Thursday 20 Feb 2014Euler systems for Rankin--Selberg convolutions of modular forms

Sarah Zerbes - UCL

Harrison 203 15:00-16:00

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 some of its
arithmetical applications. This is joint work with Antonio Lei and
David Loeffler.

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