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$p$-ADIC $L$-FUNCTIONS FOR UNITARY GROUPS

Published online by Cambridge University Press:  06 May 2020

ELLEN EISCHEN
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA; eeischen@uoregon.edu
MICHAEL HARRIS
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA; harris@math.columbia.edu
JIANSHU LI
Affiliation:
Institute for Advanced Study in Mathematics, Zhejiang University, Hangzhou, China; matom@ust.hk, jianshu@sjtu.edu.cn
CHRISTOPHER SKINNER
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA; cmcls@math.princeton.edu

Abstract

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This paper completes the construction of $p$-adic $L$-functions for unitary groups. More precisely, in Harris, Li and Skinner [‘$p$-adic $L$-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math.Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such $p$-adic $L$-functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and $p$-adic differential operators [Eischen, ‘A $p$-adic Eisenstein measure for unitary groups’, J. Reine Angew. Math.699 (2015), 111–142; ‘$p$-adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble)62(1) (2012), 177–243], Part II of the present paper provides the calculations of local $\unicode[STIX]{x1D701}$-integrals occurring in the Euler product (including at $p$). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.

Type
Number Theory
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020
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