L-functions of twisted elliptic curves over function fields

Traditionally number theorists have studied, both theoretically and computationally, elliptic curves and their L-functions over number fields, in particular over the rational numbers. Much less work has been done over function fields, especially computationally, where the underlying geometry of the function field plays an intimate role in the arithmetic of elliptic curves. We make use of this underlying geometry to develop a method to compute the L-function of an elliptic curve and its twists over the function field of the projective line over a finite field. This method requires computing the number of points on an elliptic curve over a finite field, for which we present a novel algorithm. If the j-invariant of an elliptic curve over a function field is non-constant, its L-function is a polynomial, hence its analytic rank and value at a given point can be computed exactly. We present data in this direction for a family of quadratic twists of four fixed elliptic curves over a few function fields of differing characteristic. First we present analytic rank data that confirms a conjecture of Goldfeld, in stark contrast to the corresponding data in the number field setting. Second, we present data on the integral moments of the value of the L-function at the symmetry point, which on the surface appears to refute random matrix theory conjectures.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor

The University of Texas at Austin

Year dc:date.issued

2009

Author dc:creator

• Baig, Salman Hameed

Advisor dc:contributor.advisor

• Villegas, Fernando Rodriguez

• L-functions
• Elliptic curves
• Function fields
• Finite field

Statement dc:rights

• Copyright is held by the author. Presentation of this material on the Libraries' web site by University Libraries, The University of Texas at Austin was made possible under a limited license grant from the author who has retained all copyrights in the works.

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/2152/6527

OAI identifier oai:identifier

oai:repositories.lib.utexas.edu:2152/6527