This book is a companion to the author’s previous book [11], Euler products and Eisenstein series, published by the AMS. The books’ ultimate objective is to prove algebraicity of the critical values of the zeta functions of automorphic forms on unitary and symplectic groups. In the course of the study of the zeta functions, many important results, which were obtained by the author during 1960-2...

We develop a theory of $p$-adic automorphic forms on unitary groups that allows interpolation in families and holds for all primes $p$ do not ramify the reflex field $E$ associated Shimura variety. If ordinary locus is nonempty (a condition only met if splits completely $E$), we recover Hida's forms, which defined over locus. More generally, work $\mu$-ordinary locus, open dense. By eliminating...

Let F be a non-Archimedean locally compact field, let G be a split connected reductive group over F . For a parabolic subgroup Q ⊂ G and a ring L we consider the G-representation on the L-module (∗) C∞(G/Q, L)/ ∑

Using a new definition of rank for representations of semisimple groups, sharp results are proved for the decay of matrix coefficients of unitary representations of two types of non-split p-adic simple algebraic groups of exceptional type. These sharp bounds are achieved by minimal representations. It is also shown that in one of the cases considered, the minimal representation is isolated in t...