Inertial and Hodge–Tate weights of crystalline representations

Let K be an unramified extension of $${\mathbb {Q}}_p$$Qp and $$\rho :G_K \rightarrow {\text {GL}}_n(\overline{{\mathbb {Z}}}_p)$$ρ:GK→GLn(Z¯p) a crystalline representation. If the Hodge–Tate weights of $$\rho $$ρ differ by at most p then we show that these weights are contained in a natural collection of weights depending only on the restriction to inertia of $${\overline{\rho }} = \rho \otimes _{\overline{{\mathbb {Z}}}_p} \overline{{\mathbb {F}}}_p$$ρ¯=ρ⊗Z¯pF¯p. Our methods involve the study of a full subcategory of p-torsion Breuil–Kisin modules which we view as extending Fontaine–Laffaille theory to filtrations of length p.

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

We construct 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 ℚp, over a Galois extension 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 localization at the primes above p is an extension 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.