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$p$-ADIC $L$-FUNCTIONS FOR ORDINARY FAMILIES ON SYMPLECTIC GROUPS

Part of: Discontinuous groups and automorphic forms Arithmetic algebraic geometry

Published online by Cambridge University Press:  14 January 2019

Zheng Liu
Zheng Liu*
Affiliation:
Department of Mathematics and Statistics, McGill University, Burnside Hall Room 1248, 805 Sherbrooke Street West, Montreal, QC H3A 0B9, Canada ([email protected])
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Abstract

We construct the $p$-adic standard $L$-functions for ordinary families of Hecke eigensystems of the symplectic group $\operatorname{Sp}(2n)_{/\mathbb{Q}}$ using the doubling method. We explain a clear and simple strategy of choosing the local sections for the Siegel Eisenstein series on the doubling group $\operatorname{Sp}(4n)_{/\mathbb{Q}}$, which guarantees the nonvanishing of local zeta integrals and allows us to $p$-adically interpolate the restrictions of the Siegel Eisenstein series to $\operatorname{Sp}(2n)_{/\mathbb{Q}}\times \operatorname{Sp}(2n)_{/\mathbb{Q}}$.

Keywords

p-adic L-functionsdoubling methodordinary families of Siegel modular forms

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Secondary: 11F60: Hecke-Petersson operators, differential operators (several variables) 11F30: Fourier coefficients of automorphic forms 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 4 , July 2020 , pp. 1287 - 1347
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Copyright
© Cambridge University Press 2019

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