On the values of some singular currents on Shimura varieties of orthogonal type

TL;DR

This paper evaluates a regularized theta lift on differential forms on Shimura varieties of orthogonal type, linking it to special values of L-functions and extending previous work on currents and automorphic Green functions.

Contribution

It extends prior work by explicitly computing the theta lift on differential forms and connecting it to near central L-values using the Siegel-Weil formula.

Findings

The theta lift involves near central special values of standard L-functions.

The results relate currents on Shimura varieties to automorphic L-functions.

An example with products of Shimura curves illustrates the theory.

Abstract

In a previous paper (arxiv:1409.7353), we introduced a regularized theta lift for reductive dual pairs of the form $(Sp_4,O(V))$ with $V$ a quadratic vector space over a totally real number field $F$. The lift takes values in the space of $(1,1)$-currents on the Shimura variety attached to $GSpin(V)$, and we proved that its values are cohomologous to currents given by integration on special divisors against automorphic Green functions. In this paper, we will evaluate the regularized theta lift on differential forms obtained as usual (non-regularized) theta lifts. Using the Siegel-Weil formula and ideas of Piatetskii-Shapiro and Rallis, we show that the result involves near central special values of standard $L$-functions for $Sp_{4,F}$. An example concerning products of Shimura curves will be given at the end of the paper.