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

Final version, to appear in Int. J. Number Theory

References in corpus (2)

• Critical slope p-adic L-functions of CM modular forms
• Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields

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