Gassmann class (reviewed)

For an open subgroup $H \leq \GL_2(\widehat \Z)$, the Gassmann class of $H$ is the set of open subgroups of $\GL_2(\widehat \Z)$ that are Gassmann equivalent to $H$.

Let $H,H' \leq \GL_2(\widehat \Z)$ be in the same Gassmann class. Then $H,H'$ have the same level, index and genus. Moreover, the Jacobians of $X_H$ and $X_{H'}$ are isogenous (see Corollary 4 of Prasad-Rajan [10.1016/S0022-314X(02)00053-7], applied to the covering curve $X_{\mathrm{full}}(N)$), hence $L(X_H,s)=L(X_{H'},s)$.