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de Rham \((\varphi,\Gamma)\)-modules and \(p\)-adic \(L\) functions. (\((\varphi,\Gamma)\)-modules de de Rham et fonctions \(L\) \(p\)-adiques.) (French. English summary) Zbl 1451.11132

Summary: We develop a variant of Coleman and Perrin-Riou’s methods giving, for a de Rham \(p\)-adic Galois representation, a construction of \(p\)-adic \(L\)-functions from a compatible system of global elements. As a result, we construct analytic functions on an open set of the \(p\)-adic weight space containing all locally algebraic characters of large enough conductor. Applied to Kato’s Euler system, this gives \(p\)-adic \(L\)-functions for elliptic curves with additive bad reduction and, more generally, for modular forms which are supercuspidal at \(p\). In the case of dimension \(2\), we prove a functional equation for our \(p\)-adic \(L\)-functions.

MSC:

11S40 Zeta functions and \(L\)-functions
11R23 Iwasawa theory
11S37 Langlands-Weil conjectures, nonabelian class field theory
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