Cycles on Siegel 3-folds and derivatives of Eisenstein series

TL;DR

This paper studies special cycles on Siegel 3-folds, their intersections in characteristic p, and links intersection multiplicities at isolated points to derivatives of Eisenstein series.

Contribution

It characterizes isolated intersection points of special cycles on Siegel 3-folds and relates their multiplicities to derivatives of Eisenstein series, connecting geometry and automorphic forms.

Findings

Characterization of isolated intersection points in characteristic p

Relation of intersection multiplicities to Eisenstein series derivatives

Insights into the structure of special cycles on Siegel modular varieties

Abstract

We consider the Siegel modular variety of genus 2 and a p-integral model of it for a good prime p>2, which parametrizes principally polarized abelian varieties of dimension two with a level structure. We consider cycles on this model which are characterized by the existence of certain special endomorphisms, and their intersections. We characterize that part of the intersection which consists of isolated points in characteristic p only. Furthermore, we relate the (naive) intersection multiplicities of the cycles at isolated points to special values of derivatives of certain Eisenstein series on the metaplectic group in 8 variables.