Crystalline Chebotar\"ev density theorems

TL;DR

This paper formulates and proves conjectural analogs of Chebotar"ev's Density Theorem for $F$-isocrystals over varieties over finite fields, using Tannakian formalism, algebraic group theory, and deep results from number theory and algebraic geometry.

Contribution

It introduces conjectural Chebotar"ev density analogs for $F$-isocrystals and proves them for several classes, extending classical theorems and employing advanced algebraic and number-theoretic techniques.

Findings

Proved Chebotar"ev analogs for constant and semi-simple overconvergent $F$-isocrystals.

Established a $p$-adic analog of Deligne's Equidistribution Theorem.

Developed the theory of maximal quasi-tori in algebraic groups.

Abstract

Using the Tannakian formalism, we formulate conjectural analogs of Chebotar\"ev's Density Theorem for $F$-isocrystals over a smooth geometrically irreducible variety defined over a finite field. We prove these analogs for several large classes, including (a) constant $F$-isocrystals, (b) direct sums of isoclinic convergent $F$-isocrystals, (c) semi-simple overconvergent $F$-isocrystals, and (d) semi-simple convergent $F$-isocrystals which have an overconvergent extension. Case (a) is a generalization of the Mordell-Lang Conjecture for tori and enters in the proofs of (b) and (c). For (b) we use the classical Chebotar\"ev Density Theorem, and point counting techniques in $p$-adic Lie groups building on a result of Oesterl\'e. For (c) we give two proofs. One of them uses deep input on the Langlands correspondence by Abe and Lafforgue, and the theory of Frobenius weights of Kedlaya, Abe and Caro. Building on this we formulate and prove the $p$-adic analog of Deligne's Equidistribution Theorem. Then (c) follows by applying real algebraic geometry to maximal compact subgroups in complex algebraic groups, measure theory, and a convergence result on complex hypersurfaces. For (d) we develop the theory of maximal quasi-tori (generalizing maximal tori in non-connected linear algebraic groups) and use D'Addezio's result on Crew's parabolicity conjecture to reduce to (b). These arguments also yield a second proof of (c). Besides of the deep inputs mentioned above and some Tannakian arguments, our main technique is the theory of linear algebraic groups. We include a comparison with the recent article of Cadoret and Tamagawa on the same topic.