Gehrmann, Lennart

Functoriality of Automorphic \(\mathrm{L}\)-Invariants and Applications

Doc. Math. 24, 1225-1243 (2019)
DOI: 10.25537/dm.2019v24.1225-1243
Communicated by Don Blasius

Summary

We study the behaviour of automorphic \(\mathrm{L}\)-invariants associated to cuspidal representations of \(\mathrm{GL}(2)\) of cohomological weight 0 under abelian base change and Jacquet-Langlands lifts to totally definite quaternion algebras. Under a standard non-vanishing hypothesis on automorphic \(\mathrm{L}\)-functions and some technical restrictions on the automorphic representation and the base field we get a simple proof of the equality of automorphic and arithmetic \(\mathrm{L}\)-invariants. This together with Spieß' results on \(p\)-adic \(\mathrm{L}\)-functions yields a new proof of the exceptional zero conjecture for modular elliptic curves -- at least, up to sign.

Mathematics Subject Classification

11F41, 11F67, 11F75, 11F85, 11G05

Keywords/Phrases

\(p\)-adic periods, modular forms, automorphic representations

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Affiliation

Gehrmann, Lennart
Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany

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