Independence of $\ell$ for Frobenius conjugacy classes attached to abelian varieties

Abstract

Let $A$ be an abelian variety over a number field $\mathrm{E}\subset \mathbb{C}$ and let $\mathbf{G}$ denote the Mumford--Tate group of $A$. After replacing $\mathrm{E}$ by a finite extension, the action of the absolute Galois group $\mathrm{Gal}(\overline{\mathrm{E}}/\mathrm{E})$ on the $\ell$-adic cohomology ${\mathrm{H}}_{\acute{\mathrm{e}}\mathrm{t}}^1(A_{\overline{\mathrm{E}}},{\mathbb{Q}}_{\ell })$ factors through $\mathbf{G}({\mathbb{Q}}_{\ell }).$ We show that for $v$ an odd prime of $\mathrm{E}$ where $A$ has good reduction, the conjugacy class of Frobenius ${\mathrm{Frob}}_v$ in $\mathbf{G}({\mathbb{Q}}_{\ell })$ is independent of $\ell$. Along the way we prove that every point in the $\mu$-ordinary locus of the special fiber of Shimura varieties has a special point lifting it.

63 pages. Some of the results on local models and the construction of integral models have been moved to arXiv:2409.03689

References in corpus (1)

• Independence of ell of Monodromy Groups

Cited by in corpus (1)

• Modularity of arithmetic special divisors for unitary Shimura varieties (with an appendix by Yujie Xu)