Arithmetic level raising theorem for some unitary Shimura varieties mod $p$

TL;DR

This paper investigates the geometry and stratification of certain unitary Shimura varieties mod p, constructs elements in their higher Chow groups, and proves an arithmetic level raising theorem for specific cases using an Ihara lemma.

Contribution

It introduces new elements in the higher Chow groups of supersingular loci and establishes an arithmetic level raising theorem for n=2,3, leveraging a proven Ihara lemma.

Findings

Construction of elements in higher Chow groups of supersingular loci

Proof of an Ihara lemma for specific Shimura varieties

Surjectivity of the arithmetic level raising map for n=2,3

Abstract

Let $F$ be a real quadratic field in which a fixed prime $p$ is inert, and $E_0$ be an imaginary quadratic field in which $p$ splits; put $E=E_0 F$. Let ${{\rm Sh}}_{1,n-1}$ be the special fiber over $\mathbb{F}_{p^2}$ of the Shimura variety for $G(U(1,n-1)\times U(n-1,1))$ with hyperspecial level structure at $p$ for some integer $n\geq 2$. Let ${{\rm Sh}}_{1,n-1}(K_{\mathfrak{p}}^{1})$ be the special fiber over $\mathbb{F}_{p^2}$ of a Shimura variety for $G(U(1,n-1)\times U(n-1,1))$ with parahoric level structure at $p$ for some integer $n\geq 2$. We exhibit elements in the higher Chow group of the supersingular locus of ${{\rm Sh}}_{1,n-1}$ and study the stratification of ${{\rm Sh}}_{1,n-1}.$ Moreover, we study the geometry of ${{\rm Sh}}_{1,n-1}(K_{\mathfrak{p}}^{1})$ and prove a form of Ihara lemma. With Ihara lemma, we prove the the arithmetic level raising map is surjective for $n=2,3.$