Vertical Distribution Relations For Special Cycles on Unitary Shimura Varieties

TL;DR

This paper establishes a vertical distribution relation for special cycles on unitary Shimura varieties over anticyclotomic extensions, enabling the construction of norm-compatible cycles akin to Heegner modules.

Contribution

It introduces a vertical distribution relation for cycles on unitary Shimura varieties, complementing existing horizontal relations, and constructs a universal norm family of cycles.

Findings

Proved a vertical distribution relation for special cycles.

Constructed a family of norm-compatible cycles over anticyclotomic extensions.

Enhanced the understanding of cycle relations in the context of Gan-Gross-Prasad conjectures.

Abstract

We consider cycles on a 3-dimensional Shimura varieties attached to a unitary group, defined over extensions of a CM field $E$, which appear in the context of the conjectures of Gan, Gross, and Prasad \cite{gan-gross-prasad}. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of \cite{jetchev:unitary}, and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $\Lambda$-module constructed from Heegner points.