Eisenstein Cohomology and p - adic L - Functions

§0. Introduction. In this paper, we define the module D̃(V ) of distributions with rational poles on a finite dimensional rational vector space a V . This is an infinite dimensional vector space over Q endowed with a natural action of the reductive group GV := Aut(V ). Indeed, this action extends to a natural action of the adelic group GV (AQ). For each prime p, we define we define the notion of p-adic continuity of elements of D̃(V ) and explain how p-adically continuous distributions with rational poles give rise to p-adic L-functions. In section 2, we construct a special element ξ ∈ D̃(Q), and explain its connection to the Kubota-Leopoldt p-adic L-functions. The key feature of our construction is that ξ is a global distribution, which is p-adically continuous for every prime p and therefore gives rise to p-adic L-functions for every p. In other words, we obtain a global object that specializes to the p-adic Kubota-Leopoldt p-adic L-functions. In section 3, we extend the constructions of section 2 and construct a GL(n)-symbol ψn for every n ≥ 1. When n ≥ 1, ψn is a GLn(Q)-invariant linear map from the (n− 1)dimensional cohomology of the Borel-Serre boundary of the symmetric space for GL(n) to D̃(Qn). When n = 1, ψ1 is determined by its value on the fundamental class, and this value is just the Kubota-Leopoldt distribution. When n = 2, we show that the coboundary of ψ2 vanishes identically and therefore gives rise to a classical modular symbol over GL2(Q). In section 4 (not yet included), we will explain how to associate p-adic L-functions to ξ2 and explain how these can be viewed as p-adic L-functions attached to Eisenstein series over the weight space. We will also use ξ2 to construct explicit overconvergent modular symbols, as defined by R. Pollack and G. Stevens. These are modular symbols taking values in the space of rigid analytic distributions on the p-adic upper half-plane.