Ashutosh Roy Choudhury

arXiv2026

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

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.

math.AG↗

arXiv2025

A Construction of the Symmetric Monoidal Structure of the Geometric Whittaker Model

Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $p > 0$ and let $\ell$ be a prime number different from $p$. Let $U \subseteq G$ be a maximal unipotent subgroup, $T$ a maximal torus normalizing $U$ and $W$ the Weyl group of $G$. Let $\mathcal{L}$ be a non-degenerate multiplicative $\overline{\mathbb{Q}}_{\ell} $-local system on $U$. R. Bezrukavnikov and the second author have proved that the bi-Whittaker category, namely the triangulated monoidal category of $(U, \mathcal{L})$-biequivariant $\overline{\mathbb{Q}}_{\ell}$-complexes on $G$ is monoidally equivalent to an explicit thick triangulated monoidal subcategory $\mathscr{D}_{W}^{\circ}(T) \subseteq \mathscr{D}_{W}(T)$ of "central sheaves" on the torus. In particular it has the structure of a symmetric monoidal category coming from the symmetric monoidal structure on $\mathscr{D}_W(T)$. In this paper, we give another construction of a symmetric monoidal structure on the above category and prove that it agrees with the one coming from the above construction. For this, among other things, we generalize a proof by Gelfand for finite groups to the geometric setup.

math.RT↗