Bounds on eigenvalues of Hecke operator on the Jacobian

Question

Let $p$ be a prime not dividing $N$. Consider the Hecke operator $T_p$ on the Jacobian $\text{Jac}(X_0(N))$. I'm thinking of $T_p$ as coming from a correspondence. If I understand correctly, the eigenvalues of $T_p$ are bounded by $2p^{1/2}$. What is the proof of this claim?

I'd expect to see reduction mod $p$, followed by a use of the Eichler-Shimura relation, and finally an application of the RH part of the Weil conjectures, but I'm not sure how to put these together (in particular, since $X_0(N)$ is not necessarily an elliptic curve, we can't just compare traces).

I am fine with assuming without proof that $X_0(N)$ has a good reduction mod $p$ (a theorem of Igusa).

I'm new to this subject, and so every detail in every answer helps.

Answer 1

Let $X$ be $X_0(N)$ and $\overline{X}$ its base-change to $\overline{\mathbb Q}$, let $J$ be $J_0(N)$ be its Jacobian and $\overline{J}$ its base-change to $\overline{\mathbb Q}$. For simplicity, I will suppose that $N$ is prime. I will prove the following result (or rather, I will show how it follows from Deligne's proof of the Ramanujan-Petersson conjecture):

Let $ V =(V_\ell \overline{J})^\vee$ be the dual of the $\ell$-adic Tate module of $J$. Then, for $p \nmid N$, $T_p$ acts on $V \otimes_{\mathbb Q_\ell} \overline{\mathbb Q_\ell}$ with eigenvalues which are algebraic over $\mathbb Q$, and whose absolute values are $\leq 2p^{1/2}$ at any infinite place of $\overline{\mathbb Q}$.

There is a canonical isomorphism

$$H^1_{ét}(\overline{X}, \mathbb Q_\ell) = V$$

between the first $\ell$-adic cohomology of $\overline{X}$ and the dual Tate module of its Jacobian. This is an isomorphism as $\ell$-adic Galois representations and as Hecke modules. This representation is unramified at $p$ for $p \nmid N\ell$.

Choose an isomorphism $\iota: \overline{\mathbb{Q}_\ell} \to \mathbb C$. Then we have the comparison isomorphism between de Rham cohomology and $\ell$-adic cohomology

$$H^1_{ét}(\overline{X}, \mathbb Q_\ell) \otimes_\iota \mathbb C \xrightarrow{\sim}H^1_{dR}(X(\mathbb C)/\mathbb C)$$

Hodge theory gives a decomposition

$$H^1_{dR}(X(\mathbb C)/\mathbb C) = H^0(X, \Omega^1_{X/\mathbb C}) \oplus \overline{H^0(X, \Omega^1_{X/\mathbb C})}$$

Remark that the passage to the complex numbers forces us to forget about the Galois action, but it preserves the Hecke action, because the Hecke action is defined via correspondences; morally speaking, the Hecke action occurs at the level of motives, and hence it holds in any realization, in a manner compatible with comparison isomorphisms.

Now, there is a canonical identification

$$H^0(X_0(N), \Omega^1_{X/\mathbb C}) \cong S_2(N, \mathbb C),$$

the space of weight $2$ cusp forms of level $\Gamma_0(N)$. Now the result follows from Deligne's proof of the Ramanujan-Petersson conjecture. Since $N$ is prime, we have $S_2^{\text{new}}(N, \mathbb C) = S_2(N, \mathbb C)$. The good Hecke operators are simultaneously diagonalizable on $S_2(N, \mathbb C)$, and the eigenspace attached to a system of Hecke eigenvalues has rank $1$, spanned by a normalized weight $2$ eigenform $\varphi$ such that, if $\varphi(q) = \sum_{n\geq 1} a_n q^n$, then $T_p(\varphi) = a_p\varphi$ for $p \neq N$. By Deligne, $|a_p|\leq 2p^{1/2}$. It follows that the operator $T_p$, $p \neq N$, acts on $H^0(X, \Omega^1_X)$ with eigenvalues of absolute value $\leq 2p^{1/2}$.

To get the same result on $$H^0(X, \Omega^1_{X/\mathbb C}) \oplus \overline{H^0(X, \Omega^1_{X/\mathbb C})},$$

remark that the action of $T_p$ commutes with complex conjugation, so that if $\varphi$ is an eigenform in $S_2(N, \mathbb C)$, its formal complex conjugate $\overline{\varphi}$ is still an eigenvector for the good Hecke operators; moreover, since the system of Hecke eigenvalues attached to $\varphi$ takes values in a totally real number field, it follows that the system of Hecke eigenvalues attached to $\overline{\varphi}$ is the same as the one attached to $\varphi$; in other words, this shows that the $\varphi$-isotypic component

$$H^1_{dR}(X(\mathbb C)/\mathbb C)^{\varphi}$$

is $2$-dimensional, and consists of

$$(\varphi \cdot \mathbb C) \oplus (\overline{\varphi}\cdot \mathbb C).$$

It follows that $T_p$, for $p \nmid N$, acts on $H^1_{dR}(X/\mathbb C)$ with eigenvalues of absolute value $|a_p| \leq 2p^{1/2}$. But even more is true: all of the Galois conjugates $a_p^\sigma$ of $a_p$ have absolute value $\leq 2p^{1/2}$, since $a_p^{\sigma}$ is also the $p$-th Fourier coefficient of an eigenform, namely $\varphi^\sigma$. Thus, since our isomorphism $\iota$ takes the algebraic closure of $\mathbb Q$ in $\overline{\mathbb Q_\ell}$ to the algebraic closure of $\mathbb Q$ in $\mathbb C$, the result follows.