Modular Curves and Modular Forms

Transcription

1 Modular Curves and Modular Forms Yonatan Harpaz November 12, 2008 Contents 1 Introduction 2 2 The Upper Hal Plain and its Quotients 2 3 Modular Curves Introduction Interpetation as Moduli Spaces The Hecke Operators Pulling Back and Forth Correspondences The Eichler Shimura Theorem The Action on Cusp Forms Eigenorms and neworms 14 6 Modular L-unctions 16 1

2 1 Introduction A starting motivation or studying the upper hal plain H = {z Z I(z) > 0} comes rom the amous classiication o elliptic curves over C. We can associate to a point τ H the elliptic curves corresponding to the lattice in C generated by τ and 1. Two points τ 1, τ 2 give the same elliptic curve i and only i there exist an element ρ Γ de = PSL 2 (Z) such that ρ(τ 1 ) = τ 2 where Γ acts on H via Mobius transormations. The topological space H/Γ is a (non-compact) topological maniold, and it also inherits a complex structure rom H. Ater compactiication it even admits an algebraic structure making it isomorphic to P 1. This corresponds to the act that elliptic curves over a general ield K are classiied by their j-invariant, which one can think o as a point in A 1 (K). Adding a point in ininity ( compactiication ) we can say that elliptic curves over K are classiied by P 1 (K). The j-invariant then gives us a map rom the compactiication o H/Γ to P 1 (C) which gives H/Γ an algebraic structure deined over any subield o C, i.e. deined over Q. This remarkable situation turns H/Γ (and some other quotient spaces o H) rom topological/analytical objects into arithemtic objects! This connection leads to a beautiul and surprising theory which we wish to present in these notes. 2 The Upper Hal Plain and its Quotients In the modern approach we deines a complex maniold as a pair (M, F) o a topological maniold M and a shea F o complex unctions on M (i.e. F(U) is an algebra o unctions U C with the usual restriction maps) which is locally isomorphic to the shea o holomorphic unctions on the unit disk D = {z z 1} C We then call F a complex structure on M. I X is a smooth algebraic variety over C then the set X(C) can be endowed with the topology inherited rom C and admits a natural complex structure by replacing local polynomial unctions by holomorphic unctions. We then say that the complex maniold X(C) has an algebraic structure. Note that in general, non-isomorphic algebraic varieties may give isomorphic complex maniolds, and some complex maniold may ail to have an algebraic structure. The situation is however much nicer or riemann suraces, i.e. one dimensional complex maniolds: Theorem 2.1. Let M be a compact riemann surace. Then there exists a unique projective algebraic curve X over C such that X(C) = M as complex maniolds. 2

3 Proo. (Sketch) First show that the ield C(M) o global meromorphic unctions on M is a initely generated extension o C with transcendence degree 1. C(M) then corresponds to a unique smooth projective algebraic curve X over C whose ield o rational unctions is isomorphic to C(M). By using the isomorphism on the level o unction ields one can obtain an isomorphism on the level o points M = X(C). The above theorem is not true or non-compact riemann suraces. For example, the upper hal plain H does not admit any algebraic structure. This is one o the reasons we shall usually try to compactiy the riemann suraces we stumble upon beore trying to ind algebraic structures or them. The spaces we shall be interested in are obtained rom H as quotients by an action o some inite index subgroup Γ Γ. By analyzing what happens in points with non-trivial stabilizers one can show that the resulting space is a topological maniold. Now let p : H Y (Γ ) be the quotient map. In order to deine a complex structure we declare a unction : U C (where U Y (Γ ) open) as holomorphic i p : p 1 (U) C is holomorphic on H. Note that this is deinition gives us a bijection between the set o holomorphic unctions on U and the set o Γ -invariant unctions on p 1 (U). Now, in order to ind an algebraic structure on Y (Γ ) we shall irst ind a compactiication Y (Γ ) X(Γ ) such that X(Γ ) is a compact riemann surace. How do we ind such an X(Γ )? First assume that Γ = Γ. Deine X(Γ) = Y (Γ) { } and declare U as a neighborhood o i the pullback p 1 (U\{ }) contains a set o the orm V C = {z H I(z) > C} or some 0 < C R. We then deine q(z) = e 2πiz as our local coordinate at, i.e. we declare a unction on a neighborhood U as holomorphic i the pullback p is a holomorphic unction o q, i.e. i it is o the orm g(q(z)) or some holomorphic unction g. Similarly we shall say that a meromorphic unction on a puctured neighborhood U o has a pole at i p can be expressed as g(z) = (q(z)) where has a pole at 0. In particular the global meromorphic unctions on X(Γ) are in one-to-one correspondence with meromorphic unctions on H which are invariant under Γ and have an expansion o the orm = a n q n n= N This expansion is called the Fourier expansion o. It can be shown that ater adding this single point, the space X(Γ) is already a compact riemann surace. 3

4 By analyzing this space topologically we can see that it is homeomorphic to the sphere, and thus it has to be P 1 (C), i.e. its ield o meromorphic unctions should be generated over C by a single transcedental element. This element can be taken to be the j-invariant unction. We shall return to this soon. For the case o a general inite index Γ Γ we irst ind the least h such that the map T (z) = z + h is in Γ, and add a point at ininity in the same way only using ( z q = e h) 2πiz h as a local coordinate. The rest o the points at ininity, called cusps, can be obtained by applying elements o Γ to (i.e. applying them to the deining neighborhood base and to the local coordiante) The global meromorphic unctions on X(Γ ) can now be identiied with what is called modular unctions, which we deine as: Deinition 2.2. A modular unction or a inite index subgroup Γ Γ is a meromorphic unction on H which is invariant under Γ and is meromorphic at the cusps. There are many ways to identiy X(Γ) with P 1 (C), as P 1 (C) has a large group o automorphisms (think o this is dierent choices o transcedental generators or the ield o meromorphic ucntions). However, we like to think o X(Γ) as a space which parametrizes elliptic curves over C. Thus we have a preerred choice or such an identiication - we can choose the unction j which associates to each τ H the j-invariant o the corresponding elliptic curve. j is clearly invariant under Γ and hence deines a holomorphic unction on Y (Γ). This unction can be shown to to have a simple pole at the cusp. Its Fourier expansion begins as j = q q +... Thus j deines a meromorphic unction on X(Γ) (note that the coeicients o j are integer numbers and not general complex numbers. It is not at all obvious at irst glance why this should be. We shall return to this phenomenon later). The choice o j as our identiier between X(Γ) and P 1 (C) allows us to copy any additional structure P 1 might have into X(Γ). In particular, we obtain an algebraic structure on X(Γ) which is deined over Q, and not only over C. We saw above that we can represent meromorphic unctions on X(Γ ) by Γ -invariant meromorphic unctions on H. This is a useul mechinary and we wish to generalize it a bit: Deinition 2.3. A modular orm o weight 2k or a inite index subgroup Γ Γ is a holomorphic unction on H which satisies ( ) az + b = (cz + d) 2k (z) cz + d and is holomorphic and the cusps. I is 0 at the cusps then it is called a cusp orm. 4

5 We denote by M 2k (Γ) the vector space o modular orms o weight 2k or Γ and by S 2k (Γ ) the space o cusp orms o weight 2k or Γ. Exmaples: 1. A modular unction or Γ is a modular orm or Γ o weight For k 2 we have the Eisenstein series: G k (τ) = a,b Z,(a,b) (0,0) 1 (aτ + b) 2k which are modular orms (but not cusp orms) or Γ o weight 2k. These orms appear naturally in the theory o elliptic curves as they give coeicients or a weierstrass equation or E(j(τ)): y 2 = x 3 15G 2 (τ) 35G 3 (τ) In this lecture we will be particularly interested in cusp orms o weight 1, because o the ollowing theorem: Theorem 2.4. There is a natural isomorphism between the vector space o cusp orms o weight 2 or Γ and the vector space o holomorphic 1-orms on X(Γ ). Proo. We shall describe a map Ω 1 (X(Γ )) M 2 (Γ ) and show that it is injective. It will be let as an exercise to show that its image is exactly S 2 (Γ ). Let ω Ω 1 (X(Γ )) be a 1-orm. Consider the pull back π ω o ω to H. Since H is a domain in C we can write π ω using a the global coordiante τ: π ω = (τ)dτ where is a holomorphic unction. Our map will be then ω. It is clear that i ω 0 then 0, so this is an injective map o vector spaces. We need to show that is a modular orm. π ω is invariant under Γ which are the deck transormations o π, so or ρ(τ) = aτ+c cτ+d Γ we get (τ)dτ = ρ ((τ)dτ) = Thus satisies ( aτ + c cτ + d ) aτ+c cτ+d dτ = τ ( ) aτ + c = (cτ + d) 2 (τ) cτ + d ( ) aτ + c cτ + d dτ (cτ + d) 2 It can be shown that extends holomorphically to the cusps and actually has 0 in the cusps. This gives us the desired isomorphism Ω 1 (X(Γ )) = S 2 (Γ ) 5

6 3 Modular Curves 3.1 Introduction The main riemann suraces we shall be interested in are obtained as quotients o H by one o the subgroups Γ(N) Γ deined by { } az + b Γ 0 (N) = cz + d a, b, c, d Z, c = 0( mod N) The resulting riemann surace is denoted by Y 0 (N) and its compactiiaction by X 0 (N). The algebraic curve which corresponds to X 0 (N) is called the modular curve o conductor N. The reason or the name conductor comes rom elliptic curves, as we shall see later. In particular X 0 (1) = X(Γ) = P 1 is the basic example we described beore. We shall now want to understand the ield C(X 0 (N)) o meromorphic unctions on X 0 (N). First o all note that the quotient map H X 0 (1) actors through the quotient H X 0 (N). This gives us a natural quotient map π : X 0 (N) X 0 (1). By pulling back meromorphic unctions on X 0 (1) via π we get a subield π C(X 0 (1)) o C(X 0 (N)). By abuse o notation we shall also reer to j as a unction on X 0 (N) via this pullback. This is not the whole ield o meromorphic unctions. The whole ield is in act a inite extension o π C(X 0 (1)). In order to describe this we need the ollowing observation. For a natural number m consider the (obviously holomorphic) unction ϕ m : H H deined by Then we have the ollowing ϕ m (τ) = mτ Lemma 3.1. For each k, ϕ m decends to a well-deined (holomorphic) map Proo. Let ρ Γ(mk) act on H by ϕ m : X 0 (mk) X 0 (k) ρ(τ) = aτ + b cτ + d such that c = 0( mod mk). Then there exists an integer c such that c = mkc. Now: maτ + mb a(mτ) + mb ϕ m (ρ(τ)) = = cτ + d kc (mτ) + d = ρ (ϕ m (τ)) where ρ az + mb (z) = kc z + d which is given by an element in Γ(N) since ad mbkc = ad bc = 1. Thus we have a well deined map Y 0 (mk) Y 0 (k). Note that since the holomorphic structure on Y 0 (N) is inherited rom H it is clear that this map will be holomorphic. It is let to show that it extends holomorphically to the cusps but we shall omit this part. 6

7 By choosing m = N, n = 1 we obtain a map ϕ N : X 0 (N) X 0 (1). This map is dierent rom the map π we had beore. Thus by pulling back j through ϕ N we get a new unction which turns out to generate C(X 0 (N)) over π C(X 0 (1)). This unction is naturally denoted by j(nτ). To conclude, the ield o meromorphic unctions on X 0 (N) is generated over C by the unctions j(τ) and j(nτ), which satisy a polynomial relation F N (j(τ), j(nτ)) = 0 It can be shown that this polynomial can be chosen to have coeicients in Q, which gives us an algebraic structure deined over Q. At this points, though, nothing surprises us anymore. 3.2 Interpetation as Moduli Spaces We know that or N = 1, X 0 (N) = X(Γ) can be interpreted as the compactiication o the moduli space o elliptic curves over C. Can we geenralize this interpetation to X 0 (N) or a general N? the answer is yes. The key act here is to note that i we know τ up to an element o Γ 0 (N) then we know Nτ up to an element o Γ. Thus the elliptic curve E(j(Nτ)) is well deined an since the lattice L(1, Nτ) spanned by 1 and Nτ is a sublattice o index N in the lattice L(1, τ) we get an isogeny o degree N: E(j(Nτ)) E(j(τ)) The space X 0 (N) is in act a compactiication o the moduli space o degree N isogenies o elliptic curves. Let us now interpret the natural maps π : X(Nm) X(N) and ϕ m : X(Nm) X(N) in the moduli setting. Recall that the point in X 0 (Nm) which corresponds to τ H represents the natural degree Nm isogeny E(j(Nmτ)) E(j(τ)) such an isogeny can be actored uniquely as = N m with deg( N ) = N and deg( m ) = m. In act these are the natural isogenies E(j(Nmτ)) m E(j(Nτ)) N E(j(τ)) The point in X 0 (N) corresponding to τ represents the isogeny N, so that π is the right extract map sending to N. But can also be uniquely actorized as = m N with deg( N ) = N, deg( m) = m. These are the natural isogenies E(j(Nmτ)) N E(j(mτ)) m E(j(τ)) The point in X 0 (N) corresponding to mτ represents the isogeny N, so that ϕ m is the let extract map sending to N. 7

8 4 The Hecke Operators 4.1 Pulling Back and Forth Recall that a surjective map : X Y between two smooth projective curves over a ield K induces maps on their Jacobians in both directions. On the level o divisors we deine it on points and extend linearly (P i ) = (P i ) (Q i ) = P (P )=Q To show that both these maps preserve principle divisors, recall that induces an injection i : K(Y ) K(X) and on the other direction we have a norm map: N : K (X) K(Y ). It is easy to show the commutivity relations div N = div and div i = div (where div is the unction associating to a unction its divisor). Thus we have well deined maps on the Jacobians. It is also clear that these maps are morphisms o algebraic varieties. There is an alternative way to describe these maps, which goes through the space o 1-orms. Recall that the Jacobian o a smooth projective curve X over C has a dual representation as the complex torus Ω 1 (X)/H 1 (X, Z) where Ω 1 (X) is the g-dimensional C-vector space o global 1-orms and H 1 (X, Z) is the singular integral cohomology group which is isomorphic to Z 2g and is naturally embedded as a lattice in Ω 1 (X). Now consider a surjective map : X Y o smooth projective curves. Then we have a natural pull-back map : Ω 1 (Y ) Ω 1 (X) which always exists. But in this case we can also push orward 1-orms : Ω(X) 1 Ω 1 (Y ). This push-orward can be deined as ollows. First suppose that is a galois covering (which might be ramiied - we don t care). Then the pull-back identies the space Ω 1 (Y ) with the space o G-invariant 1-orms on X. Thus we can deine by deining as (ω) = g G g ω I is not a galois covering then we can always ind a curve Z and maps Z p h X Y Such that p is a galois covering with galois group G and h is a galois covering with galois group H G such that [G : H] = deg(). To make this construction less surprising recall that on the category o smooth projective curves, the 8

9 unction ield unctor induces an equivalence o categories (with an appropriate subcategory o ields over K). The curve Z corresponds to taking the galois closure o K(Y ) over K(Y ). The group G is the galois group o K(Z) over K(Y ) and H the galois group o K(Z) over K(X). Now i we have a 1-orm ω Ω 1 (X) and we want to push it over to Y then we start by pulling it back to Z and symmetrizing by taking g h ω g S where S G is a complete set o H-coset representatives in G. Then we obtain a G-invariant orm, so it comes rom a unique orm ω Ω 1 (Y ). This is the push-orward o ω. Note the similarity with deining the norm map rom K(X) to K(Y ). It is an exercise or the reader to show that these two deinitions induce the same maps on the Jacobians. 4.2 Correspondences Now let X, Y be two complete smooth algebraic curves. A correspondence rom X to Y is a third curve Z and two surjective maps X Z g As we saw above this diagram induces a map o Jacobians Y g : J(X) J(Y ) Note that the Jacobians o curves are sel-dual abelian varieties in a natural way. This gives a duality map rom Hom(A, B) to Hom(B, A) (where A, B are Jacobians o curves). This duality already exists on the level o correspondences, as we can deine a dual correspondense by swiching the role o and g. A morphsim which is the dual o itsel is called sel-adjoint. Now let us return to our modular curves X 0 (N). For a natural number m, We know two natural (surjective) maps X 0 (Nm) X 0 (N). One is the natural projection π and the second is the map ϕ m deined above which is induced by multiplication by m in H. This gives a correspondence: X 0 (Nm) gϕ m π X 0 (N) X 0 (N) rom X 0 (N) to itsel. This correspondence is called the Hecke Correspondence o m. Let us denote by J 0 (N) the Picard variety (jacobian) o X 0 (N). 9

10 Then this correspondence induces an endomorphism T m : J 0 (N) J 0 (N) called the Hecke Operator o m. In order to understand the Hecke operators in the moduli setting recall that the maps π and ϕ m can be interpreted as right extract and let extract respectively. Use this description in order to solve the next exercise: Exercise 1. Prove that the Hecke operators commute with each other and satisy the ollowing relations 1. I gcd(m, k) = 1 then T m T k = T mk. 2. i p N then T p k T p = T p k+1 + pt p k 1 3. i p N then T p k = (T p ) k. 4.3 The Eichler Shimura Theorem We are now ready to prove the important Eichler-Shimura theorem: Theorem 4.1. For p not dividing N, let J 0 (N) be the mod-p reduction o J 0 (N) and T p the reduction o the Hecke operator. Let Φ p : J 0 (N) J 0 (N) be the Frobenious endomorphism and Φ p its dual. Then T p = Φ p + Φ p Proo. Let E, F be two elliptic curves deined over Q with smooth mod-p reductions Ẽ, F which are not supersingular. Let : E F be an isogeny o degree N. Then the isomorphism class [ : E F ] deines a point in X 0 (N) and the isomorphism class [Ẽ e F ] deines its reduction in X 0 (N). Let S 0, S 1,..., S p E be the subgroups o size p in E and T i = (S i ) their images in F. Then we have commutative diagrams o isogenies E F g i E i i F i where E i = E/S i and F i = F/T i. Then by deinition p T p ([E i F ]) = [E i Fi ] Now consider the multiplication by p map which has degree p 2. It induces a commutative diagram [p] Ẽ i=0 F Ẽ F h i [p] 10

11 For each i = 0,..., p, we can actor this diagram as Ẽ F eg i Ẽ i i Fi e hi G i Ẽ F where the G i s and H i s have degree p. Since Ẽ is not supersingular, the reduction map E Ẽ has a kernel o size p. Assume without loss o generality that it is S 0. Then the maps g 0, h 0 and G i, H i or i 0 have no kernel and hence are purely inseperable. But there is only one purly inseperable isogeny o degree p (up to maybe isomorphism, which we don t care about) - its the robenious map! To avoid conusion, let us denote this Frobenious by Ψ p, to distinguish it rom Φ p which is the Frobenious map on J 0 (N). e Now, up to maybe changing the isogenies Ẽi i Fi by an isomorphism, we get a commutative diagram H i Ẽ e F Ψ p Ẽ 0 e 0 F0 Ψ p and also commutative diagrams Ẽ i e i Fi Ψ p Ẽ e F or each i 0. This means that on X 0 (N) we have the relation Φ p ([Ẽ and or each i 0 the relation Φ p ([Ẽi Ψ p e F e 0 ]) = [Ẽ0 F0 ] e i Fi ]) = [Ẽ e F ] 11

12 e i Since the Frobenious map has no kernel, we see that the points [Ẽi Fi ] are all equal on X 0 (N). Further more since the dual o Frobenious Φ p satisies Φ p Φ p = [p] (where [p] is multiplication by [p] on J 0 (N)) we get that p [Ẽi i=1 e i Fi ] = [p][ẽ1 To conclude, we have ound that p T p ([Ẽ e F ]) = [Ẽi i=0 e 1 F1 ] = Φ p(φ p ([Ẽ1 e i Fi ] = Φ p ([Ẽ e 1 F1 ])) = Φ p([ẽ e F ]) + Φ p([ẽ e F ]) e F ]) In order to complete the proo we need to argue that in some sense, almost all rational points on X 0 (N) can be represented by an isogeny o curves having nonsupersingular smooth reduction at p, and thus T p has to be equal to Φ p +Φ p. 4.4 The Action on Cusp Forms As we saw in the previous section, a correspondence can also act on the space o 1-orms, and this action is compatible with the action on the Jacobians. We also know that the space o holomorphic 1-orms on X 0 (N) can be identiied with the space o cusp orms o weight 2 or Γ 0 (N). We shall now calculate the action o the correspondence T m on a cusp orm under the assumption that m is coprime to N. Recall the deinition in the previous section o pushing orward a 1-orm using a galois-closure curve Z: What is Z in our case? Deine Z X 0 (Np) ψ ϕ p π X 0 (N) X 0 (N) Γ Γ 0 (Nm) Γ 0 (N) to be the largest subgroup which is normal in Γ 0 (N) and Z = X(Γ ). Then ψ is a galois covering with galois group Γ 0 (N)/Γ. Let S Γ 0 (N) be a complete set o representatives o let Γ 0 (Nm)-cosets. Now take a 1-orm on X 0 (N) and represent it by a 1-orm ω(z) = (z)dz on H which is invariant under Γ 0 (N) (i.e., such that is a cusp orm o weight 2). Pulling it back via ϕ m we get the orm ϕ mω(z) = (ϕ m (z)) ϕ m dz = m(mz)dz z 12

13 We now symmetrize it by the let-coset representatives σ S: (ϕ m σ) ω Lemma 4.2. (ϕ m σ) ω = σ S σ S σ ϕ mω = σ S {a,b,d ad=m,0 b d} m d 2 ( az + b d ) dz Proo. Cosider the set O = ϕ m Γ 0 (N). The element α = ϕ m σ O is in the let coset Γ 0 (N)ϕ m i and only i σ ϕ 1 m Γ 0 (N)ϕ m. But ϕ 1 m Γ 0 (N)ϕ m = { az + m 1 b mcz + d Since m is coprime to N we see that } a, b, c, d Z, c = 0( mod N) Γ 0 (N) ϕ 1 m Γ 0 (N)ϕ m = Γ(Nm) Thus the let Γ 0 (N)-cosets o O are in one-to-one correspondence with the let Γ 0 (Nm)-cosets o Γ 0 (N). This means that the set {ϕ m σ σ S} is a complete set o representatives o let Γ 0 (N)-coset in O. But it is an easy exercise to show that the ollowing set o mobious transormations: { } az + b d ad = m, 0 b d is a complete set o representatives o let Γ 0 (N)-cosets in O. Since ω is Γ 0 (N) invariant we get that ( ) (ϕ m σ) m az + b ω = d 2 dz d σ S a,b,d ad=m,0 b<d Now i is a cusp orm o weight 2 given by Fourier expansion (z) = c n q n = c n e 2πinz n=1 n=1 Then we get the explicite expression or the action o T m on : ( ) m az + b T m ()(z) = d 2 = d ad=m,0 b<d 13

14 ad=m,0 b d ad=m,a 1 a n=1 m d 2 n=1 c n e 2πinaz d c n e 2πin(az+b) d = 1 d 1 d b=0 e 2πinb d = But the last expression is 0 i n is not divisible by d and 1 i it is. Thus by deining n = n d we get the inal answer T m () = 1 a m n =1 ac dn q an In particular or p prime not dividing N we get the ormula T p () = pc n q pn + n =1 Exercise 2. Show that or p N we have T p () = c pn q n n =1 c pn q n n =1 5 Eigenorms and neworms Our basic aim is to study the modular curves. A basic strategy or investigating a curve X is by studying the maps rom X to simpler curves. The simplest curve is P 1, and studying maps X P 1 is just like studying the ield o rational unctions on X, which we touched upon above. The next class o curves is the genus 1 curves, or elliptic curves. The way to study maps X E with E an elliptic curve is to study the Jacobian J(X). The reason is that we have a unique actorization theorem or abelian varieties: each abelian varity A is isogenous to a unique product A A ei i where each A i is a simple abelian variety, i.e. contains no subabelian varities. Now a noncontant map : X E induces a nonzero map : E J(X) and a nonzero projection : J(X) E with kernel Ê. This gives us an isogeny J(X) E Ê which means that E is one o the simple actors o J(X). In particular there is only a inite set o elliptic curves which admit a nonconstant map X E and in order to understand them we need to anaylize the Jacobian J(X). 14

15 How do we ind simple actors o J(X) which are 1-dimensional? Suppose we were working with curves over F p. Then we would have the Frobenious endomorphism Φ p : J(X) J(X). Since Φ preserves the simple actors we can look or one dimensional simple actors like looking or eigenvectors. More precisely, i one considers the Tate module M l (J(X)) = lim J(X)/lJ(X) then each simple 1-dimensional actor E o J(X) would corespond to an eigenvector in M l (J(X)) o the operator Φ p + Φ p with eigenvalue a p. Lucky or us, we don t have to reduce mod p in order to do this, because we have an endomorphism whose mod p reduction is exactly Φ p + Φ p - the Hecke operators T p! Since we are working over Q it wil be more convenient and just as eective to replace the Tate module o J 0 (N) with the vector space o holomorphic 1- orms which we identiied with the space S 2 (N) o cusp orms o weight 2. Each simple actor o J 0 (N) will correspond to a cusp orm which is an eigenvector or all the T m s with m coprime to N. With a little but o eort one can also show that i E is a simple actor o J 0 (N) and not a simple actor o any J 0 (m) or m N then the corresponding cusp orm will be a eigenvector or all the T m s. Let us make the last statement more precise. I m N then we have a natural projection map π : X 0 (N) X 0 (m) which induces a map S 2 (m) S 2 (N). We deine the subspace S 2 old (N) S2 (N) to be the space which is generated by the images o all these maps. We deine S 2 new(n) to be its orthogonal complement. The cusp orms in S 2 new(n) are called neworms. The set o operators {T m } or m coprimes to N is a commuting amily o sel adjoint operators which preserve S 2 new(n). Thus we can ind a basis or S 2 new(n) consisting o simultanous eigenvectors o all the T m s or m coprime to N. This is called a spectral decomposition. Now we have a theorem Theorem 5.1. Each sequence o eigenvalues which appears in the spectral decomposition appears with multiplicity 1, i.e. appears in a 1-dimensional subspace. Since all the T m s commute (even those which are not coprime to N) we see that these eigenvectors must actualy be simultanous eigenvectors or all the {T m } s. These correspond exactly to simple actors o J 0 (N) which don t appear in any J 0 (m) or m N. Now suppose that = n=1 c nq n Snew(N) 2 is cusp orm o weight 2 which is an eigenvector or all the T n s. Then by the ormula above we see that λ m c n = 0 a n,m a c nm a 2 and in particular λ m c 1 = c m. This means that c 1 must be non-zero, otherwize all the c m s would be zero, i.e. would be zero. Since c 1 is non-zero we can 15

16 normalize so that c 1 = 1. Such an eigenorm is called normalized. Now or a normalized eigenorm we get λ m = c m i.e. the eigenvalues o are the coeicients themselves! Now recall that (z)dz is a Γ 0 (N)-invariant 1-orm on H. mobius map ρ(τ) = 1 Nτ Consider the We can calculate and check that ρ 1 Γ 0 (N)ρ = Γ 0 (N). This implies that ρ acts on S 2 (N) and it preserves S 2 new(n). It can be shown that its action commutes with action o all the Hecke operators T m and so rom the theorem above we see that each eigenorm is a also an eigenvector o ρ. Since ρ 2 = 1 this eigenvalue is ±1. This is called the parity o the eigenorm. 6 Modular L-unctions Let = n=1 c nq n S new (N) be an eigenorm. Then we know that the eigenvalue o T n is c n. Since the T n s satisy the relations given in exercise 1 we get the same relations on the c n s, namely: 1. I gcd(m, k) = 1 then c m c k = c mk. 2. I p N then c p kc p = c p k+1 + pc p k 1 3. I p N then c p k = c k p. Deine the L-unction associated with to be the complex unction L(, s) = c n n s n=1 Then rom the properties above we can transorm this ininite sum into an Euler product: L(, s) = a p p s + p 1 2s 1 a p p s p N p N From the Eichler Shimura theorem we know that a p should also be an eigenvalue o Φ p +Φ p. Since Φ p commutes with Φ p (i.e. Φ p is a normal operator) and since Φ p Φ p = p on any abelian variety we get the Hasse-Weil bound a p 2 p. This means that the Euler product converges or Re(s) > 3 2. Theorem 6.1. Let be an eigenorm which is an ɛ-eigenvector o ρ or ɛ = ±1. Then the L-unction L(E, s) admits a holomorphic continuation to the whole plain and the unction Λ(, s) = N 1 s Γ(s)L(, s) 16

17 satisies the unctional equation Λ(, s) = ɛλ(, 2 s) Proo. The key point to showing this is that there is an analytic connection between L(, s) and via the mellin transorm, which is a particular case o a Fourier transorm or the multiplicative group R >0 with the Haar measure dt Speciically one has 0 (it)t s dt t = Γ(t)L(, s) t. 17