The Jacobians of Non‐Split Cartan Modular Curves

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. LetX split +andX non‐split +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 Cartan modular curves . The curveX split +is isomorphic to the curve X 0 ( p 2 ) . Using the Selberg trace formula for Hecke operators, we verify that the jacobian ofX non‐split +is isogenous to the new part of the jacobian of X 0 ( p 2 ) . 1991 Mathematics Subject Classification : primary 11G18; secondary 11F72.