Algebra & Number Theory

Coleman maps and the $p$-adic regulator

Antonio Lei, David Loeffler, and Sarah Livia Zerbes

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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.

Article information

Algebra Number Theory, Volume 5, Number 8 (2011), 1095-1131.

Received: 26 November 2010
Revised: 23 February 2011
Accepted: 25 March 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R23: Iwasawa theory
Secondary: 11F80: Galois representations 11S25: Galois cohomology [See also 12Gxx, 16H05]

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


Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia. Coleman maps and the $p$-adic regulator. Algebra Number Theory 5 (2011), no. 8, 1095--1131. doi:10.2140/ant.2011.5.1095.

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