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$p$-ADIC EISENSTEIN SERIES AND $L$-FUNCTIONS OF CERTAIN CUSP FORMS ON DEFINITE UNITARY GROUPS

Part of: Arithmetic algebraic geometry Algebraic number theory: local and $p$-adic fields Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  21 November 2014

Ellen Eischen  and
Xin Wan
Ellen Eischen
Affiliation:
Department of Mathematics, The University of North Carolina at Chapel Hill, CB #3250, Chapel Hill, NC 27599-3250, USA (eeischen@email.unc.edu)
Xin Wan
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10025, USA (xw2295@math.columbia.edu)
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Abstract

We construct $p$-adic families of Klingen–Eisenstein series and $L$-functions for cusp forms (not necessarily ordinary) unramified at an odd prime $p$ on definite unitary groups of signature $(r,0)$ (for any positive integer $r$) for a quadratic imaginary field ${\mathcal{K}}$ split at $p$. When $r=2$, we show that the constant term of the Klingen–Eisenstein family is divisible by a certain $p$-adic $L$-function.

Keywords

p-adic automorphic formsautomorphic forms on unitary groupsEisenstein seriesp-adic L-functionsKlingen–Eisenstein series

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms 11F03: Modular and automorphic functions
Secondary: 11F30: Fourier coefficients of automorphic forms 11S40: Zeta functions and $L$-functions 11G18: Arithmetic aspects of modular and Shimura varieties 11F55: Other groups and their modular and automorphic forms (several variables) 11F85: $p$-adic theory, local fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 3 , July 2016 , pp. 471 - 510
Copyright
© Cambridge University Press 2014 

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