Arithmetic properties of Fredholm series for -adic modular forms

Abstract

We study the relationship between recent conjectures on slopes of overconvergent -adic modular forms "near the boundary" of -adic weight space. We also prove in tame level 1 that the coeffcients of the Fredholm series of the U operator never vanish modulo , a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo using a modular interpretation of the mod reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.Accepted manuscrip