Euler systems and the BirchSwinnertonDyer conjecture
Connections for Women: New Geometric Methods in Number Theory and Automorphic Forms August 14, 2014  August 15, 2014
Location: SLMath: Eisenbud Auditorium
14042
One special case of the BirchSwinnertonDyer conjecture
is the statement that if E is an elliptic curve over a number field,
and the Lfunction of E does not vanish at s = 1, then E has only
finitely many rational points and its TateShafarevich 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 TateShafarevich 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 BirchSwinnertonDyer conjecture.
This is joint work with Antonio Lei and David Loeffler, and it has
recently been generalised by Guido Kings, David Loeffler and myself
Zerbes

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