Vol. 3, No. 4, 2021

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Square root $p$-adic $L$-functions, I: Construction of a one-variable measure

Michael Harris

Vol. 3 (2021), No. 4, 657–688

The Ichino–Ikeda conjecture, and its generalization to unitary groups by N. Harris, gives explicit formulas for central critical values of a large class of Rankin–Selberg tensor products. The latter conjecture has been proved in full generality and applies to L-values of the form L(1 2,BC(π) × BC(π)), where π and π are cohomological automorphic representations of unitary groups U(V ) and U(V ), respectively. Here V and V are hermitian spaces over a CM field, V of dimension n, V of codimension 1 in V , and BC denotes the twisted base change to GL(n) × GL(n 1).

This paper contains the first steps toward constructing a p-adic interpolation of the normalized square roots of these L-values, generalizing the construction in my paper with Tilouine on triple product L-functions. It will be assumed that the CM field is imaginary quadratic, π is a holomorphic representation and π varies in an ordinary Hida family (of antiholomorphic forms). The construction of the measure attached to π uses recent work of Eischen, Fintzen, Mantovan, and Varma.

For Jacques Tilouine

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$p$-adic $L$-function, central critical value, Shimura variety
Mathematical Subject Classification 2010
Primary: 11F55, 11F67, 11R23
Secondary: 22E47
Received: 12 September 2019
Revised: 3 June 2020
Accepted: 28 September 2020
Published: 20 October 2021
Michael Harris
Department of Mathematics
Columbia University
New York, NY
United States