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

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 $n=2,3.$