Endomorphisms of Jacobians of Modular Curves

E. Kani

Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada, K7L 3N6 Email: Kani@mast.queensu.ca

1 Introduction

Let X_{Γ} = Γ\ℌ^{∗} be the modular curve associated to a congruence subgroup Γ of level N with Γ_{1}(N) ≤ Γ ≤ Γ_{0}(N). Then X_{Γ} has a canonical model X = X_{Γ,ℚ} defined over ℚ, and its Jacobian variety J_{X} is also defined over ℚ. Let E = End_{ℚ}^{0}(J_{X}) denote the ℚ-algebra of ℚ-endomorphisms of J_{X}. It is well-known that E contains the Hecke algebra 𝕋′_{ℚ} generated (as a ℚ-algebra) by the Hecke operators T_{p} with p ∤ N.

If N is prime, then E = 𝕋′_{ℚ} (cf. Ribet[6]), but for composite N’s this is no longer true in general. One reason for this is that for each pair (M, d) with dM|N there is a “degeneracy morphism” (cf. Mazur[4]) B_{M,d} : X → X_{M}, where X_{M} is the corresponding curve of level M, and that these morphisms give rise to extra endomorphisms (called “degeneracy operators”) D_{M,d} = B_{M,1}^{∗} ∘ (B_{M,d})_{∗} and ^{t}D_{M,d} = B_{M,d}^{∗} ∘ (B_{M,1})_{∗} of J_{X}.

The purpose of this paper is to prove the following basic result which does not seem to have been recorded in the literature (but is well-known to the experts):

Theorem 1. End_{ℚ}^{0}(J_{X}) is generated as a ℚ-algebra by the Hecke algebra 𝕋′_{ℚ} and the degeneracy operators D_{M,d}, ^{t}D_{M,d}, for dM|N.

In particular, we see that the ℚ-algebra 𝔼 := ⟨𝕋′_{ℚ}, D_{M,d},^{t}D_{M,d} : dM|N⟩_{ℚ} contains the full Hecke algebra 𝕋_{ℚ} = ⟨T_{n} : n ≥ 1⟩_{ℚ} as a subalgebra; we will thus refer to 𝔼 as the extended Hecke algebra. Note that for special curves (such as X_{0}(p^{2}) and X(p)) we can use the methods below to obtain other generators of 𝔼; cf. Example 19 below.

It is immediate that the algebra 𝔼 is contained in the ℚ-algebra 𝕄 of all modular endomorphisms, i.e. 𝕄 is the ℚ-linear span of those endomorphisms of J_{X} which are defined by a double coset ΓgΓ with g ∈ GL_{2}^{+}(ℚ), as in [8]. We thus obtain:

Corollary 2. Every endomorphism f ∈ End_{ℚ}^{0}(J_{X}) is modular, i.e. f is a ℚ-linear combination of endomorphisms defined by double cosets.

Note that the analogue of this statement is no longer true for endomorphisms which are not ℚ-rational: the presence of (non-rational) CM-elliptic curves in J_{X} gives rise to endomorphisms which are not modular, as is easy to see.

Another interesting consequence of Theorem 1 is the following result which generalizes a statement proved in Mazur[4], p. 139.

Corollary 3. The centre of End_{ℚ}^{0}(J_{X}) is 𝕋′_{ℚ}, i.e. Z(E) = 𝕋′_{ℚ}.

In fact, the proof of Theorem 1 shows that a certain refinement of this corollary is also true. Let Ω^{1}(J_{X}) := H^{0}(J_{X}, Ω_{J_{X}}^{1}) denote the space of holomorphic 1-forms on J_{X}, which is naturally isomorphic to the space S_{2}(Γ, ℚ) of weight 2 cusp forms with rational Fourier expansions. Then the above proof shows that centralizer of E = 𝔼 in End_{ℚ}(Ω^{1}(J_{X})) is 𝕋′_{ℚ}, i.e. End_{E}(Ω^{1}(J_{X})) = 𝕋′_{ℚ}. Thus, by the “dictionary” of [3], we obtain the following generalization of the Shimura construction:

Corollary 4. We have End_{E}(Ω^{1}(J_{X})) = 𝕋′_{ℚ}. Thus, the map (A, p) ↦ p^{∗}Ω^{1}(A) ⊂ Ω^{1}(J_{X}) ≃ S_{2}(Γ, ℚ) induces a bijection between the set of (equivalence classes) of abelian quotients p : J_{X} → A of J_{X}/ℚ and the set of 𝕋′_{ℚ}-submodules of S_{2}(Γ, ℚ).

The first step in the proof of Theorem 1 is the following classification of the irreducible 𝔼_{ℂ}-modules, where 𝔼_{ℂ} = 𝔼 ⊗ ℂ. To state the result, recall that 𝔼_{ℂ} acts faithfully on the space S = S_{2}(Γ) of weight 2 cusp forms on Γ, and that S has (by Atkin-Lehner theory) a 𝕋′_{ℂ}-module decomposition S = ⨁_{f∈N (Γ)} S_{f}, where N (Γ) denotes the set of all normalized newforms of weight 2 (of all levels) on Γ and S_{f} denotes the χ_{f}-eigenspace with respect to the character χ_{f} : 𝕋′_{ℂ} = 𝕋′_{ℚ} ⊗ ℂ → ℂ defined by f ∈ N (Γ). We then have:

Theorem 5. Each S_{f} is an irreducible 𝔼_{ℂ}-module, and every irreducible 𝔼_{ℂ}-module is isomorphic to some S_{f}. Thus 𝔼_{ℂ} is (isomorphic to) the centralizer C_{S}(𝕋′_{ℚ}) of 𝕋′_{ℚ} in End_{ℂ}(S), and Z(𝔼_{ℂ}) = 𝕋′_{ℚ} ⊗ ℂ.

Since this theorem is of independent interest and has a natural analogue for cusp forms of arbitrary weight, we shall prove a more general version below; cf. Theorem 11.

In view of Theorem 5, Theorem 1 follows readily once the structure of E is known. For this, let n_{f} = dim S_{f} and let K_{f} = ℚ({a_{n}}) be the field generated by the Fourier coefficients of f = ∑ a_{n}q^{n}. Then we have the following theorem which is a consequence of the fundamental results of Ribet[7].

Theorem 6 (Ribet). End_{ℚ}^{0}(J_{X}) ≃ ∏_{f∈N (Γ)/G_{ℚ}} M_{n_{f}} (K_{f}).

Actually, Ribet’s results give a priori only a slightly weaker result; cf. equation (14) below. As a result, the order of the proofs of the above theorems has to be partially reversed. Indeed, after proving Theorem 5 and (14), we first deduce Corollary 3 and then Theorem 1. Finally, we use these results to prove the full version of Theorem 6.

2 The Degeneracy Operators

Throughout, we shall fix a congruence subgroup Γ with Γ_{1}(N) ≤ Γ ≤ Γ_{0}(N), so

Γ = Γ_{H}(N) := \{g ∈ SL_{2}(ℤ) : g ≡ ( \begin{matrix}a & b \\ 0 & d \end{matrix}) (mod N), a ∈ H\},

for some subgroup H ≤ (ℤ/Nℤ)^{×} ≃ Γ_{0}(N)/Γ_{1}(N). Then for any divisor M|N we have a corresponding subgroup Γ_{H}(M) := Γ\overline{H}(M), where \overline{H} ≤ (ℤ/Mℤ)^{×} is the image of H under the reduction map r_{N,M} : (ℤ/NZ)^{×} → (ℤ/Mℤ)^{×}.

For any positive integer d, let α_{d} = ( \begin{matrix}1 & 0 \\ 0 & d \end{matrix}) and β_{d} = ( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ) = dα_{d}^{−1}. Since

β_{d} ( \begin{matrix}x & y \\ z & w \end{matrix}) β_{d}^{−1} = ( \begin{matrix}x & yd \\ z/d & w \end{matrix} ), we see that if d|N, then

(1)  β_{d}Γ_{H}(N)β_{d}^{−1} = Γ_{H}(\frac{N}{d} , d) := \{( \begin{matrix}x & y \\ z & w \end{matrix}) ∈ SL_{2}(ℤ) : r_{N}(x) ∈ H, d|y, \frac{N}{d} |z\},

where r_{N} : ℤ → ℤ/Nℤ denotes the reduction (mod N) map.

In order to study the degeneracy operators mentioned in the introduction, we shall first define them as “abstract operators” via double cosets, and then (as in Shimura[8]) consider their actions on the spaces of cusp forms S_{k}(Γ). Thus, fix divisors d, M of N with dM|N and consider the double cosets

β_{M,d} = Γ_{H}(M)β_{d}Γ_{H}(N) and ^{t}β_{M,d} = Γ_{H}(N)α_{d}Γ_{H}(M).

Since α_{d}^{−1}Γ_{H}(N)α_{d} = Γ_{H}(\frac{N}{d} , d) ≤ Γ_{H}(M), we see that the degree of these double cosets (as defined in [8], p. 51) is given by

(2)  deg(β_{M,d}) = 1 and deg(^{t}β_{M,d}) = [Γ_{H}(M) : Γ_{H}(\frac{N}{d} , d)].

The (abstract) degeneracy operators are defined as the products

D_{M,d} = ^{t}β_{M,1} · β_{M,d} and ^{t}D_{M,d} = ^{t}β_{M,d} · β_{M,1}.

From the definition of the product of double cosets (cf. [8], p. 52) together with the fact that β_{d}Γ_{H}(N)β_{d}^{−1} = Γ_{H}(\frac{N}{d} , d) ≤ Γ_{H}(M) one sees easily that

(3) D_{M,d} = ∑_{γ∈R_{d}} Γ_{H}(N)γβ_{d}Γ_{H}(N) and ^{t}D_{M,d} = ∑_{γ∈R_{d}} Γ_{H}(N)α_{d}γ^{−1}Γ_{H}(N),

where R_{d} is a system of representatives of the set of double cosets (double coset space) Γ_{H}(N)\Γ_{H}(M)/Γ_{H}(\frac{N}{d} , d). Note that the above formula (together with [8], p.

70) shows that the Hecke operator T_{d} = Γ_{H}(N)α_{d}Γ_{H}(N) is a component of ^{t}D_{M,d}.

Remark 7. As the notation indicates, ^{t}D_{M,d} is the Rosati adjoint of D_{M,d}; cf. [8], p. 72 and p. 171. Similarly ^{t}β_{M,d} is the adjoint of β_{M,d}.

We next want to show that the degeneracy operators commute with the Hecke operators T_{n} = T_{n}^{N} for (n, N) = 1. (Note that the T_{n}’s generate the algebra R(Γ_{H}(N), Δ′_{N}) ⊗ ℚ in the notation of [8], p. 67.) To this end we show more generally

Proposition 8. If dM|N, then for any integer n ≥ 1 with (n, N) = 1 we have

(4)  T_{n}^{M} · β_{M,d} = β_{M,d} · T_{n}^{N} and T_{n}^{N} ·^{t}β_{M,d} = ^{t}β_{M,d} · T_{n}^{M}.

The proof of this proposition is based on the following simple fact.

Lemma 9. Let Γ_{1}, Γ_{2} be commensurable subgroups of G = GL_{2}^{+}(ℚ), and suppose that α ∈ G satisfies αΓ_{2}α^{−1} ≤ Γ_{1}. Then for any β ∈ G we have

(5)  (Γ_{1}αΓ_{2})(Γ_{2}βΓ_{2}) = c_{β}Γ_{1}αβΓ_{2}, with c_{β} = deg(Γ_{2}βΓ_{2})/ deg(Γ_{1}αβΓ_{2}).

If, in addition, there exists a subset S ⊂ Γ_{2}βΓ_{2} such that Γ_{1}βΓ_{1} = Γ_{1}αSα^{−1}, then

(6)  (Γ_{1}βΓ_{1})(Γ_{1}αΓ_{2}) = c′_{β}Γ_{1}αβΓ_{2}, with c′_{β} = deg(Γ_{1}βΓ_{1})/ deg(Γ_{1}αβΓ_{2}),

and so (Γ_{1}αΓ_{2})(Γ_{2}βΓ_{2}) = c′′_{β}(Γ_{1}βΓ_{1})(Γ_{1}αΓ_{2}), where c′′_{β} = deg(Γ_{2}βΓ_{2})/ deg(Γ_{1}βΓ_{1}).

Proof. The hypothesis on α implies that Γ_{1}αΓ_{2}βΓ_{2} = Γ_{1}(αΓ_{2}α^{−1})αβΓ_{2} = Γ_{1}αβΓ_{2}, and so (5) holds for some integer c_{β}. By taking degrees of both sides and observing that deg(Γ_{1}αΓ_{2}) = 1 by hypothesis, the given formula for c_{β} follows.

The hypotheses on S show that Γ_{1}βΓ_{1}αΓ_{2} = (Γ_{1}αSα^{−1})αΓ_{2} ⊂ Γ_{1}αΓ_{2}βΓ_{2} = Γ_{1}αβΓ_{2}, where the last equality follows from (5). Thus, Γ_{1}βΓ_{1}αΓ_{2} ⊂ Γ_{1}αβΓ_{2}, and so we must have equality since Γ_{1}βΓ_{1}αΓ_{2} is a union of (Γ_{1}, Γ_{2})-double cosets (and since double cosets are either disjoint or equal). Thus, Γ_{1}βΓ_{1}αΓ_{2} = Γ_{1}αβΓ_{2}, and so we see that (6) holds for some c′_{β}. By the same argument as for (5) we see that c′_{β} has the asserted value.

The last assertion follows immediately from (5) and (6) since c′′_{β} = c_{β}/c′_{β}. Proof of Proposition 8. Apply Lemma 9 to Γ_{1} = Γ_{H}(M), Γ_{2} = Γ_{H}(N), α = β_{d} and β = α_{p}, where p ∤ N is a prime. Thus, Γ_{1}αΓ_{2} = β_{M,d}, Γ_{1}βΓ_{1} = T_{p}^{M} and Γ_{2}βΓ_{2} = T_{p}^{N}; cf. [8], p. 71.

Since αΓ_{2}α^{−1} = Γ_{H}(\frac{N}{d} , d) ≤ Γ_{1}, we see that the first hypothesis of the lemma

holds. Moreover, take S = \{( \begin{matrix}1 & b \\ 0 & p \end{matrix} )\}_{0≤b<p} ∪ \{σ_{p}β_{p}\}, where σ_{p} ≡ ( \begin{matrix}p^{−1} & 0 \\ 0 & p \end{matrix} ) (mod N).

Since β_{d} ( \begin{matrix}1 & b \\ 0 & p \end{matrix} ) β_{d}^{−1} = ( \begin{matrix}1 & bd \\ 0 & p \end{matrix} ) = ( \begin{matrix}1 & k \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & b′ \\ 0 & p \end{matrix} ), where b′ + kp = bd, 0 ≤ b′ < p, and

since β_{d}σ_{p}β_{d}^{−1} ≡ ( \begin{matrix}p^{−1} & 0 \\ 0 & p \end{matrix} ) (mod \frac{N}{d}) (and since M|\frac{N}{d}), we see that Γ_{1}β_{d}Sβ_{d}^{−1} = Γ_{1}S = Γ_{1}βΓ_{1} = T_{p}^{M}, where the second last equality follows from [8], p. 72. Similarly, we have S ⊂ Γ_{2}βΓ_{2} = T_{p}^{N}. Thus, S satisfies the hypotheses of the lemma, and so we conclude from the lemma (together with the fact that c′′_{β} = 1 because deg(T_{p}^{M}) = deg(T_{p}^{M}) = p + 1) that the first equation of (4) holds for all primes p ∤ N.

Since we have T (n, n)β_{M,d} = β_{M,d}T (n, n) (use [8], Prop. 3.7 on p. 54), the recursion relations of the T_{n}’s (cf. [8], p. 71) show that the first formula of (4) holds for all n with (n, N) = 1.

From this it follows that we also have the formula

(7)  ^{t}T_{n}^{M} · β_{M,d} = β_{M,d} ·^{t}T_{n}^{N}, for all (n, N) = 1.

Indeed, since ^{t}T_{n} = σ_{n}T_{n} = T_{n}σ_{n} (cf. [8], p. 72), and since σ_{n}β_{M,d} = β_{M,d}σ_{n}, we see that (7) follows from the first equation of (4).

Finally, taking the Rosati adjoints of (7) yields the second equation of (4), which concludes the proof of Proposition 8.

Notation. Let 𝕋̃′ = R(Γ_{H}(N), Δ′_{N}) denote the Hecke ring generated by the (abstract) Hecke operators T_{n} with (n, N) = 1, and let 𝔼̃ = ⟨T_{n}, D_{M,d},^{t}D_{M,d} : (n, N) = 1, dM|N⟩ ⊂ 𝕄̃ := R(Γ_{H}(N), Δ) denote the Hecke ring generated by the T_{n}’s and the degeneracy operators D_{M,d} and ^{t}D_{M,d}.

Corollary 10. 𝕋̃′ is a subring of the centre of 𝔼̃, i.e. 𝕋̃′ ⊂ Z(𝔼̃).

Proof. By (4) we have T_{n}D_{M,d} = ^{t}β_{M,1}T_{n}^{M}β_{M,d} = D_{M,d}T_{n}, for any d, M with dM|N and any n with (n, N) = 1, and similarly T_{n} and ^{t}D_{m,d} commute. Thus, since 𝕋̃′ = ⟨T_{n}⟩ is commutative, we see that 𝕋̃′ ⊂ Z(𝔼̃).

3 The Structure of the Extended Hecke Algebra 𝔼_{k,ℂ}

We now fix an integer k and consider the space S = S_{k}(Γ_{H}(N)) of cusp forms of weight k on Γ_{H}(N). By [8], p. 73, each double coset in 𝕄̃ = R(Γ_{H}(N), Δ) induces a natural ℂ-linear operator on S, and S is a right 𝕄̃ ⊗ ℂ-module. In particular, since 𝔼̃ ⊂ 𝕄̃, S is also a 𝔼̃⊗ℂ-module; we let 𝔼_{k,ℂ} denote the image of 𝔼̃⊗ℂ in End_{ℂ}(S)^{op}.

Thus, by definition, 𝔼_{k,ℂ} is the subalgebra of End_{ℂ}(S)^{op} generated by the Hecke operators [T_{p}]_{k}, for p ∤ N and the degeneracy operators [D_{M,d}]_{k}, [^{t}D_{M,d}]_{k}, for dM|N. Note that the action of [D_{M,d}]_{k} on S is given by the formula

(8)  f|[D_{M,d}]_{k}(z) = d^{k−1} tr_{M}(f)(dz), for f ∈ S,

where tr_{M} = tr_{Γ_{H}(N)/Γ_{H}(M)} : S_{k}(Γ_{H}(N)) → S_{k}(Γ_{H}(M)) is the usual trace map. Indeed, since tr_{M} = [^{t}β_{M,1}]_{k} by definition, this follows immediately from the fact that f|[D_{M,d}]_{k} = f|[^{t}β_{M,1}]_{k}[β_{M,d}]_{k} together with the formula

(9)  f|[β_{M,d}]_{k}(z) = d^{k/2−1}f|_{k}β_{d}(z) = d^{k−1}f(dz), for f ∈ S_{k}(Γ_{H}(M)).

A similar formula holds for the operator [^{t}D_{M,d}]_{k}, which by [8], p. 76, is just the adjoint of [D_{M,d}]_{k} with respect to the Petersson product on S.

Let 𝕋′_{k,ℂ} be the ℂ-subalgebra of 𝔼_{k,ℂ} generated by the Hecke operators [T_{p}]_{k}, for p ∤ N. Since 𝕋′_{k,ℂ} is a commutative semi-simple algebra, S has a decomposition into 𝕋′_{k,ℂ}-eigenspaces S_{χ}. By Atkin-Lehner theory, each such character is of the form χ = χ_{f}, for a unique normalized newform f of some level N_{f}|N, and so we have the Atkin-Lehner decomposition

(10)  S = ⨁_{f∈N (Γ)} S_{f},

where S_{f} = S_{χ_{f}} and N (Γ) = N_{k}(Γ) denotes the set of all normalized newforms of weight k of level N_{f}|N on Γ = Γ_{H}(N); cf. [2], p. 28 and/or [5], §4.6). In particular, we see that dim 𝕋′_{k,ℂ} = #N (Γ).

We now prove:

Theorem 11. Each S_{f} is an irreducible 𝔼_{k,ℂ}-module, and every irreducible 𝔼_{k,ℂ}- module is isomorphic to some S_{f}. Thus 𝔼_{k,ℂ} is isomorphic to the centralizer C_{S}(𝕋′_{k,ℂ}) of 𝕋′_{k,ℂ} in End_{ℂ}(S), and its centre is Z(𝔼_{k,ℂ}) = 𝕋′_{k,ℂ}.

Proof. First note that since 𝕋′_{k,ℂ} is closed under the Petersson adjoints, the same is true for 𝔼_{k,ℂ} and hence both algebras are semi-simple (cf. [8], p. 83.) Moreover, from Corollary 10 it follows that 𝕋′_{k,ℂ} ⊂ Z(𝔼_{k,ℂ}). Thus, the operators in 𝔼_{k,ℂ} preserve the 𝕋′_{k,ℂ}-eigenspaces in S, and so each S_{f} is a 𝔼_{k,ℂ}-module.

Suppose S_{f} were reducible. Then S_{f} = V_{1} ⊕ V_{2}, where each V_{i} is a non-zero 𝔼_{k,ℂ}- module because 𝔼_{k,ℂ} is semi-simple. Consider the map tr = tr_{N_{f}} : S_{f} → S_{k}(Γ_{H}(N_{f}))∩ S_{f} = ℂf, where the last equality follows from the multiplicity 1 theorem of Atkin-Lehner theory. Now tr is non-zero because tr(f) = nf, for some n > 0. Thus, tr(V_{1}) ≠ 0 or tr(V_{2}) ≠ 0, and so there exists (wlog) g ∈ V_{1} such that tr(g) = cf, for some c ≠ 0. Then, by (8) we see that for each d|N/N_{f}, cd^{k/2−1}f|_{k}β_{d} = g|[D_{N_{f} ,d}]_{k} ∈ V_{1}, and so S_{f} ⊂ V_{1} since {f|β_{d}}_{d|N/N_{f}} is a basis of S_{f} (cf. [5], Cor. 4.6.20). Thus S_{f} = V_{1}, which means that S_{f} is irreducible.

Since S is a faithful 𝔼_{k,ℂ}-module (by construction), we see by Wedderburn that every irreducible 𝔼_{k,ℂ}-module is a submodule of S (and thus is isomorphic to some S_{f}) and that hence 𝔼_{k,ℂ} ≃ ∏_{f∈N (Γ)} M_{n_{f}} (ℂ), where n_{f} = dim S_{f} (= σ_{0}(N/N_{f})). (Note that the S_{f}’s are pairwise non-isomorphic as 𝔼_{k,ℂ}-modules because they are already non-isomorphic as 𝕋′_{k,ℂ}-modules.) Thus dim Z(𝔼_{k,ℂ}) = #N (Γ) = dim 𝕋′_{k,ℂ}, and so Z(𝔼_{k,ℂ}) = 𝕋′_{k,ℂ}.

Finally, since S has multiplicity one as a 𝔼_{k,ℂ}-module, it follows easily that 𝔼_{k,ℂ} ≃ C_{S}(𝕋′_{k,ℂ}). To see this, note first that

𝔼_{k,ℂ} ≃ 𝕊 := {φ ∈ End_{ℂ}(S)^{op} : S_{f}|φ ⊂ S_{f}, ∀f ∈ N (Γ)}

because by construction 𝔼_{k,ℂ} is isomorphic to a subring of 𝕊 and because dim 𝔼_{k,ℂ} = ∑_{f∈N (Γ)} n_{f}^{2} = dim 𝕊. Since clearly 𝕊 = C_{S}(Z(𝕊)), it follows that 𝔼_{k,ℂ} ≃ C_{S}(𝕋′_{k,ℂ}).

Corollary 12. Suppose 𝔼′ ⊂ 𝔼_{k,ℂ} is a semi-simple ℂ-algebra which contains 𝕋′_{k,ℂ}, the operators [D_{M,1}]_{k} for M|N, and for each d|N/M an operator D̃_{M,d} with the property that S_{k}(Γ_{H}(M))|β_{d} ⊂ Im([D_{M,1}]_{k}D̃_{M,d}). Then 𝔼′ = 𝔼_{k,ℂ}.

Proof. It is enough to show that each S_{f} is an irreducible 𝔼′-module, for then by the above argument we have 𝔼′ ≃ 𝔼_{k,ℂ} and hence 𝔼′ = 𝔼_{k,ℂ}.

Now if S_{f} were reducible, then as in the proof of Theorem 11 we would have S_{f} = V_{1} ⊕ V_{2} where (wlog) f ∈ V_{1}. Let d|N/N_{f}. Then by hypothesis ∃g ∈ S such that g|[D_{N_{f} ,1}]D̃_{N_{f} ,d} = f|_{k}β_{d}. By replacing g by g|ε_{f}, where ε_{f} ∈ 𝕋′_{k,ℂ} is the idempotent such that S_{f} = S|ε_{f}, we may assume that g ∈ S_{f}. Then g|[D_{N_{f} ,1}]_{k} = cf, for some c ∈ ℂ (cf. proof of Theorem 11), and so f|_{k}β_{d} = (cf)D̃_{N_{f} ,d} ∈ V_{1}. Thus as before V_{1} = S_{f}, and so S_{f} is irreducible.

Remark 13. (a) If 𝔼_{k,ℚ} denotes the ℚ-algebra generated by the operators [T_{p}]_{k} (p ∤ N), [D_{M,d}]_{k} and [^{t}D_{M,d}]_{k} (for dM|N), then we have 𝔼_{k,ℂ} = 𝔼_{k,ℚ}⊗ ℂ, provided that k ≥ 2. Indeed, by (3) we see easily that 𝔼 ⊂ B ⊂ R(Γ_{H}(N), Δ), where B is as defined on p. 83 of [8], and so the assertion follows from [8], Th. 3.48.

(b) By conjugation, the above results also hold for the groups Γ := Γ_{H}(N′, t) = β_{t}Γ_{H}(tN′)β_{t}^{−1}, where H ≤ (ℤ/N′tℤ)^{×}; in particular, they hold for the groups Γ′ considered by Shimura[8], p. 67ff. More precisely, the map Γ_{H}(N′t)αΓ_{H}(N′t) ↦ Γβ_{t}αβ_{t}^{−1}Γ defines a ring isomorphism ρ_{t} : R(Γ_{H}(N′t), Δ) →^{∼} R(Γ, β_{t}Δβ_{t}^{−1}) which identifies the Hecke algebra of Γ_{H}(N′t) with that of Γ. Furthermore, the map f ↦ f|_{k}β_{t} defines an isomorphism S_{k}(Γ) ≃ S_{k}(Γ_{H}(N′t)) which is compatible with the isomorphism ρ_{t}. Thus, the Atkin-Lehner theory for Γ_{H}(N′t) can be transported back to Γ, and thus the analogous results hold for Γ (in place of Γ_{H}(N′t)).

(c) In classical Atkin-Lehner theory (cf. Miyake[5], §4.6) one often considers submodules of S = S_{k}(Γ) with respect to the algebra 𝔸_{k} := ⟨𝕋,^{t}𝕋⟩ ⊂ End_{ℂ}(S_{k}(Γ)), where 𝕋 is the full Hecke algebra (and ^{t}𝕋 is its (Rosati) adjoint); for example, one shows that S_{f} is an 𝔸_{k}-submodule. Now it follows from the above Theorem 11 that we have the inclusion 𝔸_{k} ⊂ 𝔼_{k,ℂ} (which is not obvious from the definitions). Indeed, since C_{S}(𝔼_{k,ℂ}) = 𝕋′ ⊂ C_{S}(𝔸_{k}), the double centralizer theorem shows that 𝔸_{k} ⊂ 𝔼_{k,ℂ}.

4 Application to End_{ℚ}^{0}(J_{X})

As in the introduction, let X = X_{H}(N)/ℚ denote the canonical model (in the sense of [8], p. 152) of X_{Γ} = Γ\ℌ^{∗}, where Γ = Γ_{H}(N). Thus, X is the unique smooth, projective curve over ℚ such that its function field κ(X) is isomorphic to A_{0}(Γ_{H}(N), ℚ), the field of modular functions (of weight 0) on Γ_{H}(N) whose q-expansions have coefficients in ℚ (cf. [8], Prop. 6.9(2), p.140). Note that since H and ±H = ⟨H, −1⟩ define the same curve X_{H}(N), we may assume without loss of generality that −1 ∈ H, and we shall do so in the sequel.

Now for any two divisors d, M of N with dM|N, the map f(z) ↦ f(dz) = f|_{0}β_{d} clearly defines an injection of fields β_{d}^{∗} : A_{0}(Γ_{H}(M), ℚ) ↪ A_{0}(Γ_{H}(N), ℚ) because β_{d}^{−1}Γ_{H}(M)β_{d} ≥ β_{d}^{−1}Γ_{H}(N/d, d)β_{d} = Γ_{H}(N). We thus have a corresponding surjective morphism of curves B_{M,d} : X_{H}(N) → X_{H}(M) such that its induced pullback map on the function fields is β_{d}^{∗}. Note that the graph Γ_{B_{M,d}} ⊂ X_{H}(N) × X_{H}(M) of this morphism is precisely the correspondence X(β_{M,d}) defined by the double coset β_{M,d} = Γ_{H}(M)β_{d}Γ_{H}(N); cf. [8], p. 170. In particular, since the degree of the morphism B_{M,d} is the degree of the Rosati adjoint ^{t}β_{M,d}, we see by [8], p. 171 and (2) that deg(B_{M,d}) = deg(^{t}β_{M,d}) = [Γ_{H}(M) : Γ_{H}(N/d, d)].

Let J_{X} = J_{H}(N)/ℚ be the Jacobian variety of X_{H}(N)/ℚ. Then by functoriality and autoduality of the Jacobian we have induced homomorphisms

B_{M,d}^{∗} : J_{H}(M) → J_{H}(N) and (B_{M,d})_{∗} : J_{H}(N) → J_{H}(M).

As is explained in [8], p. 169-171, these maps are the same as the homomorphisms ξ(^{t}β_{M,d}) ∈ Hom(J_{H}(M), J_{H}(N)) and ξ(β_{M,d}) ∈ Hom(J_{H}(N), J_{H}(M)) defined by the double cosets ^{t}β_{M,d} and β_{M,d}, respectively. Thus, by [8], Prop. 7.1, we have

ξ(D_{M,d}) = ξ(^{t}β_{M,1}) ∘ ξ(β_{M,d}) = B_{M,1}^{∗} ∘ (B_{M,d})_{∗} ∈ End_{ℚ}(J_{H}(N)),

and similarly, ξ(^{t}D_{M,d}) = B_{M,d}^{∗} ∘ (B_{M,1})_{∗} ∈ End_{ℚ}(J_{H}(N)). Moreover, we also have the ℚ-endomorphisms ξ(T_{n}) ∈ End_{ℚ}(J_{H}(N)) defined by the Hecke operators T_{n}; cf. [8], p. 175. As in the introduction, we let 𝕋′_{ℚ} ⊂ End_{ℚ}^{0}(J_{H}(N)) = End_{ℚ}(J_{H}(N)) ⊗ ℚ denote the ℚ-algebra generated by the ξ(T_{p})’s, for p ∤ N.

Proposition 14. Let 𝔼 ⊂ End_{ℚ}^{0}(J_{H}(N)) denote the ℚ-algebra generated by 𝕋′_{ℚ} and the degeneracy operators ξ(D_{M,d}), ξ(^{t}D_{M,d}) with dM|N. Then 𝕋′_{ℚ} = Z(𝔼) is the centre of 𝔼 and

(11)  dim_{ℚ}(𝔼) = ∑_{f∈N_{2}(Γ)} n_{f}^{2}.

Proof. As before, write X = X_{H}(N). It is well-known that E := End_{ℚ}^{0}(J_{X}) acts faithfully on the space(s) Ω^{1}(J_{X}/ℚ) ≃ Ω^{1}(X/ℚ) = H^{0}(X, ω_{X/ℚ}) of holomorphic differentials of J_{X}/ℚ and X/ℚ (cf. e.g. [3]). Thus, the same is true for the subalgebra 𝔼 and hence 𝔼 ⊗ ℂ acts faithfully on Ω^{1}(X_{Γ}/ℂ) = Ω^{1}(X/ℚ) ⊗ ℂ. Now via the identification S_{2}(Γ) →^{∼} Ω^{1}(X_{Γ}/ℂ), the action of 𝔼 ⊗ ℂ on S_{2}(Γ) is the same as that given by the double cosets (cf. [8], p. 171), so that we have a ring isomorphism 𝔼 ⊗ ℂ ≃ 𝔼_{2,ℂ}. Thus, by Theorem 5 (or by Theorem 11) we see that dim_{ℚ} 𝔼 = dim_{ℂ} 𝔼 ⊗ ℂ = dim_{ℂ} 𝔼_{2,ℂ} = ∑_{f∈N_{2}(Γ)} n_{f}^{2}, which proves (11).

Furthermore, since by Theorem 11 we have Z(𝔼) ⊗ ℂ = Z(𝔼 ⊗ ℂ) = 𝕋′_{ℚ} ⊗ ℂ (equality as subalgebras of 𝔼 ⊗ ℂ), it follows that Z(𝔼) = 𝕋′_{ℚ}.

We can refine the above proposition by determining the isotypic 𝔼-module decomposition of the 𝔼-module Ω(J_{X}/ℚ) ≃ S_{2}(Γ, ℚ), where the latter denotes space of cusp form f ∈ S_{2}(Γ) with rational Fourier coefficients. For this, let [f] = fG_{ℚ} denote the Galois orbit of f ∈ N (Γ) and put S_{[f]} = (⊕_{g∈[f]}S_{g}) ∩ S_{2}(Γ, ℚ); cf. [2], p. 36. We then have:

Corollary 15. Each S_{[f]} is an irreducible 𝔼-module, and every irreducible 𝔼-module is of this form. Thus, the isotypic 𝔼-module decomposition of Ω^{1}(J_{X}/ℚ) is given by

(12)  Ω^{1}(J_{X}/ℚ) ≃ S_{2}(Γ, ℚ) ≃ ⨁_{f∈N (Γ)/G_{ℚ}} S_{[f]}.

Proof. It is immediate that each S_{[f]} is a 𝔼-module. Since S_{[f]} ⊗ ℂ = ⊕_{g∈[f]}S_{g}, we see from Theorem 11 (together with the fact that S_{g} ≄ S_{g′}, if g ≠ g′) that S_{[f]} ⊗ ℂ cannot have any nontrivial 𝔼 ⊗ ℂ-submodules which are G_{ℚ}-stable. Thus, S_{[f]} is an irreducible 𝔼-module.

Since Ω^{1}(J_{X}/ℚ) ≃ Ω^{1}(X/ℚ) ≃ S_{2}(Γ, ℚ) (cf. [2], p. 35), we thus see that (12) is the decomposition of Ω^{1}(J_{X}/ℚ) into pairwise non-isomorphic irreducible 𝔼-modules. In particular, every irreducible 𝔼-module is isomorphic to some S_{[f]} because Ω^{1}(J_{X}/ℚ) is a faithful 𝔼-module (and 𝔼 is semisimple).

We next study the structure of the algebra E = End_{ℚ}^{0}(J_{X}). To this end, recall that by the Shimura construction, each (weight 2) normalized newform f ∈ N (Γ) = N_{2}(Γ) of some level N_{f}|N defines an abelian variety A_{f}/ℚ of dimension [K_{f} : ℚ], where K_{f} = ℚ({a_{n}}_{n≥1}) denotes the field generated by the coefficients of f = ∑ a_{n}q^{n}; cf. [8], p. 183, [9] or [3]. (Note that A_{f} = A_{f^{σ}}, for any Galois conjugate f^{σ}, where σ ∈ G_{ℚ} = Gal(\overline{ℚ}/ℚ).) Then the abelian varieties A_{f} determine J_{X} up to ℚ-isogeny, for we have the relation

(13)  J_{X} ∼ ∏_{f∈N (Γ)/G_{ℚ}} A_{f}^{m_{f}} ,

for some positive integers m_{f}; cf. Ribet[7], Proposition (2.3). (We shall see below that m_{f} = n_{f} := dim S_{f}; cf. Remark 18(a).)

In order to deduce Theorem 6 from (13), we require the following facts which were (essentially) proven by Ribet[7]:

Theorem 16. (a) End_{ℚ}^{0}(A_{f}) ≃ K_{f}, for all f ∈ N (Γ).

(b) If f, g ∈ N (Γ), then A_{f} ∼ A_{g} if and only if g = f^{σ}, for some σ ∈ G_{ℚ}.

Proof. (a) Ribet[7], Corollary 4.2.

(b) This follows from Ribet’s results; cf. [1], Proposition 3.2.

Corollary 17. dim Z(End_{ℚ}^{0}(J_{X})) = #N (Γ) = dim 𝕋′_{ℚ}.

Proof. By Theorem 16(a) we see that each A_{f} is ℚ-simple, and hence by part (b) we have Hom_{ℚ}^{0}(A_{f}, A_{g}) = 0 if f ≠ g^{σ}, for all σ ∈ G_{ℚ}. We thus obtain from (13) that

(14)  End_{ℚ}^{0}(J_{X}) ≃ ∏_{f∈N (Γ)/G_{ℚ}} End_{ℚ}^{0}(A_{f}^{m_{f}}) ≃ ∏_{f∈N (Γ)/G_{ℚ}} M_{m_{f}} (K_{f}),

and that hence

(15) dim Z(End_{ℚ}^{0}(J_{X})) = dim ∏_{f∈N (Γ)/G_{ℚ}} K_{f} = ∑_{f∈N (Γ)/G_{ℚ}}[K_{f} : ℚ] = #N (Γ),

where the last equation follows from the fact that #(fG_{ℚ}) = [K_{f} : ℚ]. Since dim 𝕋′_{ℚ} = dim_{ℂ} 𝕋′_{2,ℂ} = #N (Γ) (cf. comment after equation (10)), the assertion follows.

Proof of Corollary 3. Since 𝔼_{ℂ} := 𝔼 ⊗ ℂ ⊂ E_{ℂ} := E ⊗ ℂ ⊂ End_{ℂ}(S), we have Z(E_{ℂ}) ⊂ C_{S}(E_{ℂ}) ⊂ C_{S}(𝔼_{ℂ}) = 𝕋′_{2,ℂ}, the latter by Theorem 11. Now by Corollary 17 we have dim_{ℂ}(E_{C}) = dim 𝕋′_{2,ℂ}, and so we have equality throughout, i.e. Z(E_{ℂ}) = C_{S}(E_{ℂ}) = C_{S}(𝔼_{ℂ}) = 𝕋′_{2,ℂ}.

Proof of Theorem 1. Since 𝔼_{ℂ} = 𝔼 ⊗ ℂ and E_{ℂ} = E ⊗ ℂ are semi-simple subalgebras of End_{ℂ}(S) with C_{S}(𝔼_{ℂ}) = C_{S}(E_{ℂ}) (cf. proof of Corollary 3), it follows from the double centralizer theorem that 𝔼_{ℂ} = E_{ℂ}. Thus 𝔼 = E, as asserted.

Proof of Corollary 2. By construction (cf. §2) we have 𝔼̃ ⊂ 𝕄̃, and so the image 𝕄 of 𝕄̃ ⊗ ℚ in End0 ; \overline{ℚ}(J_{X}) contains the image 𝔼 of 𝔼̃ ⊗ ℚ. Thus, since E = 𝔼 by Theorem 1, we have E ⊂ 𝕄, as claimed.

Proof of Corollary 4. Since S = S_{2}(Γ, ℚ) ⊗ ℂ and C_{S}(E_{ℂ}) = 𝕋′_{ℂ} by the proof of Corollary 3, the first assertion follows (via the isomorphism Ω^{1}(J_{X}) ≃ S_{2}(Γ, ℚ)). The second assertion follows from this by Theorem 4.4 (or Corollary 4.6) of [3].

Proof of Theorem 6. By (14) it is enough to show that m_{f} = n_{f}, for all f ∈ N (Γ). For this, fix f ∈ N (Γ) and let E_{f} = End^{0}(A_{f}^{m_{f}}), which by (13) (and Theorem 16) is naturally a simple two-sided ideal of E. Since 𝔼 = E by Theorem 1, we know by Corollary 15 that S_{[f]} is an irreducible (right) E-module. By the Shimura construction we know that S_{[f]}E_{f} ≠ 0, and so S_{[f]} is a faithful irreducible E_{f}-module. Thus, since E_{f} ≃ M_{m_{f}} (K_{f}), we see that dim_{ℚ} S_{[f]} = m_{f}[K_{f} : ℚ]. On the other hand, since S_{[f]} ⊗ ℂ = ⊕_{g∈[f]}S_{g}, we have dim_{ℚ} S_{[f]} = (dim_{ℂ} S_{f})[K_{f} : ℚ] = n_{f}[K_{f} : ℚ], and so m_{f} = n_{f}.

Remark 18. (a) It follows from the above proof that (13) holds with m_{f} = n_{f} = dim S_{f} = σ_{0}(N/N_{f}). This was asserted without proof in [1], equation (3.4).

(b) All the above results also hold for the curves X_{H}(N′, t) associated to the groups Γ_{H}(N′, t) = β_{t}Γ_{H}(tN′)β_{t}^{−1} of Remark 13(b); in fact, the matrix β_{t} induces a ℚ-isomorphism (β_{t})_{∗} : X_{H}(N′t) →^{∼} X_{H}(N′, t).

(c) In a letter to the author, Ken Ribet (July 2005) pointed out that it is possible to deduce Corollary 3 and the first assertion of Corollary 4 directly from Theorem 6 together with the fact that 𝕋′ ⊂ Z(E).

Example 19. (a) If J_{0}(p^{2}) is the Jacobian of the modular curve X_{0}(p^{2})/ℚ, where p is a prime, then

(16)  End_{ℚ}^{0}(J_{0}(p^{2})) = ⟨𝕋′_{ℚ}, τ, ξ(T_{p}), ξ(^{t}T_{p})⟩_{ℚ} = ⟨𝕋′_{ℚ}, τ, τ′⟩_{ℚ},

where τ = η^{∗}η_{∗} and τ′ = (η′)^{∗}η′_{∗} are the endomorphisms associated to the degeneracy maps η := B_{p,1} : X_{0}(p^{2}) → X_{0}(p) and η′ := B_{p,p} : X_{0}(p^{2}) → X_{0}(p).

(b) If J(p) is the Jacobian of the modular curve X(p)/ℚ defined by the principal congruence subgroup Γ(p), then

(17)  End_{ℚ}^{0}(J(p)) = ⟨𝕋′_{ℚ}, τ̃, ξ(T_{p}), ξ(^{t}T_{p})⟩_{ℚ} = ⟨𝕋′_{ℚ}, τ̃, τ̃′⟩_{ℚ},

where τ̃ = η̃^{∗}η̃_{∗} and τ̃′ = (η̃′)^{∗}η̃′_{∗} are the endomorphisms associated to the covers η̃ : X(p) → X^{1}(p) and η̃′ : X(p) → X_{1}(p) which are induced by the inclusions Γ^{1}(p) ⊂ Γ(p) and Γ_{1}(p) ⊂ Γ(p), respectively.

Proof. (a) Applying Theorem 1 with N = p^{2} and H = (ℤ/p^{2}ℤ)^{×} yields End_{ℚ}^{0}(J_{0}(p^{2})) = ⟨𝕋′_{ℚ}, ξ(D_{p,1}), ξ(D_{p,p}), ξ(^{t}D_{p,p})⟩_{ℚ}. (Note that ξ(D_{p^{2},1}) = id and that ξ(D_{1,d}) = 0 because X_{H}(1) = X(1) has genus 0). Now ξ(D_{p,1}) = τ by definition because Γ_{H}(p) = Γ_{0}(p). Moreover, since ^{t}T_{p} ⊂ D_{p,p} (cf. (3)) and since by (2) deg D_{p,p} = [Γ_{0}(p) : Γ_{0}(p^{2})] = p = deg^{t}T_{p}, we see that D_{p,p} = ^{t}T_{p} and hence that also ^{t}D_{p,p} = T_{p}. This proves the first equality of (16).

To prove the second equality, we shall apply Corollary 12 to 𝔼′ = ⟨𝕋′_{2,ℂ}, [τ]_{2}, [τ′]_{2}⟩_{ℂ}. Since ^{t}τ = τ and ^{t}τ′ = τ′, it is clear that 𝔼′ is semi-simple. Consider D̃_{p,p} := [ττ′]_{2} ∈ 𝔼′, and let T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). Since \{T^{a}\}_{0≤a≤p−1} is a system of coset representatives of Γ_{0}(p)/Γ_{0}(p, p), we see that f|[τ′]_{2} = ∑_{a=0}^{p−1} f|_{2}α_{p}T^{a}β_{p} = f|T_{p}β_{p}. Now since Im([τ]) = S_{2}(Γ_{0}(p)) and since T_{p} acts bijectively on S_{2}(Γ_{0}(p)) (use [5], Th. 4.6.17), we thus see that Im([ττ′]) = S_{2}(Γ_{0}(p))β_{p}. By Corollary 12 we therefore have that 𝔼′ = 𝔼_{2,ℂ}, and so 𝔼 is generated by 𝕋′_{ℚ}, τ, and τ′. This proves the second equality of (16).

(b) We observe that the proof of (a) shows more generally that if H is any subgroup with Ker(r_{p^{2},p}) ≤ H ≤ (ℤ/p^{2}ℤ)^{×}, then the analogous formula of (16) holds for the Jacobian J_{H}(p^{2}) of the curve X_{H}(p^{2}), i.e.

(18)  End_{ℚ}^{0}(J_{H}(p^{2})) = ⟨𝕋′_{ℚ}, τ_{H}, ξ(T_{p}), ξ(^{t}T_{p})⟩_{ℚ} = ⟨𝕋′, τ_{H}, τ′_{H}⟩_{ℚ},

where τ_{H} = η_{H}^{∗}(η_{H})_{∗} and τ′_{H} = (η′_{H})^{∗}(η′_{H})_{∗} are defined by the degeneracy maps η_{H} := B_{p,1} : X_{H}(p^{2}) → X_{H}(p) and η′_{H} := B_{p,p} : X_{H}(p^{2}) → X_{H}(p).

Applying this to H = Ker(r_{p^{2},p}) and noting that Γ(p) = Γ_{H}(p, p) = β_{p}Γ_{H}(p^{2})β_{p}^{−1}, we see that End_{ℚ}^{0}(J(p)) = ⟨𝕋′_{ℚ}, ρ_{p}(τ_{H}), ξ(T_{p}), ξ(^{t}T_{p})⟩_{ℚ} = ⟨𝕋′_{ℚ}, ρ_{p}(τ_{H}), ρ_{p}(τ′_{H})⟩_{ℚ}; cf. Remarks 18(b) and 13(b). Now since β_{p}Γ_{H}(p)β_{p}^{−1} = Γ_{H}(1, p) = Γ^{1}(p), we see that ρ_{p}(τ_{H}) = τ̃. Moreover, ρ_{p}(τ′_{H}) = τ̃′ because f|ρ_{p}(τ′_{H}) = f|β_{p}[τ′]_{2}β_{p}^{−1} = ∑ f|T^{a} = f|[τ̃′], and so (17) holds.

Acknowledgements. I would like to thank Ken Ribet for his helpful comments on this paper; cf. Remark 18(c). In addition, I would like to gratefully acknowledge receipt of funding from the Natural Sciences and Engineering Research Council of Canada (NSERC).

References

[1] M. Baker, E. González-Jiménez, J. Gonzalez, B. Poonen, Finiteness results for modular curves of genus at least 2. Am. J. Math. 127 (2005), 1325-1387.

[2] H. Darmon, F. Diamond and R. Taylor, Fermat’s last theorem. In: Current Developments in Mathematics, 1995 (R. Bott et al, eds.), International Press Inc., Boston, 1994, pp. 1–154.

[3] E. Kani, Abelian subvarieties and the Shimura construction. Preprint.

[4] B. Mazur, Rational isogenies of prime degree. Inv. math. 44 (1978), 129-162.

[5] T. Miyake, Modular Forms. Springer-Verlag, Berlin, 1989.

[6] K. Ribet, Endomorphisms of semi-stable abelian varieties over number fields. Ann. Math. 101 (1975), 555–562.

[7] K. Ribet, Twists of newforms and endomorphisms of abelian varieties. Math. Ann. 253 (1980), 43–62.

[8] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton, NJ, 1971.

[9] G. Shimura, On the factors of the Jacobian variety of a modular function field. J. Math. Soc. Japan 25 (1973), 523–544.