Bloch and Kato's Exponential Map: Three Explicit Formulas

The purpose of this article is to give formulas for Bloch-Kato's exponential map and its dual for an absolutely crystalline $p$-adic representation $V$, in terms of the $(\varphi,\Gamma)$-module associated to $V$. As a corollary of these computations, we can give a very simple and slightly improved description of Perrin-Riou's exponential map, which interpolates Bloch-Kato's exponentials for the twists of $V$. This new description directly implies Perrin-Riou's reciprocity formula.

2000 Mathematics Subject Classification: 11F80, 11R23, 11S25, 12H25, 13K05, 14F30, 14G20

Keywords and Phrases: Bloch-Kato's exponential, Perrin-Riou's exponential, Iwasawa theory, $p$-adic representations, Galois cohomology.

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