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Euler systems and the Birch--Swinnerton-Dyer conjecture

Connections for Women: New Geometric Methods in Number Theory and Automorphic Forms August 14, 2014 - August 15, 2014

August 15, 2014 (09:30 AM PDT - 10:30 AM PDT)
Speaker(s): Sarah Zerbes (University College)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

14042

Abstract

One special case of the Birch--Swinnerton-Dyer conjecture

is the statement that if E is an elliptic curve over a number field,

and the L-function of E does not vanish at s = 1, then E has only

finitely many rational points and its Tate-Shafarevich group is

finite. This is known to be true for elliptic curves over Q by a

theorem of Kolyvagin.

 

Kolyvagin's proof relies on an object called an 'Euler system' -- a

system of elements of Galois cohomology groups -- in order to control

the Tate-Shafarevich group. It has long been conjectured that Euler

systems should exist in other contexts, and these should have

similarly rich arithmetical applications; but only a very small number

of examples have so far been found. In this talk I'll describe the

construction of a new Euler system attached to pairs of elliptic

curves -- or more generally pairs of modular forms -- and its

arithmetical applications to the Birch--Swinnerton-Dyer conjecture.

This is joint work with Antonio Lei and David Loeffler, and it has

recently been generalised by Guido Kings, David Loeffler and myself

Supplements
21324?type=thumb Zerbes 373 KB application/pdf Download
22361?type=thumb Zerbes 2 245 KB application/pdf Download
Video/Audio Files

14042

H.264 Video 14042.mp4 317 MB video/mp4 rtsp://videos.msri.org/data/000/021/296/original/14042.mp4 Download
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