Regularized theta lifts and (1,1)-currents on GSpin Shimura varieties. I

TL;DR

This paper develops a regularized theta lift for certain dual pairs over totally real fields, producing (1,1)-currents on GSpin Shimura varieties and relating them to special divisors and automorphic Green functions.

Contribution

It introduces a novel regularized theta lift for $(Sp_4,O(V))$ dual pairs over totally real fields, linking currents to special divisors on GSpin Shimura varieties.

Findings

Currents are cohomologous to integrals over special divisors.

The lift's values relate to automorphic Green functions.

Method enables evaluation on non-regularized theta lifts.

Abstract

We introduce 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 prove that its values are cohomologous to currents given by integration on special divisors against automorphic Green functions. In the second part to this paper, we will show how to evaluate the regularized theta lift on differential forms obtained as usual (non-regularized) theta lifts.