Shimura Degrees for Elliptic Curves over Number Fields | Posted on:2015-10-22 | Degree:Ph.D | Type:Dissertation | University:University of Washington | Candidate:Deines, Alyson | Full Text:PDF | GTID:1478390017998083 | Subject:Mathematics | Abstract/Summary: | | A crowning achievement of Number theory in the 20th century is a theorem of Wiles which states that for an elliptic curve E over Q 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 Q 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 Q. 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. | Keywords/Search Tags: | Degrees, Curve, Shimura, Elliptic, Over | | Related items |
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