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The Rosetta stone of \(L\)-functions. (English) Zbl 1171.11321

This is a survey on motivic \(L\)-functions, associated to algebraic varieties over number fields, Artin \(L\)-functions, associated to finite dimensional representations of the absolute Galois group of a number field, and automorphic \(L\)-functions, associated to automorphic forms on algebraic groups modulo discrete subgroups. Conjecturally, the three classes are equal, but this is known only in very few cases.

MSC:

11M41 Other Dirichlet series and zeta functions
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11R42 Zeta functions and \(L\)-functions of number fields
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