#### 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
##### Abstract

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 $L\left(\frac{1}{2},BC\left(\pi \right)×BC\left({\pi }^{\prime }\right)\right)$, where $\pi$ and ${\pi }^{\prime }$ are cohomological automorphic representations of unitary groups $U\left(V\right)$ and $U\left({V}^{\prime }\right)$, respectively. Here $V$ and ${V}^{\prime }$ are hermitian spaces over a CM field, $V$ of dimension $n$, ${V}^{\prime }$ of codimension 1 in $V$, and $BC$ denotes the twisted base change to $GL\left(n\right)×GL\left(n-1\right)$.

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, $\pi$ is a holomorphic representation and ${\pi }^{\prime }$ varies in an ordinary Hida family (of antiholomorphic forms). The construction of the measure attached to $\pi$ 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