On two arithmetic theta lifts

TL;DR

This paper investigates the relationship between two types of Green functions associated with special cycles on Shimura varieties, demonstrating their difference forms a modular function and exploring implications for arithmetic geometry and Kudla's conjecture.

Contribution

It establishes the modularity of the difference of two Green function families and constructs a new section of the Maass lowering operator using a regularized theta lift.

Findings

The difference of the Green functions is a modular form with trivial holomorphic projection.

The difference of the arithmetic theta functions is proven to be modular.

Relations between arithmetic heights of cycles and derivatives of Eisenstein series are clarified.

Abstract

Our aim is to clarify the relationship between Kudla's and Bruinier's Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type. These functions play a key role in the arithmetic geometry of the special cycles in the context of Kudla's program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a modular form with trivial holomorphic projection. Along the way, we construct a section of the Maass lowering operator for moderate growth modular forms valued in the Weil representation using a regularized theta lift, which may be of independent interest. We also consider arithmetic-geometric applications to integral models of U(n, 1) Shimura varieties. Each family of Green functions gives rise to a formal arithmetic theta function, valued in an arithmetic Chow group, that is conjectured to be modular; our main result is the modularity of the difference of the two arithmetic theta functions. Finally, we relate the arithmetic heights of the special cycles to special derivatives of Eisenstein series, as predicted by Kudla's conjecture, and describe a refinement of a theorem of Bruinier-Howard-Yang on arithmetic intersections against CM points.