Arithemetic level raising theorem for some unitary Shimura varieties mod $p$
Zijie Tao
TL;DR
The paper develops a framework to relate unitary Shimura varieties of different signatures through higher Chow groups on supersingular loci, establishing an Ihara lemma and proving an arithmetic level raising theorem for n=2,3. Central to the approach are explicit cycle correspondences Y_j,BC-Deligne–Lusztig data, Dieudonné-module descriptions, and Grothendieck–Messing deformation theory, which together enable precise control of l-adic cohomology and its Galois actions. By analyzing Newton and Ekedahl–Oort stratifications and employing weight spectral sequences on parahoric and Iwahori level models, the authors derive surjectivity results for level-raising maps and provide a pathway toward generalizations via conjectural Galois-automorphic correspondences. The work advances the arithmetic Langlands program for unitary Shimura varieties by linking geometric cycles to automorphic/cohomological phenomena at inert primes and offering concrete, verifiable n=2,3 cases with explicit geometric constructions.
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 $n=2,3.$