Endomorphisms of Jacobians of Modular Curves

Transcription

1 Enomorphisms of Jacobians of Moular Curves E. Kani Department of Mathematics an Statistics, Queen s University, Kingston, Ontario, Canaa, K7L 3N6 Kani@mast.queensu.ca 1 Introuction Let X Γ = Γ\H be the moular curve associate to a congruence subgroup Γ of level N with Γ 1 (N) Γ Γ 0 (N). Then X Γ has a canonical moel X = X Γ,Q efine over Q, an its Jacobian variety J X is also efine over Q. Let E = En 0 Q(J X ) enote the Q-algebra of Q-enomorphisms of J X. It is well-known that E contains the Hecke algebra T Q generate (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, ) with M N there is a egeneracy morphism (cf. Mazur[4]) B M, : X X M, where X M is the corresponing curve of level M, an that these morphisms give rise to extra enomorphisms (calle egeneracy operators ) D M, = BM,1 (B M,) an t D M, = BM, (B M,1) of J X. The purpose of this paper is to prove the following basic result which oes not seem to have been recore in the literature (but is well-known to the experts): Theorem 1. En 0 Q(J X ) is generate as a Q-algebra by the Hecke algebra T Q an the egeneracy operators D M,, t D M,, for M N. In particular, we see that the Q-algebra E := T Q, D M,, t D M, : M 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 extene Hecke algebra. Note that for special curves (such as X 0 (p 2 ) an X(p)) we can use the methos below to obtain other generators of E; cf. Example 19 below. It is immeiate that the algebra E is containe in the Q-algebra M of all moular enomorphisms, i.e. M is the Q-linear span of those enomorphisms of J X which are efine by a ouble coset ΓgΓ with g GL + 2 (Q), as in [8]. We thus obtain: Corollary 2. Every enomorphism f En 0 Q(J X ) is moular, i.e. f is a Q-linear combination of enomorphisms efine by ouble cosets. Note that the analogue of this statement is no longer true for enomorphisms which are not Q-rational: the presence of (non-rational) CM-elliptic curves in J X gives rise to enomorphisms which are not moular, as is easy to see. Another interesting consequence of Theorem 1 is the following result which generalizes a statement prove in Mazur[4], p. 139.

2 2 E. Kani Corollary 3. The centre of En 0 Q(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 1 J X ) enote 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 En Q (Ω 1 (J X )) is T Q, i.e. En E(Ω 1 (J X )) = T Q. Thus, by the ictionary of [3], we obtain the following generalization of the Shimura construction: Corollary 4. We have En E (Ω 1 (J X )) = T Q. Thus, the map (A, p) p Ω 1 (A) Ω 1 (J X ) S 2 (Γ, Q) inuces a bijection between the set of (equivalence classes) of abelian quotients p : J X A of J X /Q an the set of T Q -submoules of S 2(Γ, Q). The first step in the proof of Theorem 1 is the following classification of the irreucible E C -moules, 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, an that S has (by Atkin-Lehner theory) a T C -moule ecomposition S = f N (Γ) S f, where N (Γ) enotes the set of all normalize newforms of weight 2 (of all levels) on Γ an S f enotes the χ f -eigenspace with respect to the character χ f : T C = T Q C C efine by f N (Γ). We then have: Theorem 5. Each S f is an irreucible E C -moule, an every irreucible E C -moule is isomorphic to some S f. Thus E C is (isomorphic to) the centralizer C S (T Q ) of T Q in En C (S), an Z(E C ) = T Q C. Since this theorem is of inepenent interest an 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 reaily once the structure of E is known. For this, let n f = im S f an let K f = Q({a n }) be the fiel generate by the Fourier coefficients of f = a n q n. Then we have the following theorem which is a consequence of the funamental results of Ribet[7]. Theorem 6 (Ribet). En 0 Q(J X ) f N (Γ)/G Q M nf (K f ). Actually, Ribet s results give a priori only a slightly weaker result; cf. equation (14) below. As a result, the orer of the proofs of the above theorems has to be partially reverse. Inee, after proving Theorem 5 an (14), we first euce Corollary 3 an 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 ( a 0 b ) (mo N), a H},

3 Enomorphisms of Jacobians of Moular Curves 3 for some subgroup H (Z/NZ) Γ 0 (N)/Γ 1 (N). Then for any ivisor M N we have a corresponing subgroup Γ H (M) := Γ H (M), where H (Z/MZ) is the image of H uner the reuction map r N,M : (Z/NZ) (Z/MZ). For any positive integer ), let α = ( ) an β = ( ) = α 1. Since β ( x y z w ) β 1 = ( x y z/ w (1) β Γ H (N)β 1, we see that if N, then = Γ H ( N, ) := {( x y z w ) SL 2 (Z) : r N (x) H, y, N z}, where r N : Z Z/NZ enotes the reuction (mo N) map. In orer to stuy the egeneracy operators mentione in the introuction, we shall first efine them as abstract operators via ouble cosets, an then (as in Shimura[8]) consier their actions on the spaces of cusp forms S k (Γ). Thus, fix ivisors, M of N with M N an consier the ouble cosets β M, = Γ H (M)β Γ H (N) an t β M, = Γ H (N)α Γ H (M). Since α 1 Γ H(N)α = Γ H ( N, ) Γ H(M), we see that the egree of these ouble cosets (as efine in [8], p. 51) is given by (2) eg(β M, ) = 1 an eg( t β M, ) = [Γ H (M) : Γ H ( N, )]. The (abstract) egeneracy operators are efine as the proucts D M, = t β M,1 β M, an t D M, = t β M, β M,1. From the efinition of the prouct of ouble cosets (cf. [8], p. 52) together with the fact that β Γ H (N)β 1 = Γ H ( N, ) Γ H(M) one sees easily that (3) D M, = γ R Γ H (N)γβ Γ H (N) an t D M, = γ R Γ H (N)α γ 1 Γ H (N), where R is a system of representatives of the set of ouble cosets (ouble coset space) Γ H (N)\Γ H (M)/Γ H ( N, ). Note that the above formula (together with [8], p. 70) shows that the Hecke operator T = Γ H (N)α Γ H (N) is a component of t D M,. Remark 7. As the notation inicates, t D M, is the Rosati ajoint of D M, ; cf. [8], p. 72 an p Similarly t β M, is the ajoint of β M,. We next want to show that the egeneracy operators commute with the Hecke operators T n = Tn N for (n, N) = 1. (Note that the T n s generate the algebra R(Γ H (N), N ) Q in the notation of [8], p. 67.) To this en we show more generally Proposition 8. If M N, then for any integer n 1 with (n, N) = 1 we have (4) T M n β M, = β M, T N n an T N n tβ M, = t β M, T M n.

4 4 E. Kani The proof of this proposition is base on the following simple fact. Lemma 9. Let Γ 1, Γ 2 be commensurable subgroups of G = GL + 2 (Q), an 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 = eg(γ 2 βγ 2 )/ eg(γ 1 αβγ 2 ). If, in aition, 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 βγ 1 αβγ 2, with c β = eg(γ 1 βγ 1 )/ eg(γ 1 αβγ 2 ), an so (Γ 1 αγ 2 )(Γ 2 βγ 2 ) = c β (Γ 1βΓ 1 )(Γ 1 αγ 2 ), where c β = eg(γ 2βΓ 2 )/ eg(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, an so (5) hols for some integer c β. By taking egrees of both sies an observing that eg(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, an so we must have equality since Γ 1 βγ 1 αγ 2 is a union of (Γ 1, Γ 2 )-ouble cosets (an since ouble cosets are either isjoint or equal). Thus, Γ 1 βγ 1 αγ 2 = Γ 1 αβγ 2, an so we see that (6) hols for some c β. By the same argument as for (5) we see that c β has the asserte value. The last assertion follows immeiately from (5) an (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 an β = α p, where p N is a prime. Thus, Γ 1 αγ 2 = β M,, Γ 1 βγ 1 = Tp M an Γ 2 βγ 2 = Tp N ; cf. [8], p. 71. Since αγ 2 α 1 = Γ H ( N, ) Γ 1, we see that the first hypothesis of the lemma hols. Moreover, take S = {( )} ( ) 1 b 0 p {σ p 0 b<p pβ p }, where σ p p (mo N). ( Since β 1 b ) 0 p β 1 = ( ) 1 b 0 p = ( 1 k 0 1 ) ( ) 1 b ( ) 0 p, where b + kp = b, 0 b < p, an since β σ p β 1 p p (mo N ) (an since M N ), we see that Γ 1β Sβ 1 = Γ 1 S = Γ 1 βγ 1 = Tp M, where the secon last equality follows from [8], p. 72. Similarly, we have S Γ 2 βγ 2 = Tp N. Thus, S satisfies the hypotheses of the lemma, an so we conclue from the lemma (together with the fact that c β = 1 because eg(t p M ) = eg(tp M ) = p + 1) that the first equation of (4) hols for all primes p N. Since we have T (n, n)β M, = β M, 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) hols for all n with (n, N) = 1. From this it follows that we also have the formula (7) t T M n β M, = β M, tt N n, for all (n, N) = 1. Inee, since t T n = σ n T n = T n σ n (cf. [8], p. 72), an since σ n β M, = β M, σ n, we see that (7) follows from the first equation of (4).

5 Enomorphisms of Jacobians of Moular Curves 5 Finally, taking the Rosati ajoints of (7) yiels the secon equation of (4), which conclues the proof of Proposition 8. Notation. Let T = R(Γ H (N), N ) enote the Hecke ring generate by the (abstract) Hecke operators T n with (n, N) = 1, an let Ẽ = T n, D M,, t D M, : (n, N) = 1, M N M := R(Γ H (N), ) enote the Hecke ring generate by the T n s an the egeneracy operators D M, an t D M,. Corollary 10. T is a subring of the centre of Ẽ, i.e. T Z(Ẽ). Proof. By (4) we have T n D M, = t β M,1 T M n β M, = D M, T n, for any, M with M N an any n with (n, N) = 1, an similarly T n an t D m, commute. Thus, since T = T n is commutative, we see that T Z(Ẽ). 3 The Structure of the Extene Hecke Algebra E k,c We now fix an integer k an consier the space S = S k (Γ H (N)) of cusp forms of weight k on Γ H (N). By [8], p. 73, each ouble coset in M = R(Γ H (N), ) inuces a natural C-linear operator on S, an S is a right M C-moule. In particular, since Ẽ M, S is also a Ẽ C-moule; we let E k,c enote the image of Ẽ C in En C(S) op. Thus, by efinition, E k,c is the subalgebra of En C (S) op generate by the Hecke operators [T p ] k, for p N an the egeneracy operators [D M, ] k, [ t D M, ] k, for M N. Note that the action of [D M, ] k on S is given by the formula (8) f [D M, ] k (z) = k 1 tr M (f)(z), for f S, where tr M = tr ΓH (N)/Γ H (M) : S k (Γ H (N)) S k (Γ H (M)) is the usual trace map. Inee, since tr M = [ t β M,1 ] k by efinition, this follows immeiately from the fact that f [D M, ] k = f [ t β M,1 ] k [β M, ] k together with the formula (9) f [β M, ] k (z) = k/2 1 f k β (z) = k 1 f(z), for f S k (Γ H (M)). A similar formula hols for the operator [ t D M, ] k, which by [8], p. 76, is just the ajoint of [D M, ] k with respect to the Petersson prouct on S. Let T k,c be the C-subalgebra of E k,c generate by the Hecke operators [T p ] k, for p N. Since T k,c is a commutative semi-simple algebra, S has a ecomposition into T k,c -eigenspaces S χ. By Atkin-Lehner theory, each such character is of the form χ = χ f, for a unique normalize newform f of some level N f N, an so we have the Atkin-Lehner ecomposition (10) S = S f, f N (Γ)

6 6 E. Kani where S f = S χf an N (Γ) = N k (Γ) enotes the set of all normalize newforms of weight k of level N f N on Γ = Γ H (N); cf. [2], p. 28 an/or [5], 4.6). In particular, we see that im T k,c = #N (Γ). We now prove: Theorem 11. Each S f is an irreucible E k,c -moule, an every irreucible E k,c - moule 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 En C(S), an its centre is Z(E k,c ) = T k,c. Proof. First note that since T k,c is close uner the Petersson ajoints, the same is true for E k,c an 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, an so each S f is a E k,c -moule. Suppose S f were reucible. Then S f = V 1 V 2, where each V i is a non-zero E k,c - moule because E k,c is semi-simple. Consier the map tr = tr Nf : 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, an so there exists (wlog) g V 1 such that tr(g) = cf, for some c 0. Then, by (8) we see that for each N/N f, c k/2 1 f k β = g [D Nf,] k V 1, an so S f V 1 since {f β } N/Nf is a basis of S f (cf. [5], Cor ). Thus S f = V 1, which means that S f is irreucible. Since S is a faithful E k,c -moule (by construction), we see by Weerburn that every irreucible E k,c -moule is a submoule of S (an thus is isomorphic to some S f ) an that hence E k,c f N (Γ) M n f (C), where n f = im S f (= σ 0 (N/N f )). (Note that the S f s are pairwise non-isomorphic as E k,c -moules because they are alreay non-isomorphic as T k,c -moules.) Thus im Z(E k,c) = #N (Γ) = im T k,c, an so Z(E k,c ) = T k,c. Finally, since S has multiplicity one as a E k,c -moule, it follows easily that E k,c C S (T k,c ). To see this, note first that E k,c S := {ϕ En C (S) op : S f ϕ S f, f N (Γ)} because by construction E k,c is isomorphic to a subring of S an because im E k,c = f N (Γ) n2 f = im 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, an for each N/M an operator D M, with the property that S k (Γ H (M)) β Im([D M,1 ] k DM, ). Then E = E k,c. Proof. It is enough to show that each S f is an irreucible E -moule, for then by the above argument we have E E k,c an hence E = E k,c. Now if S f were reucible, then as in the proof of Theorem 11 we woul have S f = V 1 V 2 where (wlog) f V 1. Let N/N f. Then by hypothesis g S

7 Enomorphisms of Jacobians of Moular Curves 7 such that g [D Nf,1] D Nf, = f k β. By replacing g by g ε f, where ε f T k,c is the iempotent such that S f = S ε f, we may assume that g S f. Then g [D Nf,1] k = cf, for some c C (cf. proof of Theorem 11), an so f k β = (cf) D Nf, V 1. Thus as before V 1 = S f, an so S f is irreucible. Remark 13. (a) If E k,q enotes the Q-algebra generate by the operators [T p ] k (p N), [D M, ] k an [ t D M, ] k (for M N), then we have E k,c = E k,q C, provie that k 2. Inee, by (3) we see easily that E B R(Γ H (N), ), where B is as efine on p. 83 of [8], an so the assertion follows from [8], Th (b) By conjugation, the above results also hol for the groups Γ := Γ H (N, t) = β t Γ H (tn )βt 1, where H (Z/N tz) ; in particular, they hol for the groups Γ consiere by Shimura[8], p. 67ff. More precisely, the map Γ H (N t)αγ H (N t) Γβ t αβt 1 Γ efines a ring isomorphism ρ t : R(Γ H (N t), ) R(Γ, β t βt 1 ) which ientifies the Hecke algebra of Γ H (N t) with that of Γ. Furthermore, the map f f k β t efines 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 transporte back to Γ, an thus the analogous results hol for Γ (in place of Γ H (N t)). (c) In classical Atkin-Lehner theory (cf. Miyake[5], 4.6) one often consiers submoules of S = S k (Γ) with respect to the algebra A k := T, t T En C (S k (Γ)), where T is the full Hecke algebra (an t T is its (Rosati) ajoint); for example, one shows that S f is an A k -submoule. Now it follows from the above Theorem 11 that we have the inclusion A k E k,c (which is not obvious from the efinitions). Inee, since C S (E k,c ) = T C S (A k ), the ouble centralizer theorem shows that A k E k,c. 4 Application to En 0 Q(J X ) As in the introuction, let X = X H (N)/Q enote the canonical moel (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 fiel κ(x) is isomorphic to A 0 (Γ H (N), Q), the fiel of moular 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 an ±H = H, 1 efine the same curve X H (N), we may assume without loss of generality that 1 H, an we shall o so in the sequel. Now for any two ivisors, M of N with M N, the map f(z) f(z) = f 0 β clearly efines an injection of fiels β : A 0(Γ H (M), Q) A 0 (Γ H (N), Q) because β 1 Γ H(M)β β 1 Γ H(N/, )β = Γ H (N). We thus have a corresponing surjective morphism of curves B M, : X H (N) X H (M) such that its inuce pullback map on the function fiels is β. Note that the graph Γ B M, X H (N) X H (M) of this morphism is precisely the corresponence X(β M, ) efine by the ouble coset β M, = Γ H (M)β Γ H (N); cf. [8], p In particular, since the egree of the morphism

8 8 E. Kani B M, is the egree of the Rosati ajoint t β M,, we see by [8], p. 171 an (2) that eg(b M, ) = eg( t β M, ) = [Γ H (M) : Γ H (N/, )]. Let J X = J H (N)/Q be the Jacobian variety of X H (N)/Q. Then by functoriality an autouality of the Jacobian we have inuce homomorphisms B M, : J H (M) J H (N) an (B M, ) : J H (N) J H (M). As is explaine in [8], p , these maps are the same as the homomorphisms ξ( t β M, ) Hom(J H (M), J H (N)) an ξ(β M, ) Hom(J H (N), J H (M)) efine by the ouble cosets t β M, an β M,, respectively. Thus, by [8], Prop. 7.1, we have ξ(d M, ) = ξ( t β M,1 ) ξ(β M, ) = B M,1 (B M, ) En Q (J H (N)), an similarly, ξ( t D M, ) = B M, (B M,1) En Q (J H (N)). Moreover, we also have the Q-enomorphisms ξ(t n ) En Q (J H (N)) efine by the Hecke operators T n ; cf. [8], p As in the introuction, we let T Q En0 Q(J H (N)) = En Q (J H (N)) Q enote the Q-algebra generate by the ξ(t p ) s, for p N. Proposition 14. Let E En 0 Q(J H (N)) enote the Q-algebra generate by T Q an the egeneracy operators ξ(d M, ), ξ( t D M, ) with M N. Then T Q = Z(E) is the centre of E an (11) im Q (E) = n 2 f. f N 2 (Γ) Proof. As before, write X = X H (N). It is well-known that E := En 0 Q(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 ifferentials of J X /Q an X/Q (cf. e.g. [3]). Thus, the same is true for the subalgebra E an hence E C acts faithfully on Ω 1 (X Γ /C) = Ω 1 (X/Q) C. Now via the ientification 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 ouble 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 im Q E = im C E C = im C E 2,C = f N 2 (Γ) n2 f, 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 etermining the isotypic E-moule ecomposition of the E-moule Ω(J X /Q) S 2 (Γ, Q), where the latter enotes space of cusp form f S 2 (Γ) with rational Fourier coefficients. For this, let [f] = fg Q enote the Galois orbit of f N (Γ) an 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 irreucible E-moule, an every irreucible E-moule is of this form. Thus, the isotypic E-moule ecomposition of Ω 1 (J X /Q) is given by (12) Ω 1 (J X /Q) S 2 (Γ, Q) S [f]. f N (Γ)/G Q

9 Enomorphisms of Jacobians of Moular Curves 9 Proof. It is immeiate that each S [f] is a E-moule. 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-submoules which are G Q -stable. Thus, S [f] is an irreucible E-moule. 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 ecomposition of Ω 1 (J X /Q) into pairwise non-isomorphic irreucible E-moules. In particular, every irreucible E-moule is isomorphic to some S [f] because Ω 1 (J X /Q) is a faithful E-moule (an E is semisimple). We next stuy the structure of the algebra E = En 0 Q(J X ). To this en, recall that by the Shimura construction, each (weight 2) normalize newform f N (Γ) = N 2 (Γ) of some level N f N efines an abelian variety A f /Q of imension [K f : Q], where K f = Q({a n } n 1 ) enotes the fiel generate 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(Q/Q).) Then the abelian varieties A f etermine J X up to Q-isogeny, for we have the relation (13) J X A mf f, f N (Γ)/G Q for some positive integers m f ; cf. Ribet[7], Proposition (2.3). (We shall see below that m f = n f := im S f ; cf. Remark 18(a).) In orer to euce Theorem 6 from (13), we require the following facts which were (essentially) proven by Ribet[7]: Theorem 16. (a) En 0 Q(A f ) K f, for all f N (Γ). (b) If f, g N (Γ), then A f A g if an 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. im Z(En 0 Q(J X )) = #N (Γ) = im T Q. Proof. By Theorem 16(a) we see that each A f is Q-simple, an hence by part (b) we have Hom 0 Q(A f, A g ) = 0 if f g σ, for all σ G Q. We thus obtain from (13) that (14) En 0 Q(J X ) En 0 Q(A mf f ) M mf (K f ), f N (Γ)/G Q f N (Γ)/G Q an that hence (15) im Z(En 0 Q(J X )) = im f N (Γ)/G Q K f = f N (Γ)/G Q [K f : Q] = #N (Γ),

10 10 E. Kani where the last equation follows from the fact that #(fg Q ) = [K f : Q]. Since im T Q = im 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 En 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 im C (E C ) = im T 2,C, an 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 an E C = E C are semi-simple subalgebras of En C (S) with C S (E C ) = C S (E C ) (cf. proof of Corollary 3), it follows from the ouble centralizer theorem that E C = E C. Thus E = E, as asserte. Proof of Corollary 2. By construction (cf. 2) we have Ẽ M, an so the image M of M Q in En 0 (J Q X) contains the image E of Ẽ Q. Thus, since E = E by Theorem 1, we have E M, as claime. Proof of Corollary 4. Since S = S 2 (Γ, Q) C an 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 secon 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) an let E f = En 0 (A m f f ), which by (13) (an Theorem 16) is naturally a simple two-sie ieal of E. Since E = E by Theorem 1, we know by Corollary 15 that S [f] is an irreucible (right) E-moule. By the Shimura construction we know that S [f] E f 0, an so S [f] is a faithful irreucible E f -moule. Thus, since E f M mf (K f ), we see that im Q S [f] = m f [K f : Q]. On the other han, since S [f] C = g [f] S g, we have im Q S [f] = (im C S f )[K f : Q] = n f [K f : Q], an so m f = n f. Remark 18. (a) It follows from the above proof that (13) hols with m f = n f = im S f = σ 0 (N/N f ). This was asserte without proof in [1], equation (3.4). (b) All the above results also hol for the curves X H (N, t) associate to the groups Γ H (N, t) = β t Γ H (tn )βt 1 of Remark 13(b); in fact, the matrix β t inuces a Q-isomorphism (β t ) : X H (N t) X H (N, t). (c) In a letter to the author, Ken Ribet (July 2005) pointe out that it is possible to euce Corollary 3 an the first assertion of Corollary 4 irectly from Theorem 6 together with the fact that T Z(E). Example 19. (a) If J 0 (p 2 ) is the Jacobian of the moular curve X 0 (p 2 )/Q, where p is a prime, then (16) En 0 Q(J 0 (p 2 )) = T Q, τ, ξ(t p ), ξ( t T p ) Q = T Q, τ, τ Q, where τ = η η an τ = (η ) η are the enomorphisms associate to the egeneracy maps η := B p,1 : X 0 (p 2 ) X 0 (p) an η := B p,p : X 0 (p 2 ) X 0 (p).

11 Enomorphisms of Jacobians of Moular Curves 11 (b) If J(p) is the Jacobian of the moular curve X(p)/Q efine by the principal congruence subgroup Γ(p), then (17) En 0 Q(J(p)) = T Q, τ, ξ(t p ), ξ( t T p ) Q = T Q, τ, τ Q, where τ = η η an τ = ( η ) η are the enomorphisms associate to the covers η : X(p) X 1 (p) an η : X(p) X 1 (p) which are inuce by the inclusions Γ 1 (p) Γ(p) an Γ 1 (p) Γ(p), respectively. Proof. (a) Applying Theorem 1 with N = p 2 an H = (Z/p 2 Z) yiels En 0 Q(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) = i an that ξ(d 1, ) = 0 because X H (1) = X(1) has genus 0). Now ξ(d p,1 ) = τ by efinition because Γ H (p) = Γ 0 (p). Moreover, since t T p D p,p (cf. (3)) an since by (2) eg D p,p = [Γ 0 (p) : Γ 0 (p 2 )] = p = eg t T p, we see that D p,p = t T p an hence that also t D p,p = T p. This proves the first equality of (16). To prove the secon equality, we shall apply Corollary 12 to E = T 2,C, [τ] 2, [τ ] 2 C. Since t τ = τ an t τ = τ, it is clear that E is semi-simple. Consier D p,p := [ττ ] 2 E, an let T = ( ). 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 = p 1 a=0 f 2α p T a β p = f T p β p. Now since Im([τ]) = S 2 (Γ 0 (p)) an since T p acts bijectively on S 2 (Γ 0 (p)) (use [5], Th ), we thus see that Im([ττ ]) = S 2 (Γ 0 (p))β p. By Corollary 12 we therefore have that E = E 2,C, an so E is generate by T Q, τ, an τ. This proves the secon 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) hols for the Jacobian J H (p 2 ) of the curve X H (p 2 ), i.e. (18) En 0 Q(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 = ηh (η H) an τ H = (η H ) (η H ) are efine by the egeneracy maps η H := B p,1 : X H (p 2 ) X H (p) an η H := B p,p : X H (p 2 ) X H (p). Applying this to H = Ker(r p 2,p) an noting that Γ(p) = Γ H (p, p) = β p Γ H (p 2 )βp 1, we see that En 0 Q(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) an 13(b). Now since β p Γ H (p)βp 1 = Γ H (1, p) = Γ 1 (p), we see that ρ p (τ H ) = τ. Moreover, ρ p (τ H ) = τ because f ρ p (τ H ) = f β p[τ ] 2 βp 1 = f T a = f [ τ ], an so (17) hols. Acknowlegements. I woul like to thank Ken Ribet for his helpful comments on this paper; cf. Remark 18(c). In aition, I woul like to gratefully acknowlege receipt of funing from the Natural Sciences an Engineering Research Council of Canaa (NSERC).

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