Square root $p$-adic $L$-functions

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 version for unitary groups is now a theorem, and expresses the central critical value of $L$-functions of the form $L(s,\Pi \times \Pi')$ in terms of squares of automorphic periods on unitary groups. Here $\Pi \times \Pi'$ is an automorphic representation of $\mathrm{GL}(n,F)\times\mathrm{GL}(n-1,F)$ that descends to an automorphic representation of $\mathrm{U}(V) \times \mathrm{U}(V')$, where $V$ and $V'$ are hermitian spaces over $F$, with respect to a Galois involution $c$ of $F$, of dimension $n$ and $n-1$, respectively.

I will report on the construction of a $p$-adic interpolation of the automorphic period — in other words, of the square root of the central values of the $L$-functions — when $\Pi'$ varies in a Hida family. The construction is based on the theory of $p$-adic differential operators due to Eischen, Fintzen, Mantovan, and Varma. Most aspects of the construction should generalize to higher Hida theory. I will explain the archimedean theory of the expected generalization, which is the subject of work in progress with Speh and Kobayashi.

number theory

Audience: researchers in the topic

Harvard number theory seminar