# Lecture 17: Eichler--Shimura

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In the first part of this lecture, we prove the Eichler--Shimura theorem. In the second part, we study Shimura's construction, and use it to construct the Galois representation associated to a weight 2 cusp form.

## The Eichler--Shimura theorem

### Statement

In the previous lecture, we defined the Hecke correspondence $T_p \colon X_0(N) \dashrightarrow X_0(N)$ over the complex numbers (for $p \nmid N$). The definition (which we review below) makes sense over the rational numbers. Thus $T_p$ induces an endomorphism of the Jacobian $J_0(N)$ of $X_0(N)$, over $\bQ$. As $J_0(N)$ has good reduction at $p$, it extends uniquely to an abelian scheme over $\bZ_p$ (which we still denote by $J_0(N)$), and $T_p$ extends uniquely as well. The Eichler--Shimura theorem concerns the reduction of this endomorphism at $p$:

Theorem. We have $T_p=F+V$ on $J_0(N)_{\bF_p}$, where $F$ is Frobenius and $V$ is Verscheibung.

### A lemma on correspondences

Lemma. Let $\cO$ be a complete DVR with field of fractions $K$ and residue field $k$. Let $X/\cO$ be a proper smooth curve, and let $f,g \colon Y \rightrightarrows X$ be finite flat maps giving a correspondence from $X$ to itself. Let $J_K=\Jac(X_K)$ and let $h_K \colon J_K \to J_K$ be the map induces by $(f,g)$. Let $J/\cO$ be the Néron model of $J$, let $h \colon J \to J$ be the map induced by $h_K$, and let $h_k \colon J_k \to J_k$ be its base change. Let $D_0$ be a degree 0 divisor on $X_k$ defining a point $x_0 \in J(k)$. Then $h_k(x_0)$ is represented by $g_*(f^*(D_0))$.

Proof. Lift $D_0$ to a relative divisor $D$ on $X$, which is possible since $X$ is smooth over $\cO$. Let $D'=g_*(f^*(D))$, which is again a relative divisor on $X$. Then $D$ and $D'$ define $\cO$-points $x$ and $y$ of $J$. Then $h(x)$ is the unique $\cO$-point of $J$ extending the $K$-point $h_K(x_K)$. However, $h_K(x_K)$ is by definition the point represented by $g_*(f^*(D_K))$, and is therefore equal to $y_K$. Since $y$ extends $y_K$, we have $h(x)=y$. It follows that $h_k(x_0)=y_0$, the reduction of $y$. But $y_0$ is computed by $g_*(f^*(D_0))$, since these operations commute with base change.

### Hecke correspondences, integrally

Recall that $\ol{\fM}_0(N)$ is the proper Deligne--Mumford stack over $\bZ$ parametrizing generalized elliptic curves with $\Gamma_0(N)$-structure and $\ol{M}_0(N)$ is its coarse space. We also use the notation $X_0(N)$ when $N$ is inverted. Let $\wt{f} \colon \ol{\fM}_0(Np) \to \ol{\fM}_0(N)$ be the map which forgets the level $p$ structure, and let $\wt{g} \colon \ol{\fM}_0(Np) \to \ol{\fM}_0(N)$ be the composition of $f$ with the Atkin--Lehner involution at $p$. Precisely, $\wt{g}(E,H)=E/H$, where $E$ is an elliptic curve with $\Gamma_0(N)$-structure and $H$ is a $\Gamma_0(p)$-structure. Then $\wt{f}$ and $\wt{g}$ induce maps $f,g \colon \ol{M}_0(Np) \to \ol{M}_0(N)$ on the coarse spaces, which over $\bC$ is the usual Hecke correspondence $T_p$. Moreover, $f$ and $g$ are finite and flat. It follows (from the previous lemma) that we can compute $T_p$, as an endomorphism of $J_0(N)_{\bF_p}$, by the formula $g_*f^*$ on divisors.

### Proof of the theorem

We work over $\bF_p$. Let $\wt{i} \colon \ol{\fM}_0(N) \to \ol{\fM}_0(Np)$ be the map defined by $\wt{i}(E)=(E,\ker{F})$, where $F \colon E \to E^{(p)}$ is Frobenius. Let $\wt{j} \colon \ol{\fM}_0(N) \to \ol{\fM}_0(Np)$ be the composition of $\wt{i}$ and the Atkin--Lehner involution at $p$. Precisely, $\wt{j}(E)=(E^{(p)}, \ker{V})$, where $V$ is Verscheibung. Let $i$ and $j$ be the maps induced on coarse spaces. As we explained in the previous lecture, we have

$$fi = \id, \quad gj=\id, \quad fj=F, \quad gi=F,$$

where $F \colon \ol{M}_0(N) \to \ol{M}_0(N)$ is Frobenius.

Let $M_0(N)^{\ord}$ and $M_0(Np)^{\ord}$ be the ordinary locus, i.e., the coarse moduli space parametrizing ordinary elliptic curves with appropriate level structure. The above picture, combined with the analysis of $\Gamma_0(p)$-structures on ordinary curves, shows that $M_0(Np)^{\ord}$ is isomorphic, via $i$ and $j$, to $M_0(N)^{\ord} \amalg M_0(N)^{\ord}$. Furthermore, on the first copy $f$ restricts to the identity and $g$ to Frobenius, while on the second copy, $f$ restricts to Frobenius and $g$ to the identity.

The above discussion shows that, on $M_0(N)^{\ord}$, the correspondence $T_p$ is the sum (disjoint union) of the self-correspondences $(1,F)$ and $(F,1)$ of $M_0(N)^{\ord}$. The effect of $(1,F)$ on divisors is simply Frobenius, while the effect of $(F,1)$ is the dual of $(1,F)$, i.e., Verscheibung. We thus see that if $D$ is a divisor supported on the ordinary locus, then $T_p(D)=F(D)+V(D)$, as divisors. Since every divisor is linearly equivalent to one supported on the ordinary locus (this is true whenever a finite number of points are removed from a curve), it follows that $T_p=F+V$ holds on all of $J_0(N)$.

### The Tate module of J_0(N)

Let $V_{\ell}$ be the rational Tate module of $J_0(N)_{\bF_p}$. This carries an action of Frobenius $F$. Since $\bT$ acts by endomorphisms of $J_0(N)$, it acts on $V_{\ell}$, and so we can regard $V_{\ell}$ as a module for $\bT \otimes \bQ_{\ell}$. As such, it is free of rank 2. We can therefore regard the action of $F$ as a matrix in $\GL_2(\bT \otimes \bQ_{\ell})$.

Proposition. We have $\tr(F \mid V_{\ell})=T_p$ and $\det(F \mid V_{\ell})=p$

Proof. Recall that we have the Weil pairing $\langle, \rangle \colon V_{\ell} \times V_{\ell} \to \bQ_{\ell}(1)$. For any $\ell \nmid N$, the endomorphism $T_{\ell}$ of $J_0(N)$ is induced by a correspondence $(f_{\ell}, g_{\ell})$ as above. The adjoint of $T_{\ell}$ is induced by the transposed correspondence $(g_{\ell}, f_{\ell})$. However, these are easily seen to be equal: $(f_{\ell}, g_{\ell})$ sums over $p$-isogenies $E \to E'$, while $(g_{\ell}, f_{\ell})$ sums over $p$-isogenies $E' \to E$, and these sets are in bijection via the dual isogeny. Thus each $T_{\ell}$ is its own adjoint under $\langle, \rangle$.

We thus see that the isomorphism $\phi \colon V_{\ell} \to V_{\ell}^*$ defined by $x \mapsto \langle -, x \rangle$ commutes with the natural action of $\bT$ on each side. As $\phi(Fx)=V\phi(x)$, we see that $\tr(F \mid V_{\ell})=\tr(V \mid V_{\ell}^*)$. But the matrix for $V$ on $V_{\ell}^*$ is just the transpose of the matrix of $V$ on $V_{\ell}$ (in appropriate bases), and so $\tr(F \mid V_{\ell})=\tr(V \mid V_{\ell})$. But now appealing to $T_p=F+V$ again, we see that $2 T_p=2\tr(F)$, and so $\tr(F)=T_p$. Multiplying the Eichler--Shimura relation by $F$, we find $F^2-T_pF+p=0$. Since $\tr(F | V_{\ell})=T_p$, it follows that $\det(F | V_{\ell})=p$.

Remark. It is very important in the above theorem that the trace and determinant are taken in the sense of $\bT \otimes \bQ_{\ell}$ modules. If we just think of $F$ as an endomorphism of the $\bQ_{\ell}$ vector space $V_{\ell}$, its determinant is $p^g$, where $g=\dim(J_0(N))$.

## The Shimura Construction

### The construction

Fix a prime number $N$ and a normalized weight 2 cuspidal eigenform $f \in S_2(N)$. Let $\alpha \colon \bT \to \bC$ be the homomorphism giving the eigenvalues, i.e., $\alpha(T_p)$ is the $T_p$-eigenvalue of $f$. Then $\alpha(\bT \otimes \bQ)$ is a number field $K \subset \bC$ (in fact, the field generated by the $a_p(f)$), while $\alpha(\bT)$ is an order $\cO$ in $K$. Let $\fa \subset \bT$ be the kernel of $\alpha$. Define

$$A_f = J_0(N)/\fa J_0(N).$$

By $\fa J_0(N)$ we mean $\sum_{T \in \fa} T J_0(N)$, which is an abelian subvariety of $J_0(N)$. Thus $A_f$ is an abelian variety over $\bQ$. This is the Shimura construction. In the rest of this section, we'll study $A_f$.

Remark. Everything we'll do only depends on $\fa$: the embedding of $K=(\bT/\fa) \otimes \bQ$ into $\bC$ will not be used.

Remark. This can be done for composite $N$ as well as long as one works with newforms.

### Dimension

The tangent space to $J_0(N)$ at the identity is $\rH^0(X_0(N), \Omega^1)$. Over the complex numbers, this $\rH^0$ is identified with $S_2(N)$, which have shown is a free module of rank 1 over $\bT_{\bC}$. It follows that $T_0(J_0(N))$ is a free module of rank 1 over $\bT_{\bQ}$. We thus see that

$$T_0(J_0(N))/\fa T_0(J_0(N)) = T_0(A_f)$$

is a one dimensional vector space of $K$. We have thus shown:

Proposition. $A_f$ is an abelian variety of rank $[K:\bQ]$.

Corollary. If $K=\bQ$ then $A_f$ is an elliptic curve.

### Good reduction

We have the following result on the good reduction of $A_f$:

Proposition. $A_f$ has good reduction away from $N$.

In fact, this is an immediate corollary of the following general result:

Proposition. Let $B$ be an abelian variety over a DVR with good reduction, and let $A$ be a subquotient of $B$. Then $A$ has good reduction.

Proof. Let $\ell$ be a prime different from the residue characteristic. Since $B$ has good reduction, $V_{\ell}(B)$ is an unramified representation, by Néron--Ogg--Shafarevich. As $V_{\ell}(A)$ is a subquotient of $V_{\ell}(B)$, it too is unramified. Another application of Néron--Ogg--Shafarevich shows that $A$ has good reduction.

### Structure of the Tate module

We have a natural map $\cO=\bT/\fa \to \End(A_f)$. We can therefore regard $V_{\ell}(A_f)$ as $K \otimes \bQ_{\ell}$-module. With this in mind, we have:

Proposition. Let $p \nmid \ell N$ be a prime, and let $F_p \in \Gal(\ol{\bQ}/\bQ)$ be the Frobenius at $p$. Then $\tr(F_p \mid V_{\ell})=a_p$ and $\det(F_p \mid V_{\ell})=p$.

Proof. It suffices to prove this over $\bF_p$. Here it follows immediately from our above results on the Eichler--Shimura theorem. (Just apply the idempotent of $\bT$ that projects onto $K$.)

### The Galois representation associated to f

Choose an embedding $K \to \ol{\bQ}_{\ell}$. We have the following extremely important result:

Theorem. There exists a unique semi-simple representation $\rho \colon \Gal(\ol{\bQ}/\bQ) \to \GL_2(\ol{\bQ}_{\ell})$ satisfying the following conditions:

• $\rho$ is unramified away from $N\ell$.

• $\tr(\rho(F_p))=a_p$ for $p \nmid N\ell$.

• $\det(\rho)=\chi_{\ell}$.

Proof. We first prove existence. The choice of embedding $K \to \ol{\bQ}_{\ell}$ determines a place of $K$ above $\ell$, i.e., an idempotent $e$ in $K \otimes \bQ_{\ell}$. Simply take $\rho$ to be the semi-simplification of the representation on the space $eV_{\ell}(A_f)$. The first two points follow from the previous section. That section also shows that $\det(\rho(F_p))=p$ for all $p \nmid N\ell$, which implies that $\det(\rho)=\chi_{\ell}$.

Uniqueness follows from Chebotarev and an exercise in group theory. Precisely, suppose that $\rho'$ were a second representation satisfying the same conditions. Then $\tr(\rho(F_p))=\tr(\rho'(F_p))$ holds for all $p \nmid N\ell$. Chebotarev then implies that $\tr(\rho(g))=\tr(\rho'(g))$ for all $g$ in the Galois group, since the $F_p$ are dense. It now follows that $\rho$ and $\rho'$ are equivalent, as they are semi-simple representations with the same character.

Remark. In fact, the representation $\rho$ constructed above is absolutely irreducible.

Remark. Instead of taking our data to be a form $f$ and an embedding of its coefficient field into $\ol{\bQ}_{\ell}$, we could simply have considered a homomorphism $\bT \to \ol{\bQ}_{\ell}$. In other words, for any homomorphism $\alpha \colon \bT \to \ol{\bQ}_{\ell}$ we get a representation $\rho_{\alpha}$ as above. Furthermore, one has a decomposition

$$\rH^1_{\et}(X_0(N)_{\ol{\bQ}}, \ol{\bQ}_{\ell}) = \bigoplus \rho_{\alpha},$$

where the sum is over all $\alpha \colon \bT \to \ol{\bQ}_{\ell}$.

### Application: strong multiplicity 1

Theorem. Let $f,g \in S_2(N)$ (with $N$ prime) be normalized (i.e., $a_1=1$). Suppose there is a density 1 set of primes $S$ such that $f$ and $g$ are eigenvectors of $T_p$ for all $p \in S$ with the same eigenvalues. Then $f=g$.

Proof. Let $\alpha \colon S \to \bC$ be the function taking $p$ to the $T_p$-eigenvalue of $f$ and $g$. Let $V \subset S_2(N)$ be the space of forms $h$ satisfying $T_ph=\alpha(p) h$. It suffices to show that $V$ is one-dimensional. Now, $V$ has a basis consisting of eigenforms for the full Hecke algebra $\bT$. It thus suffices to show that if $h,h' \in V$ are normalized eigenforms for the full $\bT$ then $h=h'$.

Let $K \subset \bC$ contain the eigenvalues of $h$ and $h'$, and choose an embedding of $K$ into $\ol{\bQ}_{\ell}$. We then get semi-simple representations $\rho,\rho' \colon \Gal(\ol{\bQ}/\bQ) \to \GL_2(\ol{\bQ}_{\ell})$ satisfying $\tr(\rho(F_p))=a_p(h)$ and $\tr(\rho'(F_p))=a_p(h')$ for all $p \nmid N\ell$. Thus $\tr(\rho(F_p))=\tr(\rho'(F_p))$ holds for all $p \in S$. However, by Chebotarev then $F_p$ with $p \in S$ are dense in the Galois group, and so $\rho \cong \rho'$. We thus see that $a_p(h)=a_p(h')$ for all $p \nmid N\ell$. By using two different $\ell$'s, we get this for all $p \ne N$. But now $h=h'$ by the version of multiplicity one we previously proved.