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The Jacquet-Langlands correspondence, Eisenstein congruences, and integral \(L\)-values in weight \(2\). (English) Zbl 1429.11090

Summary: We use the Jacquet-Langlands correspondence to generalize well-known congruence results of Mazur on Fourier coefficients and \(L\)-values of elliptic modular forms for prime level in weight \(2\) both to nonsquare level and to Hilbert modular forms.

MSC:

11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F30 Fourier coefficients of automorphic forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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