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Summation formulae, automorphic realizations and a special value of Eisenstein series. (English) Zbl 0788.11017

The self-duality (with respect to the Fourier-transform) of the theta distribution \(\theta\) on the adeles gives rise to the fact that automorphic forms can be constructed by applying \(\theta\) to coefficients of the metaplectic representations. This is the theory of classical theta series.
In the paper under consideration the author defines a “multiplicative version” of the theta distribution, which, restricted to a suitable space of test functions, also is self-dual. This distribution is used to derive analogs for the classical Weil-Siegel formula in the case of the groups \(\text{SL}_ 2\) and \(\text{GL}_ 2\). The Weil-Siegel formula expresses a special value of an Eisenstein series (which still is a function on the group) as an integral of the automorphic form used to define the Eisenstein series against a theta series [S. Kudla and S. Rallis, J. Reine Angew. Math. 387, 1-68 (1988; Zbl 0644.10021)].
In the present context the analog of this is a formula that gives a special value of an Eisenstein series at the identity element as the multiplicative theta-distribution applied to the automorphic form.

MSC:

11F27 Theta series; Weil representation; theta correspondences
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F70 Representation-theoretic methods; automorphic representations over local and global fields

Citations:

Zbl 0644.10021
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