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.