# Automorphic Forms

## Automorphy: Galois Representations Attached to Automorphic Forms

## Eisenstein Congruences and Euler Systems

**GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS SEMINAR**

## Completed Cohomology of Shimura Curves and a p-Adic Jacquet-Langlands Correspondence

In this talk, I will describe a construction of a geometric realisation of a p-adic Jacquet-Langlands correspondence for certain forms of GL(2) over a totally real field. The construction makes use of the completed cohomology of Shimura curves, and a study of the bad reduction of Shimura curves due to Rajaei (generalising work of Ribet for GL(2) over the rational numbers). Along the way I will also describe a p-adic analogue of Mazur's principle in this setting.

## Automorphy Lifting for Galois Representations With Small Residual Image

**GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS SEMINAR**

## Algebraic Cycles on Picarad Moduli Spaces of Abelian Varieties

Picard moduli spaces parametrize principally polarized abelian varieties with complex multiplication by the ring of integers in an imaginary-quadratic field. The loci where the abelian varieties split off an elliptic curve in a controlled way are divisors on this moduli space. We study the intersection behaviour of these divisors and prove in the non-degenerate case a relation between their intersection numbers and Fourier coefficients of the derivative at s=0 of a certain incoherent Eisenstein series for the unitary group. This is joint work with Kudla.

## On the Realization of Some Degenerate Automorphic Forms on Certain Griffiths-Schmid Varieties

GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS SEMINAR

Some automorphic forms, despite the fact they are algebraic, do not have any interpretation as cohomology classes on a Shimura variety: therefore nothing is known at present on their expected arithmetic properties. I shall explain how such forms appear to be related to more general objects (Griffiths-Schmid varieties) and discuss some related rationality questions.