Workshop

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

One approach to constructing certain p-adic L-functions relies on construction of a p-adic family of Eisenstein series. I will explain how to construct such p-adic families for certain unitary groups. As part of the talk, I will explain how to p-adically interpolate certain values of both holomorphic and non-holomorphic Eisenstein series. I will also mention some applications to number theory and beyond.

Symplectic Langlands parameters of dimension 2n naturally arise from l-adic realization of symplectic motives like H_1(X), X curve of genus n. It corresponds to automorphic representations of odd special orthogonal group SO(2n+1) under the Langlands conjecture. It has been known that when n=1 the Langlands correspondence associates a specific automorphic form, the newform, to a 2-dimensional Galois representation attached to an elliptic curve. Such a newform has the property that it is fixed under an arithmetic group of prescribed level which is equal to the Artin conductor of the Galois representation. We will generalize the notion of newforms associated to symplectic motives of higher ranks by determining its local components, the new vectors, associated to the symplectic Langlands parameters.