Iwasawa theory and p-adic L-functions over $\mathbf{Z}_p^2$-extensions

TL;DR

This paper develops a two-variable p-adic regulator map for crystalline Galois representations over -dimensional p-adic Lie extensions and explores its implications for p-adic L-functions and Iwasawa theory.

Contribution

It introduces a novel two-variable regulator map for crystalline representations over -dimensional p-adic Lie extensions and links it to conjectures on p-adic L-functions.

Findings

Formulation of a conjecture on the existence of a zeta element related to p-adic L-functions.

Demonstration that the conjecture implies known properties of 2-variable p-adic L-functions.

Extension of Perrin-Riou's regulator theory to higher-dimensional p-adic Lie extensions.

Abstract

In this paper, we define a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of $\mathbf{Q}_p$, over a Galois extension of $\mathbf{Q}_p$ whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localisation at the primes above p are extensions of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.