Vol. 5, No. 8, 2011

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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

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 V . Denote by (Γ) the algebra of p-valued distributions on Γ = Gal(p(μp)p). Our first result determines the (Γ)-elementary divisors of the quotient of Dcris(V ) (Brig,p+)ψ=0 by the (Γ)-submodule generated by (φ(V ))ψ=0, where (V ) is the Wach module of V . By comparing the determinant of this map with that of 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.

$p$-adic regulator, Wach module, Selmer groups of modular forms
Mathematical Subject Classification 2010
Primary: 11R23
Secondary: 11F80, 11S25
Received: 26 November 2010
Revised: 23 February 2011
Accepted: 25 March 2011
Published: 5 June 2012
Antonio Lei
School of Mathematical Sciences
Monash University
VIC 3800
Department of Mathematics and Statistics
Burnside Hall
McGill University
805 Rue Sherbrooke Ouest
Montréal, QC
H3A 0B9
David Loeffler
Mathematics Institute
Zeeman Building
University of Warwick
United Kingdom
Sarah Livia Zerbes
Mathematics Research Institute
Harrison Building
University of Exeter
United Kingdom