Automorphic spectral identities and applications to automorphic \(L\)-functions on \(\mathrm{GL}_2\). (English) Zbl 1300.11051
Summary: We prove an automorphic spectral identity on \(\mathrm{GL}_2\) involving second moments. From it we obtain an asymptotic, with power-saving error term, for (non-archimedean) conductor-aspect integral moments, twisting by \(\mathrm{GL}_1\) characters ramifying at a fixed finite place. The strength of the spectral identity, and of the resulting asymptotics, is illustrated by extracting a subconvex bound in conductor aspect at a fixed finite prime.
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 |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
Keywords:
automorphic \(L\)-functions; integral moments; conductor aspect; spectral decomposition; Poincaré series; subconvexityReferences:
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