The Jacobian of modular curves associated to Cartan subgroups

Résumé 0

The mod p representation associated to an elliptic curve is called split/non-split dihedral if its image lies in the normaliser of a split/non-split Cartan subgroup of GL2(Fp). Let X+split(p) and X+non-split(p) denote the modular curves which classify elliptic curves with dihedral split and non-split mod p representation, respectively. We call such curves (split/non-split) Cartan modular curves. It is well known that X+split(p) is isomorphic to the curve X+0(p2). On the other hand, the curve X+non-split(p) is distinctly different from any of the classical modular curves. Despite this apparent disparity, it is shown in this thesis that the jacobian of X+non-split(p) is isogenous to the new part of the jacobian of X+0(p2). The method of proof uses the Selberg trace formula. An explicit formula for the trace of Hecke operators is derived for both split and non-split Cartan modular curves. Comparing these two trace formulae, one obtains a trace relation, which in combination with the Eichler-Shimura relations allows us to conclude that the L-series of the two abelian varieties in question are the same, up to finitely many L-factors. The result then follows by Faltings' isogeny theorem.