Artin vanishing along the $\ell$-adic tower of an abelian variety

Abstract

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field and let $\ell$ be a prime invertible in the field. For every constructible $\mathbb{F}_{\ell}$-sheaf $F$ on $A$ we prove that there is an integer $e$, depending on $F$, such that the pullback map $[\ell^{e}]^{*}\colon\mathrm{H}^{i}(A,[\ell^{n}]^{*}F)\to\mathrm{H}^{i}(A,[\ell^{n+e}]^{*}F)$ is zero for all $n\geq0$ and all $i>\dim\operatorname{Supp}F$. In particular $\varinjlim\limits_{n}\mathrm{H}^{i}(A,[\ell^{n}]^{*}F)=0$ for $i>\dim\operatorname{Supp}F$, which answers a question of Bhatt--Schnell--Scholze. Our methods also give sharp codimension estimates for the supports of the cohomology sheaves of the Fourier--Mellin transform of a perverse sheaf. With $\ell$-adic coefficients, we remove the arithmeticity hypothesis from the estimates of Esnault--Kerz. The corresponding estimates for the completed transform with $\mathbb{F}_{\ell}$-coefficients are new and hold even when Hard Lefschetz fails.