Patrick Allen ( U. Illinois Urbana-Champaign)
Automorphic points in polarized deformation space

Abstract: Mazur’s deformation theory of Galois representations has played a central role in the study of Langlands reciprocity for number fields. Some of the very basic properties of universal deformation space, e.g. its dimension, still remain unproven in general. In the case of modular forms and under some technical conditions, Böckle showed that every component of deformation space contains a smooth modular point, which allows one to determine much of its basic geometry and, when coupled with the infinite fern of Gouêa and Mazur, proves that the modular points are Zariski dense. I will discuss an improvement and generalization of Böckle’s result to arbitrary dimensions in the polarized case. This yields new results on the geometry of universal polarized deformation space, and, when combined with work of Chenevier, new results on the Zariski density of automorphic points in dimension three. br>