The Jacobians of non-split Cartan modular curves

Abstract

The mod $p$ representation associated to an elliptic curve is called split or non-split dihedral if its image lies in the normaliser of a split or non-split Cartan subgroup of $\GL_2(\f_p)$, respectively. Let $\xsplit$ and $\xnonsplit$ denote the modular curves which classify elliptic curves with split and non-split dihedral mod $p$ representation, respectively. We call such curves split and non-split{\it Cartan modular curves}. The curve $\xsplit$ is isomorphic to the curve $X_0^+(p^2)$. Using the Selberg trace formula for Hecke operators, we verify that the jacobian of $\xnonsplit$ is isogenous to the new part of the jacobian of $X_0^+(p^2)$.

1991 Mathematics Subject Classification: primary 11G18; secondary 11F72.