Lecture 17: EichlerShimura
In the first part of this lecture, we prove the EichlerShimura 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 EichlerShimura 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 EichlerShimura 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 DeligneMumford 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 AtkinLehner 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 AtkinLehner 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 selfcorrespondences $(1,F)$ 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 EichlerShimura relation by $F$, we find $F^2T_pF+p=0$. 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 NéronOggShafarevich. As $V_{\ell}(A)$ is a subquotient of $V_{\ell}(B)$, it too is unramified. Another application of NéronOggShafarevich 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 EichlerShimura 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 semisimple 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 semisimplification 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 semisimple 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 onedimensional. 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 semisimple 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. ◾