Shimura Degrees for Elliptic Curves over Number Fields

Abstract

A crowning achievement of Number theory in the 20th century is a theorem of Wiles which states that for an elliptic curve E over <bold>Q</bold> of conductor N, there is a non-constant map from the modular curve of level N to E. For some curve isogenous to E, the degree of this map will be minimal; this is the modular degree. Generalizing to number fields, we no longer always have a modular curve. In the totally real number field case, the modular curve is replaced with a variety of dimension the same as the number field. It is only in the special case of <bold>Q</bold> that this variety happens to be a curve. The Jacquet-Langlands correspondence allows us to parameterize elliptic curves by Shimura curves. In this case we have several different Shimura curve parameterizations for a given isogeny class. I generalize to totally real number fields some of the results of Ribet and Takahashi over <bold>Q</bold>. I further discuss finding the curve in the isogeny class parameterized by a given Shimura curve and how this relates to pairs of isogenous curves with the same discriminant. Finally, I use my algorithm to compute new data about degrees. Then I compare it with D-new modular degrees and D-new congruence primes. This data indicates that there is a strong relationship between Shimura degrees and new modular degrees and congruence primes. These connections with D-new degrees lead me to the conjecture that they are the same.