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Relative modular symbols and \(p\)-adic Rankin-Selberg convolutions. (English) Zbl 0810.11036

Für cuspidale automorphe Darstellungen \(\pi\) von \(\text{GL}_ 3\) und \(\sigma\) von \(\text{GL}_ 2\) über dem Adelring von \(\mathbb{Q}\) wird nach H. Jacquet, I. I. Piatetski-Shapiro and J. A. Shalika [Am. J. Math. 105, 367-464 (1983; Zbl 0525.22018)] eine \(L\)-Funktion \(L(\pi, \sigma,s)\) definiert. Unter gewissen – zumindest in Spezialfällen erfüllten – Zusatzvoraussetzungen an \(\pi\) und \(\sigma\) leitet der Autor arithmetische Eigenschaften für \(L(\pi, \sigma, {1\over 2})\) her. Diese stehen in enger Beziehung zu modularen Symbolen, zu automorphen Periodenintegralen, \(p\)-adischen Integralen über eine Distribution auf \(\mathbb{Z}_ p^*\), die mit Hilfe Heckescher Eigenformen definiert sind. Die entstehenden \(p\)-adischen Maße erlauben die Definition neuer \(p\)-adischer \(L\)-Funktionen. Es ist kaum möglich, die Fülle der ebenso technisch komplizierten wie tiefliegenden Resultate in Kürze angemessen zu beschreiben.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F85 \(p\)-adic theory, local fields

Citations:

Zbl 0525.22018
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References:

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