Diagonal restriction of Eisenstein series and Kudla-Millson theta lift

TL;DR

This paper links Kudla-Millson theta series for signature (N,N) quadratic spaces to Eisenstein series via a see-saw argument and Siegel-Weil formula, extending intersection number results to totally real fields.

Contribution

It establishes that the regularized integral of Kudla-Millson theta series along a torus equals the diagonal restriction of an Eisenstein series, generalizing previous results.

Findings

Fourier coefficients are expressed as intersection numbers.

Diagonal restriction of Eisenstein series is characterized.

Extension of Darmon-Pozzi-Vonk results to totally real fields.

Abstract

We consider the Kudla-Millson theta series associated to a quadratic space of signature $(N,N)$. By combining a `see-saw' argument with the Siegel-Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree $N$ is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon-Pozzi-Vonk to totally real fields.