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Triple product $$p$$-adic $$L$$-functions for balanced weights. (English) Zbl 07159950
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.
##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols
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##### References:
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