Crystalline Chebotarëv density theorems
Urs Hartl, Ambrus Pal
TL;DR
This work develops crystalline analogues of Chebotarëv’s Density Theorem for $F$-isocrystals on a smooth variety over a finite field, recasting the problem in neutral Tannakian terms and defining Frobenius conjugacy classes via monodromy groups. The authors prove density statements for several cases: constant $F$-isocrystals, direct sums of isoclinic $F$-isocrystals, and semi-simple overconvergent $F$-isocrystals, with a deeper treatment of semi-simple convergent $F$-isocrystals having overconvergent extensions. A central strategy combines Tannakian reductions, the theory of maximal quasi-tori and compact subgroups, and $p$-adic equidistribution results inspired by Deligne, Kedlaya, Abe, and Lafforgue; they also provide an effective Chebotarëv-type density theorem and relate their results to Cadoret-Tamagawa’s work. The paper culminates in establishing an $p$-adic equidistribution theorem for overconvergent $F$-isocrystals, and a Parabolicity Conjecture framework that yields further density consequences. Overall, the results connect deep $p$-adic Hodge theory, Langlands correspondences, and algebraic group theory to crystalline density phenomena with implications for understanding Frobenius images in monodromy groups.
Abstract
Using the Tannakian formalism, we formulate conjectural analogs of Chebotarëv'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 Chebotarëv Density Theorem, and point counting techniques in $p$-adic Lie groups building on a result of Oesterlé. 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.