Empirical evidence for the Birch and Swinnerton-Dyer conjecture
by Miller, Robert L., Ph.D., UNIVERSITY OF WASHINGTON, 2010, 57 pages; 3431591

Abstract:

The current state of knowledge about the Birch and Swinnerton-Dyer conjecture relies on some of the deepest and most difficult mathematical endeavors, including the modularity theorem of Wiles, Breuil, Conrad, Diamond and Taylor, which was instrumental in the proof of Fermat's last theorem. There are also the Euler systems of Kato and Kolyvagin, Rubin's work on curves with complex multiplication, Néron's classification and Tate's algorithm, and the formula of Cross and Zagier. Despite all of this mathematical energy there is still much to be learned. Many facts about the conjecture only become clear one case at a time, after hard computation. We prove the full Birch and Swinnerton-Dyer conjecture for many specific elliptic curves of analytic rank zero and one and conductor up to 5000 by combining theoretical and computational methods.

 
AdviserWilliam Stein
SchoolUNIVERSITY OF WASHINGTON
SourceDAI/B 71-12, p. , Dec 2010
Source TypeDissertation
SubjectsMathematics
Publication Number3431591
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