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Period invariants of Hilbert modular forms. I: Trilinear differential operators and L-functions. (English) Zbl 0716.11020
Cohomology of arithmetic groups and automorphic forms, Proc. Conf., Luminy/Fr. 1989, Lect. Notes Math. 1447, 155-202 (1990).
[For the entire collection see Zbl 0706.00009.]
The paper under review concerns Shimura’s conjecture, which says that the periods of holomorphic Hilbert modular forms factor, up to algebraic numbers, as products of simpler quantities. In particular the author introduces a set of invariants, which are defined in terms of nonholomorphic Hilbert modular forms and satisfy certain of Shimura’s axioms. Shimura’s approach to special values of Rankin-Selberg L- functions is reinterpreted in the language of coherent cohomology. Then the results of the author’s joint work with S. Kudla on triple product L- functions are rewritten in cohomological terms. A combination leads to nontrivial relations among the period invariants of binary theta functions.
Reviewer: A.Krieg

MSC:
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
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