Coleman maps and the $p$-adic regulator

Abstract

We study the Coleman maps for a crystalline representation $V$ with non-negative Hodge–Tate weights via Perrin-Riou’s $p$-adic “regulator” or “expanded logarithm” map ${\mathcal{ℒ}}_V$. Denote by $\mathcal{\mathscr{H}}\left(\Gamma \right)$ the algebra of ${\mathbb{Q}}_p$-valued distributions on $\Gamma =Gal\left({\mathbb{Q}}_p\left({\mu }_{p^{\infty }}\right)∕{\mathbb{Q}}_p\right)$. Our first result determines the $\mathcal{\mathscr{H}}\left(\Gamma \right)$-elementary divisors of the quotient of ${\mathbb{D}}_{cris}\left(V\right)\otimes {\left({\mathbb{B}}_{rig,{\mathbb{Q}}_p}^{+}\right)}^{\psi =0}$ by the $\mathcal{\mathscr{H}}\left(\Gamma \right)$-submodule generated by ${\left({\phi }^{\ast }\mathbb{N}\left(V\right)\right)}^{\psi =0}$, where $\mathbb{N}\left(V\right)$ is the Wach module of $V$. By comparing the determinant of this map with that of ${\mathcal{ℒ}}_V$ (which can be computed via Perrin-Riou’s explicit reciprocity law), we obtain a precise description of the images of the Coleman maps. In the case when $V$ arises from a modular form, we get some stronger results about the integral Coleman maps, and we can remove many technical assumptions that were required in our previous work in order to reformulate Kato’s main conjecture in terms of cotorsion Selmer groups and bounded $p$-adic $L$-functions.