Coleman maps and the p-adic regulator

Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia

doi:10.2140/ant.2011.5.1095

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Coleman maps and the $p$-adic regulator

Antonio Lei, David Loeffler and Sarah Livia Zerbes

Vol. 5 (2011), No. 8, 1095–1131

DOI: 10.2140/ant.2011.5.1095

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 $ℒ_{V}$ . Denote by $ℋ (Γ)$ the algebra of $ℚ_{p}$ -valued distributions on $Γ = Gal (ℚ_{p} (μ_{p^{\infty}}) ∕ ℚ_{p})$ . Our first result determines the $ℋ (Γ)$ -elementary divisors of the quotient of $D_{cris} (V) \otimes {(B_{rig, ℚ_{p}}^{+})}^{ψ = 0}$ by the $ℋ (Γ)$ -submodule generated by ${(φ^{*} ℕ (V))}^{ψ = 0}$ , where $ℕ (V)$ is the Wach module of $V$ . By comparing the determinant of this map with that of $ℒ_{V}$ (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.

Keywords

$p$-adic regulator, Wach module, Selmer groups of modular forms

Mathematical Subject Classification 2010

Primary: 11R23

Secondary: 11F80, 11S25

Milestones

Received: 26 November 2010

Revised: 23 February 2011

Accepted: 25 March 2011

Published: 5 June 2012

Authors

Antonio Lei
School of Mathematical Sciences Monash University VIC 3800 Australia	Department of Mathematics and Statistics Burnside Hall McGill University 805 Rue Sherbrooke Ouest Montréal, QC H3A 0B9 Canada

David Loeffler
Mathematics Institute Zeeman Building University of Warwick Coventry CV4 7AL United Kingdom

Sarah Livia Zerbes
Mathematics Research Institute Harrison Building University of Exeter Exeter EX4 4QF United Kingdom

Vol. 5, No. 8, 2011