On local representation densities of hermitian forms and special cycles

TL;DR

This paper reformulates conjectural formulas for arithmetic intersection numbers of special cycles on unitary Shimura varieties using lattice counting methods, providing new insights into their local representation densities.

Contribution

It introduces a novel approach to express intersection numbers via weighted lattice counts, connecting geometric and arithmetic aspects of hermitian forms.

Findings

Reformulation of conjectural formulas for intersection numbers

Connection established between lattice counts and special cycle intersections

Potential implications for understanding hermitian form representations

Abstract

In this paper, we reformulate conjectural formulas for the arithmetic intersection numbers of special cycles on unitary Shimura varieties with minuscule parahoric level structure in terms of weighted counting of lattices containing special homomorphisms.