Modular degree of elliptic curves over function fields in relation to Jacquet-Langlands

Abstract:

In this thesis, we study the geometry of automorphic forms in the function field setting. The primary goal of this thesis prove a formula relating degrees of modular parametrization of an elliptic curve by different Drinfeld modular curves. This is analogous to the similar result of Ribet-
Takahashi [RT97] in the number field setting, which relates degrees of modular parameterizations of an elliptic curve over Q by varying Shimura curves. To prove this result, I prove a result analogous to Ribet’s short exact sequence [Rib90a] which relates the special fibers of Shimura varieties at different primes. Using this technical result I deduce (in the function field case) level-lowering results akin to those of Ribet [Rib90a] and relations between Petersson inner products of modular forms that are related by the Jacquet-Langlands correspondence similar to the work of K. Prasanna [Pra03].