Overconvergent Families of p-adic Representations
- Author(s): Upton, James Thomas
- Advisor(s): Wan, Daqing
- et al.
Let X be a variety over a finite field of characteristic p. The purpose of this dissertation is to extend many known results about p-adic representations of the fundamental group π1(X) to families of p-adic representations. The notion of a family of p-adic representations parameterized by a rigid analytic space arises naturally in many contexts, including geometric Iwasawa theory and the theory of p-adic modular forms. In either context there is significant interest in understanding the variation of the p-adic L-functions L(ρ,s) as ρ moves through a given family. It seems unlikely that we can say much in general, as there are far too many p-adic representations of the group π1(X). In this dissertation we restrict our attention to so-called overconvergent representations, which have the property that the L-function L(ρ,s) is always a p-adic meromorphic function in s. Thus for overconvergent families of representations, the question of understanding the p-adic variation of the L(ρ,s) reduces to the understanding of the variation of their zeroes and poles. Our main theorem is a relative version of the Dwork-Monsky trace formula, which says that these zeroes and poles are are naturally interpolated by rigid analytic objects which we call spectral varieties. In general, the geometry of these spectral varieties is quite mysterious: in the context of p-adic modular forms, the question is the subject of Coleman’s well known spectral halo conjecture. For a few specific examples of overconvergent families, analogues of Coleman’s conjecture have been proven by studying suitable integral models of the parameter space. For this reason, we choose to work primarily with formal schemes and their overconvergent analogues. We hope that this paves the way to a greater understanding of the arithmetic of these overconvergent families.