Title:

The Jacobian of modular curves associated to Cartan subgroups

The mod p representation associated to an elliptic curve is called split/nonsplit dihedral if its image lies in the normaliser of a split/nonsplit Cartan subgroup of GL_{2}(F_{p}). Let X^{+}_{split}(p) and X^{+}_{nonsplit}(p) denote the modular curves which classify elliptic curves with dihedral split and nonsplit mod p representation, respectively. We call such curves (split/nonsplit) Cartan modular curves. It is well known that X^{+}_{split}(p) is isomorphic to the curve X^{+}_{0}(p^{2}). On the other hand, the curve X^{+}_{nonsplit}(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^{+}_{nonsplit}(p) is isogenous to the new part of the jacobian of X^{+}_{0}(p^{2}). The method of proof uses the Selberg trace formula. An explicit formula for the trace of Hecke operators is derived for both split and nonsplit Cartan modular curves. Comparing these two trace formulae, one obtains a trace relation, which in combination with the EichlerShimura relations allows us to conclude that the Lseries of the two abelian varieties in question are the same, up to finitely many Lfactors. The result then follows by Faltings' isogeny theorem.
