Siegel Functions, Modular Curves, and Serre's Uniformity Problem
Digital Document
Handle |
Handle
http://hdl.handle.net/11134/20002:860634162
|
||||||
---|---|---|---|---|---|---|---|
Persons |
Persons
Author (aut): Daniels, Harris B.
Thesis advisor (ths): Lozano-Robledo, Alvaro
Associate Advisor (asa): Holsinger, Kent
Associate Advisor (asa): Lee, Kyu-Hwan
|
||||||
Title |
Title
Title
Siegel Functions, Modular Curves, and Serre's Uniformity Problem
|
||||||
Origin Information |
Origin Information
|
||||||
Parent Item |
Parent Item
|
||||||
Resource Type |
Resource Type
|
||||||
Digital Origin |
Digital Origin
born digital
|
||||||
Description |
Description
Serre’s uniformity problem asks whether there exists a bound k such that for any p > k, the Galois representation associated to the p-torsion of an elliptic curve E/Q is surjective independent of the choice of E. Serre showed that if this representation is not surjective, then it has to be contained in either a Borel subgroup, the normalizer of a split Cartan subgroup, the normalizer of a non-split Cartan subgroup, or one of a finite list of “exceptional” subgroups. We will focus on the case when the image is contained in the normalizer of a split Cartan subgroup. In particular, we will show that the only elliptic curves whose Galois representation at 11 is contained in the normalizer of a split Cartan have complex multiplication. To prove this we compute X_s^+(11) using modular units, use the methods of Poonen and Schaefer to compute its jacobian, and then use the method of Chabauty and Coleman to show that the only points on this curve correspond to CM elliptic curves.
|
||||||
Genre |
Genre
|
||||||
Organizations |
Organizations
Degree granting institution (dgg): University of Connecticut
|
||||||
Held By | |||||||
Rights Statement |
Rights Statement
|
||||||
Use and Reproduction |
Use and Reproduction
These Materials are provided for educational and research purposes only. The University of Connecticut Library holds the copyright except where noted. Permission must be obtained in writing from the University of Connecticut Library and/or the owner(s) of the copyright to publish reproductions or quotations beyond "fair use."
These materials are provided for educational and research purposes only.
|
||||||
Degree Name |
Degree Name
Doctor of Philosophy
|
||||||
Degree Level |
Degree Level
Doctoral
|
||||||
Degree Discipline |
Degree Discipline
Mathematics
|
||||||
Local Identifier |
Local Identifier
ASC Theses 18401
842150676
OC_d_67
|