The Jacobian of modular curves associated to Cartan subgroups

Thesis

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(F_p). 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.

Authors: Chen, I

• Peer review status: Peer reviewed
• DOI: 10.5287/ora-br00ev6oq
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford