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In this paper, we define a two-variable analogue of Perrin-Riou's p-adicregulator map for the Iwasawa cohomology of a crystalline representation of theabsolute 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 dimension2.  We use this regulator map to study p-adic representations of global Galoisgroups over certain abelian extensions of number fields whose localisation atthe primes above p are extensions of the above type. In the example of therestriction to an imaginary quadratic field of the representation attached to amodular 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 thisconjecture implies the known properties of the 2-variable p-adic L-functionsconstructed by Perrin-Riou and Kim.

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