cusps, called P_{1}, P_{N}, P_{M} and P_{NM} in the notation of

[18]. The morphism π_{1} : X_{0}(NM) ) X_{0}(N) sends P_{1} and P_{M} to the cusp 0 on X_{0}(N), with ramification indices M and 1, respectively. The map π_{1} also sends P_{N} and P_{NM} to the cusp of X_{0}(N), with ramification indices M and 1, respectively. The morphism w_{M} swaps the points P_{1} and P_{M}, and it also swaps the points P_{N} and P_{NM} .

Now we can define the degeneracy map

π_{M} : X_{0}(NM) ) X_{0}(N)

by setting π_{M} = π_{1} ∘ w_{M}. This allows us to define the morphisms

α = (π_{1}^{∗}, π_{M}^{∗}) : J_{0}(N) × J_{0}(N) ) J_{0}(NM)

and

β = (π_{1}^{∗}, π_{N}^{∗}) : J_{0}(M) × J_{0}(M) ) J_{0}(NM).

Since w_{M}^{2} is the identity morphism on X_{0}(NM), we can see that w_{M}^{∗}α(R_{1}, R_{2}) = α(R_{2}, R_{1}). (Note that we shall use w_{M} to denote w_{M}^{∗} where it is not likely to cause confusion.)

Definition 6.1 Let A = im(α) and B = im(β). The old part of J_{0}(NM) is the abelian subvariety of J_{0}(NM) that is generated by A and B.

Since J_{0}(M) has good reduction everywhere away from M and β is defined over ℚ,

we can conclude that B has good reduction away from M. Similarly, A has good reduction away from N. Therefore, A ∩ B has good reduction everywhere and is finite (see [18, Theorem 3] and [5]).

It was proved in [16] that

ker(α) = {(P, −P) ∈ J_{0}(N)^{2} : P ∈ Σ}.

We shall paraphrase this fact by saying that “the kernel of α is Σ, embedded anti-diagonally”. The analogous result of course also holds for β.

This completes the description of A and B. To completely describe the old part

of J_{0}(NM), we need to find A ∩ B. The odd part of A ∩ B was found in [18]. We shall complete that description by proving Theorem 1.2. We shall proceed through a sequence of lemmas.

Lemma 6.2 The preimage of the group A ∩ B under the map α is contained in J_{0}(N)[I]^{2}.

Proof. Let x ∈ J_{0}(N)^{2} be such that α(x) ∈ A ∩ B. Then there is some y ∈ J_{0}(M)^{2}

such that β(y) = α(x). The point β(y) = α(x) has finite order, since it is contained in the finite group A ∩ B. Since B has good reduction at N, the point β(y) must be unramified at N unless its order is divisible by N. But that cannot happen, since, by [18, Corollary 1], the odd part of the order of β(y) must divide the quantity

gcd(num((N − 1)/24), num((M − 1)/24)),

which is not divisible by N. Therefore, β(y) = α(x) is unramified at N. Now, since ker(α) = Σ (embedded anti-diagonally), this implies, as in [11, II, Lemma (16.5)], that x is unramified at N. This in turn implies by [20, Prop. 3.3] that x ∈ J_{0}(N)[I]^{2} as required.

■

Lemma 6.3 The image of the divisor c = 0 − on X_{0}(N) under the map π_{1}^{∗} − π_{M}^{∗} is the divisor (M − 1)D^{−,−} = (M − 1)(P_{1} − P_{N} − P_{M} + P_{NM} ).

Proof. This is immediate using the action of w_{M} on X_{0}(NM) and the ramification

indices of the cusps of X_{0}(NM) over the cusps of X_{0}(N). Indeed,

π_{1}^{∗}(c) = π_{1}^{∗}(0 − ) = MP_{1} + P_{M} − MP_{N} − P_{NM}

and

π_{M}^{∗}(c) = \overline{w}_{M}^{∗}(π_{1}^{∗}(c)) = MP_{M} + P_{1} − MP_{NM} − P_{N}.

The difference of the two displayed lines yields the result. ■

If M < 5 then B = 0 and Theorem 1.2 is true. Therefore, we may assume from

now on that M ≥ 5.

Let P be a point that maps to the generator of J_{0}(N)[I]/Σ that we constructed

before. Thus, P can be chosen as the point e_{2} in Theorem 1.1. If N ≡ 1 (mod 8), let z be the non-trivial 2-torsion element in Σ.

If N ≡ 1 (mod 8) then Σ[2] = C[2]. Thus, we can express z as

z = \frac{n}{2} · c.  (6.1)

This relation, which usually plays the role of an obstacle to be circumvented, now will help us get a handle on the behavior of z instead.

Now that we have established that the image of J_{0}(N)[I]^{2} is the only thing we need

to consider when determining A ∩ B, we prove a lemma that allows us to focus on an even smaller set.

Lemma 6.4 Let R ∈ J_{0}(N)[I]^{2}. If α(R) ∈ A ∩ B, then

• if N ≢ 1 (mod 8) then R lies in the ℤ-linear span of π_{1}^{∗}(P) − π_{M}^{∗}(P);

• if N ≡ 1 (mod 8) then R lies in the ℤ-linear span of π_{1}^{∗}(P) − π_{M}^{∗}(P) and π_{1}^{∗}(z).

Proof. Assume that R = (R_{1}, R_{2}) maps into A ∩ B under α. Note that w_{M} acts

on A ∩ B as multiplication by −1. Indeed, it is well known that β ∘ w_{M} = w_{M} ∘ β (see for example [18, “Formulaire”]), and by (the M-analog of) Lemma 6.2, we know that w_{M} acts as −1 on β^{−1}(A ∩ B). Therefore,

−α(R_{1}, R_{2}) = w_{M}α(R_{1}, R_{2}) = α(R_{2}, R_{1})

and hence

0 = α(R_{1} + R_{2}, R_{1} + R_{2}).

Once again using the fact that ker(α) = Σ (embedded anti-diagonally), we con-

clude that R_{1} + R_{2} is an element of Σ[2]. We now complete the proof by examining each of the possibilities that this allows for R_{1} + R_{2}.

In case R_{1} + R_{2} = 0, we can write R_{1} = aP + σ for some integer a and some σ ∈ Σ.

Then

α(R_{1}, R_{2}) = α(aP + σ, −aP − σ) = aα(P, −P) = a(π_{1}^{∗}(P) − π_{M}^{∗}(P)).
If N ≡ 1 (mod 8) then Σ[2] ≠ 0, and therefore it can also happen that R_{1} + R_{2} = z.

Then we can write R_{1} = z + aP + σ for some integer a and some σ ∈ Σ. Then R_{2} = −aP − σ and

α(R_{1}, R_{2}) = α(z, 0) + aα(P, −P) = π_{1}^{∗}(z) + a(π_{1}^{∗}(P) − π_{M}^{∗}(P)).

In either case, the lemma is proved. ■ Let us assume from now on that

N ≡ 1 (mod 8).

In the rest of this chapter, we shall frequently use the following argument to prove

that various points on J_{0}(NM) are not equal. Let J be an abelian variety defined over ℚ (for example, J_{0}(NM)). Given a prime number p, we can consider J to be defined over ℚ_{p} so that we can look at its Néron model J /ℤ_{p}.

Given a point Q in J(ℚ), we can consider it an element of J(ℚ_{p}). By the Néron

mapping property, Q even extends to an element of J (ℤ_{p}). Reducing modulo p, we can then regard Q as an element of J_{p}(𝔽_{p}), where J_{p} denotes the special fiber of J. The scheme J_{p} is a group scheme over 𝔽_{p}, but it is not necessarily irreducible. Let Φ_{p}(J) denote its group of irreducible components. Given our point Q ∈ J_{p}(𝔽_{p}), we can check which element of Φ_{p}(J) it maps to. The argument we shall often use proceeds as follows. Consider the function

ϖ : J(ℚ) ) Φ_{p}(J)

described above. Let P, Q ∈ J(ℚ). If ϖ(P) ≠ ϖ(Q) then we must also have P ≠ Q. Usually we shall show that ϖ(P) ≠ ϖ(Q) by first showing that ϖ(P) = 0 (that is to say, P maps to the component of the identity under reduction modulo p), and then showing that ϖ(Q) ≠ 0.

This method can work only if Φ_{p}(J) has more than one element, which in turn can

happen only if J has bad reduction at p. Since J_{0}(NM) has good reduction away from N and M, we will always choose p = N or p = M.

Now we prepare the ground for this method by proving that various points are

defined over ℚ and by studying the components they map to when reduced modulo N or modulo M.

Lemma 6.5 The points π_{1}^{∗}(P) − π_{M}^{∗}(P) and π_{1}^{∗}(z) are defined over ℚ, and hence so is every point of A ∩ B.

Proof. By Lemma 6.4, (and the fact that J_{0}(NM) is defined over ℚ), π_{1}^{∗}(P)−π_{M}^{∗}(P)

and π_{1}^{∗}(z) being defined over ℚ does imply that every point of A ∩ B is defined over ℚ.

The point c is defined over ℚ, and hence so is z = (n/2)c. Since the degeneracy

map π_{1} is also defined over ℚ, we conclude that π_{1}^{∗}(z) is defined over ℚ.

By [16, Theorem 4.3], the map π_{1}^{∗} − π_{M}^{∗} : J_{0}(N) ) J_{0}(NM) (which is defined

over ℚ) factors through J_{0}(N)/Σ ) J_{0}(NM) (also defined over ℚ). Since the image of P in J_{0}(N)/Σ is also defined over ℚ by Theorem 1.1, it follows that π_{1}^{∗}(P) − π_{M}^{∗}(P) is defined over ℚ. ■

Lemma 6.6 Every point of A ∩ B reduces to the identity component of J_{0}(NM) modulo N (and modulo M).

Proof. By symmetry, it suffices to prove the claim for the reduction modulo N. The variety B has good reduction modulo N, therefore any point in A ∩ B ⊆ B(ℚ)

reduces to the identity component modulo N. ■

Lemma 6.7 The point π_{1}^{∗}(z) does not reduce to the identity component of J_{0}(NM) modulo N.

Proof. Recall from (6.1) that z = (n/2)c. This will enable us to use the results of

[11, Appendix I] to determine exactly where in the component group of J_{0}(N) modulo N the point π_{1}^{∗}(z) will map. We have

π_{1}^{∗}(z) = π_{1}^{∗}(\frac{n}{2}(0 − )) = \frac{n}{2}(MP_{1} + P_{M} − MP_{N} − P_{NM} ) =
= \frac{n}{2}(M + 1)(\overline{0} − \overline{}).

Here \overline{0} (respectively \overline{}) means a cusp of X_{0}(NM) that reduces to the same irreducible component as the cusp 0 (respectively ) modulo N.

Using the notation of [11, Appendix I], we have u ∈ {0, 1} and v ∈ {0, 1}, with

u = 1 {}  N ≡ 7 or 11 (mod 12) and M ≡ 1 (mod 4)
v = 1 {}  N ≡ 5 or 11 (mod 12) and M ≡ 1 (mod 3).

We have assumed that N ≡ 1 (mod 8), which excludes the possibilities N ≡ 7, 11

(mod 12). Therefore we must have u = 0, and

v = 1 {} N ≡ 2 (mod 3) and M ≡ 1 (mod 3).

Now that we have a good grip on the pair (u, v), we can look up in the table

of [11, Appendix I] the order of the divisor \overline{0} − \overline{}in the component group of J_{0}(NM) modulo N. The table below summarizes the possibilities (for brevity, we use the notation x = (N − 1)(M + 1)).

N (mod 3) M (mod 3)  n  ord(\overline{0} − \overline{})  π_{1}^{∗}(z)
1  1, 2  (N − 1)/12  x/12  x(\overline{0} − \overline{})/24
2  1  (N − 1)/4  x/4  x(\overline{0} − \overline{})/8
2  2  (N − 1)/4  x/12  x(\overline{0} − \overline{})/8

In each case, we can see that π_{1}^{∗}(z) does not reduce to the identity. This completes

the proof. ■

Lemma 6.8 The point D^{−,−} reduces to the identity component of J_{0}(NM) modulo N and modulo M.

Proof. Considering D^{−,−} in the group of components modulo N, we have

D^{−,−} = P_{1} − P_{N} − P_{M} + P_{NM} = \overline{0} − \overline{}− \overline{0} + \overline{}= 0.

The same proof works modulo M. ■

Lemma 6.9 The point (π_{1}^{∗} − π_{M}^{∗})P reduces to the identity component of J_{0}(NM) modulo N.

Proof. This is a direct consequence of the fact that π_{1}^{∗} − π_{M}^{∗} annihilates the com-

ponent group of J_{0}(N). (See [17, Theorem 2] and [4, proof of Théorème 1].) ■

In view of Lemma 6.4, the following Theorem shall bring us closer to our goal of

proving Theorem 1.2.

Theorem 6.10 The divisor class of (π_{1}^{∗} − π_{M}^{∗})(vP) on X_{0}(NM) is equal to the divisor class of \frac{v(M−1)}{2} D^{−,−}.

Remark. Note that since π_{1}^{∗} − π_{M}^{∗} annihilates Σ ⊂ J_{0}(N) and P is “half of c mod-

ulo Σ”, the Theorem 6.10 is consistent with the statement of the Lemma 6.3. Unfortunately, the Theorem does not follow yet, since there are a lot of ways in which we could take half of v(M − 1)D^{−,−}, only one of which is consistent with the Theorem.

Since the definition of P involved the cover X_{0}^{#} (N) of X_{0}(N), we must pause here

to define a cover of X_{0}(NM) that will allow us to determine (π_{1}^{∗} − π_{M}^{∗})P.

Let X_{1,0}(N, M) denote the modular curve corresponding to the congruence sub-

group Γ_{1}(N)∩Γ_{0}(M). The curve X_{1,0}(N, M) is defined over ℚ and corresponds to the moduli problem of classifying elliptic curves equipped with a point of order N and a subgroup of order M. The natural degeneracy map

X_{1,0}(N, M) ) X_{0}(NM)

will be denoted by β, by an abuse of notation that is intended to remind the reader to the similarity of this map to the previously defined β : X_{1}(N) ) X_{0}(N). Specifically, both maps β have the moduli-theoretic interpretation of taking a point of order N and replacing it by the subgroup of order N that it generates.

There is also a degeneracy map

π_{1} : X_{1,0}(N, M) ) X_{1}(N),

corresponding to the natural transformation “forget the level M structure” between the corresponding moduli functors. Then we have a commutative diagram of curves and maps defined over ℚ as follows:

X_{1,0}(N, M) −)^{β} X_{0}(NM)
↓^{π_{1}} ↓^{π_{1}}
X_{1}(N) −)^{β} X_{0}(N).

The curve X_{1,0}(N, M) has 2N − 2 cusps, which shall be denoted P_{i}^{0}, P_{i}^{}, Q_{i}^{0}, Q_{i}^{},

where the index i is allowed to range over 1, 2, …, r. The map π_{1} : X_{1,0}(N, M) ) X_{1}(N) sends the points P_{i}^{0} and P_{i}^{}to the point P_{i}, with ramification indices M and 1, respectively. Similarly, the same map π_{1} sends the points Q_{i}^{0} and Q_{i}^{}to the point Q_{i}, with ramification indices M and 1, respectively.

The map β : X_{1,0}(N, M) ) X_{0}(NM) takes the cusps P_{1}^{0}, P_{2}^{0}, …, P_{r}^{0} to the cusp P_{1}

of X_{0}(NM); P_{1}^{}, P_{2}^{}, …, P_{r}^{}are sent to P_{M}; Q_{1}^{0}, Q_{2}^{0}, …, Q_{r}^{0} go to the cusp P_{N}; Q_{1}^{}, Q_{2}^{}, …, Q_{r}^{}all map to P_{NM} . None of the four cusps of X_{0}(NM) is a branch point of β.

Since β : X_{1,0}(N, M) ) X_{0}(NM) is a cyclic Galois covering of degree r, it has a

unique intermediate covering of X_{0}(NM) of any degree dividing r (and this intermediate covering is also defined over ℚ). As in the definition of X_{0}^{#} (N), we can use this fact to define the curve X_{0}^{#} (N, M).

Definition 6.11 Let

ϕ : X_{0}^{#} (N, M) ) X_{0}(NM)

be the unique covering of degree 2^{k} that factors through β : X_{1,0}(N, M) ) X_{0}(NM). Let J_{0}^{#} (N, M) = Jac(X_{0}^{#} (N, M)).

The curve X_{0}^{#} (N, M) has a moduli interpretation similar to that of X_{0}^{#} (N), with an

extra subgroup of order M thrown in.

We have again a degeneracy map π_{1} : X_{0}^{#} (N, M) ) X_{0}^{#} (N) (corresponding to

forgetting the level M structure) that makes the following diagram commute

\begin{matrix}X_{0}^{#} (N, M) & −)^{ϕ} & X_{0}(NM) \\ ↓^{π_{1}} & & ↓^{π_{1}} \\ X_{0}^{#} (N) & −)^{ϕ} & X_{0}(N). \end{matrix}  (6.2)

The curve X_{0}^{#} (N, M) has 2^{k+2} cusps. Every cusp of X_{0}(NM) has 2^{k} cusps of

X_{0}^{#} (N, M) lying over it. To simplify the notation, we shall refer to a cusp of X_{0}^{#} (N, M) by the name of any cusp of X_{1,0}(N, M) that lies over it.

We can now use X_{0}^{#} (N, M) to deal with π_{1}^{∗}P, but in order to study π_{M}^{∗}P = w_{M}^{∗}π_{1}^{∗}P,

we shall need an analog of w_{M} on the curve X_{0}^{#} (N, M). Let

λ : X_{0}^{#} (N, M) ) X_{0}^{#} (N, M)

be the morphism induced by the action of the matrix ({Mb \atop aNM} {1 \atop M} ) on the complex upper half plane (here again, a and b are integers chosen so that bM − aN = 1). The morphism λ has the following moduli interpretation: if (E, P_{N}, C_{M}) is a triplet of an elliptic curve E, a point P ∈ E of order N, and a cyclic subgroup C_{M} ⊂ E[M] of order M corresponding to a point Q of X_{0}^{#} (N, M), then λQ corresponds to the triplet (E/C_{M}, P_{N}/C_{M}, E[M]/C_{M}).

We now let \overline{w}_{M} = λ^{−1}. The morphism \overline{w}_{M} covers w_{M} in the sense that the

following diagram is commutative:

X_{0}^{#} (N, M) −)^{\overline{w}_{M}} X_{0}^{#} (N, M)
↓^{π_{1}} ↓^{π_{1}}
X_{0}(NM) −)^{w_{M}} X_{0}(NM).

(One might notice that λ also covers w_{M} and wonder why we set \overline{w}_{M} = λ^{−1} instead of \overline{w}_{M} = λ. The reason is that although \overline{w}_{M} = λ would also work, the function we would have to consider to prove Theorem 6.13 would be much more complicated.)

The morphism \overline{w}_{M} sends the cusps P_{i}^{0}, P_{i}^{}, Q_{i}^{0}, Q_{i}^{}(respectively) to the cusps

P_{i}^{}, P_{i/M}^{0} , Q_{Mi}^{}, Q_{i}^{0} (respectively), where i/M is understood to mean division modulo N and where all indices are to be taken in C_{N}.

We now have the tools to deal with (π_{1}^{∗} − π_{M}^{∗})P. Before we move on to the proof

of Theorem 6.10, we shall need the following lemma.

Lemma 6.12 The unique non-trivial 2-torsion element in the kernel of

ϕ^{∗} : J_{0}(NM) ) J_{0}^{#} (N, M)

is the point π_{1}^{∗}(z).

Proof. We can use the fact that z is a 2-torsion element of the kernel ϕ^{∗} : J_{0}(N) )

J_{0}^{#} (N) and the commutativity of the diagram (6.2) to conclude that π_{1}^{∗}(z) is a 2-torsion element of the kernel of ϕ^{∗} : J_{0}(NM) ) J_{0}^{#} (N, M).

The fact that π_{1}^{∗}(z) is not trivial follows from Lemma 6.7.

Since the map ϕ : X_{0}^{#} (N, M) ) X_{0}(NM) is Galois with cyclic Galois group, there

are no other non-trivial 2-torsion points in its kernel. ■

Let us now take for granted the following theorem, the proof of which shall be the

subject of Chapter 7.

Theorem 6.13 The divisors

(1 − \overline{w}_{M}^{∗})π_{1}^{∗}(d) and ϕ^{∗} (\frac{v(M − 1)}{2} D^{−,−})

on X_{0}^{#} (N, M) are linearly equivalent.

This theorem allows us to conclude our proof as follows.

Proof of Theorem 6.10. Consider the following commutative diagram.

J_{0}(NM) −)^{ϕ^{∗}} J_{0}^{#} (N, M)
↑^{1−w_{M}^{∗}} ↑^{1−\overline{w} _{M}^{∗}}
J_{0}(NM) −)^{ϕ^{∗}} J_{0}^{#} (N, M)
↑^{π_{1}^{∗}} ↑^{π_{1}^{∗}}
J_{0}(N) −)^{ϕ^{∗}} J_{0}^{#} (N).

By definition, the point vP maps to d ∈ J_{0}^{#} (N). The left hand vertical map (1 − w_{M}^{∗}) ∘ π_{1}^{∗} = π_{1}^{∗} − π_{M}^{∗} sends the point vP to (π_{1}^{∗} − π_{M}^{∗})(vP). By the commutativity of the diagram and Theorem 6.13, we may therefore deduce that (π_{1}^{∗} − π_{M}^{∗})(vP) − (\frac{v(M−1)}{2} D^{−,−}) lies in the kernel of the top map ϕ^{∗}. Furthermore, by Lemma 6.3 and the remark after it, we can see that

2 ((π_{1}^{∗} − π_{M}^{∗})(vP) − (\frac{v(M − 1)}{2} D^{−,−})) = 0.

By Lemma 6.12, (π_{1}^{∗} − π_{M}^{∗})(vP) − (\frac{v(M−1)}{2} D^{−,−}) is equal to either 0 or π_{1}^{∗}(z).

However, reduction modulo N lands in the identity component of J_{0}(NM) for (π_{1}^{∗}−π_{M}^{∗})(vP) (by Lemma 6.9), and for D^{−,−} (by Lemma 6.8), and away from the identity component for π_{1}^{∗}(z) (by Lemma 6.7). Therefore

(π_{1}^{∗} − π_{M}^{∗})(vP) = (\frac{v(M − 1)}{2} D^{−,−})

and we have now completed the proof of Theorem 6.10. ■

Proof of Theorem 1.2. In this proof, allow N and M to be any pair of distinct

primes. Observe that if N < 11 then A = 0 and Theorem 1.2 is true. Similarly, we may assume that M ≥ 11.

If N ≢ 1 (mod 8) (equivalently, n is odd), then by Lemmas 6.2 and 6.4, the group

A ∩ B is spanned by a multiple of (π_{1}^{∗} − π_{M}^{∗})(P).

On the other hand, if N ≡ 1 (mod 8) (equivalently, n is even), then by Lemmas 6.2

and 6.4, the group A ∩ B is spanned by a multiple of (π_{1}^{∗} − π_{M}^{∗})(P) and π^{∗}(z). However, by Lemmas 6.6 and 6.7, in fact the group A ∩ B is spanned by multiples of (π_{1}^{∗}−π_{M}^{∗})(P) alone.

The multiples of (π_{1}^{∗}−π_{M}^{∗})(P) are exactly the image of the group J_{0}(N)[I]/Σ under

the map π_{1}^{∗} − π_{M}^{∗}. Since π_{1}^{∗} − π_{M}^{∗} is injective on J_{0}(N)[I]/Σ, we can see that the image X_{N} has order n. We will now show that this image lies entirely in the cyclic group spanned by D^{−,−}.

We shall consider separately the image of the 2-primary part and the odd part of

J_{0}(N)[I]/Σ. The odd part is generated by 2^{k}P = 2^{k−1}c, and its image does lie among the multiples of D^{−,−} by Lemma 6.3. The 2-primary part of J_{0}(N)[I]/Σ (which is non-trivial only if n is even) is generated by vP. An application of Theorem 6.13 completes our argument.

Running the same argument again, but exchanging the roles of N and M, we find

that A ∩ B can also be found in the subgroup X_{M} of order m of the multiples of D^{−,−}. Therefore, A ∩ B must be the intersection of our groups X_{N} and X_{M}. This completes the proof. ■

Chapter 7

A unit calculation on X_{0}^{#} (N, M)

We now proceed to give a proof of Theorem 6.13.

Let us use the notation of Chapter 2 to identify our Siegel units. Note that (K1)

and (K2) are still valid (with N replaced by NM), as are (2.2) and (2.3). At the appropriate points, we shall still massage the indices of our Siegel units into the array E′ (with N, r replaced by NM, (NM − 1)/2, respectively). Note that [8, Chapter 4, Theorem 1.3] does not apply to a level that is not a prime power; thus our Theorem 2.1 does not apply in this case. However, we shall use the half of Theorem 2.1 which remains valid by [8, Chapter 3, Theorem 5.2]—namely, a function satisfying conditions (U1–4) is a unit on X(NM). (Note that (U2–4) are now congruences modulo NM.)

Definition 7.1 Recall from Chapter 4 that Ω denotes the set of 2^{k}th powers in C_{N}. For any integer y that is not divisible by N,

• let Ω_{y} denote the representatives in the interval [1, r] of the elements of the coset yΩ of Ω;

• let J_{y} denote the set of integers x in the interval [1, (NM − 1)/2] such that

– M does not divide x, and

– x maps to Ω_{y} under the natural surjection ℤ ) C_{N};

• let

e(x) = ∏_{j∈J_{M}} g_{(0,j)}^{q}.

We shall need a definition and some lemmas before we can show that div(e) is the

divisor mentioned in Theorem 6.13.

Definition 7.2 For any integer x ∈ [1, r], let ϕ(x) be the element of [1, r] that satisfies the congruence

Mϕ(x) ≡ ±x (mod N).

(In other words, ϕ(x) is the representative for x/M in C_{N}.)

The following two lemmas will be useful later. Their proofs are very easy and will

not be given here.

Lemma 7.3 For any integer y that is not divisible by N,

(a)

∑_{j∈J_{y}} 1 = 2zv(M − 1),

(b)

∑_{j∈J_{y}} j = \frac{1}{2}zvN(M − 1)(M + 1) + ∑_{b∈Ω_{y}} (b̃ − Mϕ(b̃)),

(c)

∑_{j∈J_{y}} j^{2} = \frac{1}{6}zvN^{2}(M − 1)(M + 1)M + ∑_{b∈Ω_{y}} (Mb̃^{2} − M^{2}ϕ(b̃)^{2}).

Lemma 7.4 Let y be a real number. Then

M (∑_{i=0}^{M−1} B_{2} (\frac{y + iN}{NM} ))− B_{2} (\frac{y}{N}) = 0.

We shall need one last lemma.

Lemma 7.5 Let c and d be relatively prime integers. Assume that c is divisible by NM and that d maps into Ω under the map ℤ ) C_{N}. If y is any integer that is not divisible by N, we have

∑_{j∈J_{y}} ⌊\frac{dj}{NM} ⌋ ≡ (1 + d) ∑_{j∈J_{y}} j (mod 2).

Proof. In this proof only, let {·} denote {·}_{NM} , as given in Definition 3.4. All the

congruences below are modulo 2. Proceeding as in the proof of Lemma 4.6, we obtain

∑_{j∈J_{y}} ⌊\frac{dj}{NM} ⌋ ≡ −d ∑_{j∈J_{y}} j + d ∑_{j∈J_{y}} j − NM ∑_{j∈J_{y}} ⌊\frac{dj}{NM} ⌋
= −d ∑_{j∈J_{y}} j + ∑_{j∈J_{y}}(dj mod NM)
= −d ∑_{j∈J_{y}} j + d ∑_{j∈J_{y}}{dj} + ∑_{j∈J_{y}}[dj mod NM > NM/2](−{dj} + NM − {dj})
≡ (1 + d) ∑_{j∈J_{y}} j + m,

where m = ∑ _{j∈J_{y}} [dj mod NM > NM/2]. Then

(−1)^{m} ∏_{j∈J_{y}} j ≡ ∏_{j∈J_{y}}{dj} ≡ ∏_{j∈J_{y}}(dj) (mod NM),

which we can divide through by ∏ _{j∈J_{y}} j, since no element of J_{y} is divisible by either N or M, to obtain

(−1)^{m} ≡ d^{#J_{y}} (mod NM).

Since M − 1 divides #J_{y}, we get that

d^{#J_{y}} ≡ 1 (mod M)

by Fermat’s Little Theorem. On the other hand, since 2(#Ω) divides #J_{y}, we obtain

d^{#J_{y}} ≡ 1 (mod N).

(Already raising to the (#Ω)th power will send d ∈ Ω to 1 ∈ C_{N}, which corresponds to d ≡ ±1 (mod N).) Therefore, m must be even and the proof of the lemma is complete. ■

Claim 7.6 The function e(τ) is a unit on X_{0}^{#} (N, M).

Proof. We just need to check conditions (U1–4) of Theorem 2.1, and that e(τ)

remains invariant under the action of any matrix ({a \atop c} {b \atop d}) ∈ SL_{2}ℤ with c ≡ 0 (mod NM) and d mapping to Ω under the map ℤ ) C_{N}.

By the definition of the function e and Lemma 7.3(a), (U1) is equivalent to

2zvq(M − 1) ≡ 0 (mod 12).

This is always satisfied since 2zq = 6 and M − 1 is even.

It is clear from the definition of e that conditions (U2) and (U3) are satisfied,

because 0 is divisible by NM.

Lemma 7.3(c) shows us that condition (U4) is equivalent to

\frac{1}{6}zvqN^{2}(M − 1)(M + 1)M + q ∑_{b∈Ω_{y}} (Mb̃^{2} − M^{2}ϕ(b̃)^{2}) ≡ 0 (mod NM).

The condition modulo M is obviously satisfied. For the condition modulo N, observe that the first term is clearly divisible by N, whereas the second term is divisible by N since for any b ∈ Ω_{y},

b̃^{2} ≡ M^{2}ϕ(b̃)^{2} (mod N),

and hence

∑_{b∈Ω_{y}} (Mb̃^{2} − M^{2}ϕ(b̃)^{2}) ≡ ∑_{b∈Ω_{y}} (Mb̃^{2} − b̃^{2}) ≡ (M − 1) ∑_{b∈Ω_{y}} b̃^{2} ≡ 0 (mod N),

where the last congruence used Lemma 4.3.

To check invariance under ({a \atop c} {b \atop d}), we proceed as in the proof of Theorem 3.2. First

observe that ({a \atop c} {b \atop d}) indeed permutes the indices of functions in e(τ) (after reducing the indices to E′). The only thing we need to show is that the transformation factor arising from reducing the indices to E′ is 1. Using (K2), this reduces to showing that

c ∑_{j∈J_{M}} j − d\frac{c}{NM} ∑_{j∈J_{M}} j^{2} + NM ∑_{j∈J_{M}} ⌊\frac{dj}{NM} ⌋ ≡ 0 (mod 2NM).

(The floor function in ⌊dj/NM ⌋ arises in a fashion entirely analogous to how it came up in the proof of Theorem 3.5.) Since NM divides each of c, ∑ j^{2} (see above in our proof that (U4) is satisfied) and NM (respectively), it must divide the first, second and third term (respectively) of the above expression. Therefore, we need only check that

c ∑_{j∈J_{M}} j − d\frac{c}{NM} ∑_{j∈J_{M}} j^{2} + NM ∑_{j∈J_{M}} ⌊\frac{dj}{NM} ⌋ ≡ 0 (mod 2).

We have, using Lemma 7.5 and Lemma 7.3 parts (b) and (c),

c ∑_{j∈J_{M}} j − d\frac{c}{NM} ∑_{j∈J_{M}} j^{2} + NM ∑_{j∈J_{M}} ⌊\frac{dj}{NM} ⌋

≡ c ∑_{b∈Ω_{M}} (b̃ − ϕ(b̃)) − dc ∑_{b∈Ω_{M}} (b̃ − ϕ(b̃)) + (1 + d) ∑_{b∈Ω_{M}} (b̃ − ϕ(b̃))
≡ (d + 1)(c + 1) ∑_{b∈Ω_{M}} (b̃ − ϕ(b̃)) (mod 2).

Since c and d are relatively prime integers, at least one of them is odd, and therefore the number (d + 1)(c + 1) must be even. This concludes our proof of the claim. ■

Claim 7.7 The divisor of e(τ) is

(1 − \overline{w}_{M}^{∗})π_{1}^{∗}(d) − ϕ^{∗} (\frac{v(M − 1)}{2} D^{−,−}) .

Proof. As was mentioned before, we shall prove here that the divisor above is

equal to the divisor of the function e(τ).

The proof of this claim is going to be similar to that of Theorem 4.9(c), but more

complex since there are more kinds of cusps and the objects we are considering are more complicated.

First, we shall identify the divisors π_{1}^{∗}(d) ∈ J_{0}^{#} (N, M) and \overline{w}_{M}^{∗}π_{1}^{∗}(d) ∈ J_{0}^{#} (N, M).

We do this by first recalling what the divisor d ∈ J_{0}^{#} (N) is. For any 1 ≤ t ≤ r, let

A_{t} = \frac{q}{2^{k}} (∑_{b∈Ω} \frac{2^{k}N}{2}B_{2} (\frac{\{b̃t\}}{N}) + ∑_{b∈C_{N}} \frac{−N}{2}B_{2} (\frac{\{b̃t\}}{N})).

Then, by the definition of d, for any 1 ≤ t ≤ r we have

ord_{P_{t}}(d) = A_{t},  ord_{Q_{t}} (d) = 0,

and therefore

ord_{P_{t}^{0}}(π_{1}^{∗}(d)) = MA_{t},  ord_{Q_{t}^{0}} (π_{1}^{∗}(d)) = 0
ord_{P_{t}} (π_{1}^{∗}(d)) = A_{t},  ord_{Q_{t}} (π_{1}^{∗}(d)) = 0.

By considering the action of \overline{w}_{M}^{∗} on the cusps, we also obtain that

ord_{P_{t}^{0}}(\overline{w}_{M}^{∗}π_{1}^{∗}(d)) = A_{Mt} ,  ord_{Q_{t}^{0}} (\overline{w}_{M}^{∗}π_{1}^{∗}(d)) = 0
ord_{P_{t}} (\overline{w}_{M}^{∗}π_{1}^{∗}(d)) = MA_{t},  ord_{Q_{t}} (\overline{w}_{M}^{∗}π_{1}^{∗}(d)) = 0.

Before we turn our attention to div(e), we note that

ord_{P_{t}^{0}}(D^{−,−}) = 1,  ord_{Q_{t}^{0}} (D^{−,−}) = −1
ord_{P_{t}} (D^{−,−}) = −1,  ord_{Q_{t}} (D^{−,−}) = 1.

We shall now show for each cusp of J_{0}^{#} (N, M) that the divisor mentioned in the

statement of the claim and div(e) have the same order.

For a cusp labeled Q_{t}^{}for some 1 ≤ t ≤ r (having ramification index 1 over the

cusp of X(1)), we observe that it can be represented by \binom{t}{NM} in Shimura’s notation and therefore

ord_{Q_{t}} (div(e)) = q(#J_{M})\frac{1}{2}B_{2}(0) = \frac{2zvq(M − 1)}{12} = \frac{v(M − 1)}{2} .

Since π_{1}^{∗}(d) and \overline{w}_{M}^{∗}π_{1}^{∗}(d) have order 0 at this cusp, this proves our claim for the cusps Q_{t}^{}.

For a cusp labeled Q_{t}^{0} for some 1 ≤ t ≤ r (having ramification index M over

the cusp of X(1)), we observe that it can be represented by \binom{t}{N} in Shimura’s notation and therefore

ord_{Q_{t}^{0}} (div(e)) = q ∑_{j∈J_{M}} \frac{M}{2} B_{2} (\frac{Nj mod NM}{NM} ) = q ∑_{j∈J_{M}} \frac{M}{2} B_{2} (\frac{j mod M}{M} )
= qM(#Ω_{M})\frac{1}{2} ∑_{i=1}^{M−1} B_{2} (\frac{i}{M} ) = \frac{−v(M − 1)}{2} ,

where we used the fact that ∑ _{i=1}^{M−1} B_{2}(i/M) is equal to (1 − M)/(6M). Since π_{1}^{∗}(d) and \overline{w}_{M}^{∗}π_{1}^{∗}(d) have order 0 at this cusp, this proves our claim for the cusps Q_{t}^{0}.

For a cusp labeled P_{t}^{}for some 1 ≤ t ≤ r (having ramification index N over

the cusp of X(1)), we observe that it can be represented by \binom{1}{Mϕ(t)} in Shimura’s notation. Therefore, as above,

ord_{P_{t}} (div(e)) = q ∑_{j∈J_{M}} \frac{N}{2}B_{2} (\frac{ϕ(t)j mod N}{N} ) .

Therefore,

ord_{P_{t}} (div(e) − π_{1}^{∗}(d) + \overline{w}_{M}^{∗}π_{1}^{∗}(d))

= \frac{qN}{2} ∑_{j∈J_{M}} B_{2} (\frac{ϕ(t)j mod N}{N} ) + (1 − M) (q ∑_{b∈Ω} \frac{N}{2}B_{2} (\frac{\{b̃t\}}{N}) + \frac{qr}{12 · 2^{k}} )
= \frac{qN}{2} (1 − M) ( ∑_{b∈Ω_{Mϕ(t)}} B_{2} (\frac{b̃}{N}) − ∑_{b∈Ω_{t}} B_{2} (\frac{b̃}{N}))+ \frac{qr(1 − M)}{12 · 2^{k}}
= \frac{qN}{2} (1 − M) · 0 + \frac{−2qzv(M − 1)}{12} = \frac{−v(M − 1)}{2} ,

where the above argument used the fact that modulo N, the set J_{M} can be considered as M − 1 copies of the set Ω_{M}. Then we used the fact that Mϕ(t) and t map to the same element in C_{N} to cancel out the last two nasty-looking sums. Thereby we completed the proof for the cusps P_{t}^{}.

It remains for us to consider the cusps labeled P_{t}^{0} for some 1 ≤ t ≤ r (having ram-

ification index NM over the cusp of X(1)). We observe that such a cusp can be represented by \binom{0}{t} in Shimura’s notation (where we add N to t if it is otherwise divisible by M). Then

ord_{P_{t}^{0}}(div(e)) = q ∑_{j∈J_{M}} \frac{NM}{2} B_{2} (\frac{tjM mod NM}{NM} ) .

Therefore

ord_{P_{t}^{0}}(div(e) − π_{1}^{∗}(d) + \overline{w}_{M}^{∗}π_{1}^{∗}(d)) = \frac{qNM}{2} ∑_{j∈J_{M}} B_{2} (\frac{tjM mod NM}{NM} )

+ \frac{qNM}{2} ∑_{b∈Ω} B_{2} (\frac{\{b̃t\}}{N}) − \frac{qNM}{2^{k+1}} ∑_{b∈C_{N}} B_{2} (\frac{b̃}{N})

− \frac{qN}{2} ∑_{b∈Ω} B_{2} (\frac{\{b̃tM\}}{N} ) + \frac{qN}{2^{k+1}} ∑_{b∈C_{N}} B_{2} (\frac{b̃}{N}) .

We shall separate the above sum of five terms into groups. First of all, note that the sum of the third and the fifth terms is equal to

\frac{qN}{2} (1 − M)\frac{1}{2^{k}} ∑_{b∈C_{N}} B_{2} (\frac{b̃}{N}) = \frac{qN}{2} (1 − M)\frac{1}{2^{k}} \frac{−r}{6N} = \frac{v(M − 1)}{2} .

Therefore, to complete the proof of our claim, it suffices to show that the sum of the first, second and fourth terms is zero. This boils down (after division by qN/2) to showing that

M ∑_{j∈J_{Mt}} B_{2} (\frac{j}{NM} ) + M ∑_{b∈Ω_{t}} B_{2} (\frac{b̃}{N}) − ∑_{b∈Ω_{Mt}} B_{2} (\frac{b̃}{N}) = 0.

However, the above equation is just the sum of the conclusions of Lemma 7.4 when we allow the y of the Lemma to run over all elements of Ω_{M}. ■

Proof of Theorem 6.13. This theorem follows from Claims 7.6 and 7.7. ■ We shall now make some general comments about studying the cuspidal divisor

group on X_{0}^{#} (N, M) in the style of our Chapter 3. We find a number of differences from the case of prime level considered in Chapter 3.

First of all, it is still the case (as can be seen from [8, Chapter 3, Theorem 5.2])

that the functions of the form specified by Theorem 2.1 are units (they are the so-called Siegel units), but they do not exhaust all the units of X(NM) any more. Instead, the Siegel units form a subgroup of the group of all units, in such a way that the quotient group is an elementary 2-abelian group. Therefore, the divisor mentioned in Theorem 6.13 might have been principal without this fact being revealed by an analysis of Siegel units on X_{0}^{#} (N, M).

Secondly, we cannot expect the analog of our Fact 2.2 to hold in the present case,

since X(NM) has (N^{2} − 1)(M^{2} − 1)/2 cusps, whereas there are (N^{2}M^{2} − 1)/2 essentially different Siegel units. Therefore, there will be a lot of extra relations between the divisors of Siegel units that will make it significantly harder to emulate the proof of our Theorem 3.2 to establish the group of Siegel units on X_{1}(NM).

Fortunately, both of the above difficulties can probably be overcome, and we shall

outline a suggested solution now.

Firstly, the non-Siegel units of X(NM) have also been studied extensively. An

analysis of [7] and [23] should probably allow a sufficiently explicit description of all units and their divisors.

Secondly, although we have too many Siegel units, the relations among them are

well-understood. They are discussed in [8] under the name “distribution relations”. These relations generally take the form of an “old” unit (i.e., one coming from a lower level) giving a divisor that is linear combination of divisors of various other Siegel units (some of which might also be old). It turns out that by the time we descend all the way to X_{1,0}(N, M), we have 2N − 2 cusps and 2N + 1 Siegel units. Instead of having just two kinds of functions as in Definition 3.1, we now have eight different kinds of functions. (There are two types each coming from levels N and M, as well as four kinds of new functions.) Nevertheless, the relations each work out to be in the form where the product of a subset of the Siegel units is a constant. Since the three subsets arising form a partition of the set of Siegel units, it turns out that the arguments of Theorem 3.2 still go through.

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