Explicit 8-descent on elliptic curves

  • In this thesis I describe an explicit method for performing an 8-descent on an elliptic curve E over the rational numbers. An 8-descent is the next step after a 2- and a 4-descent. The method of 2-descent belongs to the classical tools in arithmetic geometry. More recent is the exploration of higher 2-descent. In 1996 Merriman, Siksek, and Smart found a method for doing 4-descent, and a different method for 4-descent was presented in 1998 by Cassels. I have been working on extending these methods to an 8-descent. I present a method for computing the 8-Selmer group of E as an abstract group, and next, for representing its elements as 8-coverings of E. With this method I was able to compute some examples, one of which is a curve of order 8 in the Shafarevich-Tate group of an elliptic curve. This is the first explicit example of a curve of such high order in Sha.

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Publishing Institution:IRC-Library, Information Resource Center der Jacobs University Bremen
Granting Institution:Internat. Univ.
Author:Sebastian Karl Michael Stamminger
Referee:Michael Stoll, Dierk Schleicher, John E. Cremona
Advisor:Michael Stoll
Persistent Identifier (URN):urn:nbn:de:101:1-201305171186
Document Type:PhD Thesis
Date of Successful Oral Defense:2005/12/09
Year of First Publication:2005
PhD Degree:Mathematics
Library of Congress Classification:Q Science / QA Mathematics (incl. computer science)
School:SES School of Engineering and Science
Call No:Thesis 2005/10

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