A reciprocity map and the two variable p-adic L-function

TL;DR

This paper proposes a conjecture linking Galois cohomology cup product values with two-variable p-adic L-functions of modular forms, using reciprocity maps and towers in cyclotomic fields.

Contribution

It introduces a new conjecture connecting Galois cohomology, reciprocity maps, and two-variable p-adic L-functions, expanding understanding of their interrelations.

Findings

Constructs an isomorphism comparing reciprocity map values to p-adic L-functions.

Proposes a conjecture relating cup products in Galois cohomology to p-adic L-values.

Develops a framework passing through cyclotomic and Hida towers for these comparisons.

Abstract

For primes p greater than 3, we propose a conjecture that relates the values of cup products in the Galois cohomology of the maximal unramified outside p extension of a cyclotomic field on cyclotomic p-units to the values of p-adic L-functions of cuspidal eigenforms that satisfy mod p congruences with Eisenstein series. Passing up the cyclotomic and Hida towers, we construct an isomorphism of certain spaces that allows us to compare the value of a reciprocity map on a particular norm compatible system of p-units to what is essentially the two-variable p-adic L-function of Mazur and Kitagawa.