# Coleman maps and the p-adic regulator - Mathematics > Number Theory

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Abstract: This paper is a sequel to our earlier paper -Wach modules and Iwasawa theoryfor modular forms- arXiv: 0912.1263, where we defined a family of Colemanmaps for a crystalline representation of the Galois group of Qp withnonnegative Hodge-Tate weights. In this paper, we study these Coleman mapsusing Perrin-Riou-s p-adic regulator L V. Denote by H\Gamma the algebra ofQp-valued distributions on \Gamma = GalQp\mu p^\infty - Qp. Our firstresult determines the H\Gamma-elementary divisors of the quotient ofD {cris}V \otimes H\Gamma by the H\Gamma-submodule generated by \phi *NV^{\psi = 0}, where NV is the Wach module of V. By comparing thedeterminant of this map with that of L V which can be computed viaPerrin-Riou-s explicit reciprocity law, we obtain a precise description of theimages of the Coleman maps. In the case when V arises from a modular form, weget some stronger results about the integral Coleman maps, and we can removemany technical assumptions that were required in our previous work in order toreformulate Kato-s main conjecture in terms of cotorsion Selmer groups andbounded p-adic L-functions.

Author: ** Antonio Lei, David Loeffler, Sarah Livia Zerbes**

Source: https://arxiv.org/