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
Location: MSRI: Simons Auditorium
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
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