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Greenberg, Matthew; Seveso, Marco Adamo

Triple product \(p\)-adic \(L\)-functions for balanced weights. (English) Zbl 1475.11093

Math. Ann. 376, No. 1-2, 103-176 (2020).
Summary: We construct \(p\)-adic triple product \(L\)-functions that interpolate (square roots of) central critical \(L\)-values in the balanced region. Thus, our construction complements that of Harris and Tilouine. There are four central critical regions for the triple product \(L\)-functions and two opposite settings, according to the sign of the functional equation. In the first case, three of these regions are of interpolation, having positive sign; they are called the unbalanced regions and one gets three \(p\)-adic \(L\)-functions, one for each region of interpolation (this is the Harris-Tilouine setting). In the other setting there is only one region of interpolation, called the balanced region: we produce the corresponding \(p\)-adic \(L\)-function. Our triple product \(p\)-adic \(L\)-function arises as \(p\)-adic period integrals interpolating normalizations of the local archimedean period integrals. The latter encode information about classical representation theoretic branching laws. The main step in our construction of \(p\)-adic period integrals is showing that these branching laws vary in a \(p\)-adic analytic fashion. This relies crucially on the Ash-Stevens theory of highest weight representations over affinoid algebras.

Cited in 7 Documents

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields

Keywords:

\(p\)-adic \(L\)-functions; balanced region; affinoid algebras
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References:

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