Tamagawa Numbers

Let $\Frob_p:X_J\rightarrow X_J$ denote the map induced by the Frobenius automorphism. We have $\Frob_p=-W_p$, where $W_p$ is the map induced by the Atkin-Lehner involution on $J_0(p)$. Let $f$ be a newform, $A=A_f$ the corresponding optimal quotient, and $w_p$ the sign of the eigenvalue of $W_p$ on $f$.

Proposition 7.1  

$$\Phi_A(\mathbf{F}_p) = \begin{cases} \Phi_A(\overline{\mathbf{F}... ...p=-1$},\\ \Phi_A(\overline{\mathbf{F}}_p)[2] & \text{if $w_p=1$.} \end{cases}$$

Proof. If $w_p=-1$, then $\Frob_p=1$ and the $\Gal(\overline{\mathbf{F}}_p/\mathbf{F}_p)$-action of $\Phi_A(\overline{\mathbf{F}}_p)$ is trivial. In this case $\Phi(\mathbf{F}_p)=\Phi(\overline{\mathbf{F}}_p)$. Next suppose $w_p=1$. Recall that we have an exact sequence

$$0\rightarrow X_{A^{\vee}} \rightarrow \Hom(X_A,\mathbf{Z}) \rightarrow \Phi_A \rightarrow 0.$$

Since $W_p$ acts as $+1$ on $f$, it also acts as $+1$ on each of the modules $A$, $X_A$,  $\Hom(X_A,\mathbf{Z})$, and $\Phi_A$. Thus $\Frob_p=-W_p$ acts as $-1$ on $\Phi_A$. Since the subgroup of $2$-torsion elements of a finite abelian group equals the subgroup of elements fixed under $-1$, it follows that $\Phi_A(\mathbf{F}_p) = \Phi_A(\overline{\mathbf{F}}_p)[2]$. $\qedsymbol$

Warning: When extending this result to the whole of $J_0(N)$, be careful. The action of $\Frob_p=T_p$ need not be by $\pm 1$, even though it must be by an involution of order $2$. For example, the component group of $J_0(65)$ at $5$ is cyclic of order $42$. The action of $\Frob_5$ is by multiplication by $-13$. Note that $(-13)^2 = 1 \pmod{42}$. The fixed points of multiplication by $-13$ is the order $14$ subgroup of $\mathbf{Z}/42\mathbf{Z}$.