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Mollification of the fourth moment of automorphic \(L\)-functions and arithmetic applications. (English) Zbl 1054.11026

Summary: We compute the asymptotics of twisted fourth power moments of modular \(L\)-functions of large prime level near the critical line. This allows us to prove some new non-vanishing results on the central values of automorphic \(L\)-functions, in particular those obtained by base change from \(GL_2(\mathbb Q)\) to \(GL_2(K)\) for \(K\) a cyclic field of low degree.

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11M41 Other Dirichlet series and zeta functions