Square root p-adic L-functions

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, Π × Π′) in terms of squares of automorphic periods on unitary groups. Here Π×Π′ is an automorphic representation of GL(n, F) × GL(n − 1, F) that descends to an automorphic representation of U(V) × 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 Π′ 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.