Rank 2 Local Systems, Barsotti-Tate Groups, and Shimura Curves

We develop a descent criterion for $K$-linear abelian categories. Using recent advances in the Langlands correspondence due to Abe, we build a correspondence between certain rank 2 local systems and certain Barsotti-Tate groups on complete curves over a finite field. We conjecture that such Barsotti-Tate groups "come from" a family of fake elliptic curves. As an application of these ideas, we provide a criterion for being a Shimura curve over $\mathbb{F}_q$. Along the way, we formulate a conjecture on the field-of-coefficients of certain compatible systems.Comment: 30 pages. Part of author's PhD thesis. Comments welcome

Periodic de Rham bundles over curves

In this article, we introduce the notion of periodic de Rham bundles over smooth complex curves. We prove that motivic de Rham bundles over smooth complex curves are periodic. We conjecture the converse, that is, that periodic de Rham bundles over smooth complex curves are motivic. The conjecture holds for rank one objects and certain rigid objects.Comment: 41 page

Periodic Higgs bundles over curves

In this article, we study periodic Higgs bundles and their applications. We obtain the following results: i). an elliptic curve has infinitely many primes of supersingular reduction if and only if any periodic Higgs bundle over it is a direct sum of torsion line bundles; ii). the uniformizing de Rham bundle attached to a generic projective hyperbolic curve is not one-periodic, and it is motivic iff it admits a modular embedding (e.g. Shimura curves, triangle curves); iii). there is an explicit Deuring-Eichler mass formula for the Newton jumping locus a Shimura curve of Hodge type. We propose the periodic Higgs conjecture, which would imply an arithmetic Simpson correspondence. The conjecture holds in rank one case.Comment: 36 page

Frobenius trace fields of cohomologically rigid local systems

Let $X/\mathbb{C}$ be a smooth projective variety and let $L$ be an irreducible $\overline{\mathbb{Q}}_{\ell}$-local system on $X$ with torsion determinant. Suppose $L$ is cohomologically rigid. The pair $(X, L)$ may be spreaded out to a finitely generated base, and therefore reduced modulo $p$ for almost all $p$; the Frobenius traces of this mod $p$ reduction lie in a number field $F_p$, by a theorem of Deligne. We investigate to what extent $F_p$ depends on $p$. We prove that for a positive density of primes $p$, the $F_p$'s are contained in a fixed number field. More precisely, we prove that $F_p$ is unramified at primes $\ell$ such that $\ell\neq p$ and $\ell$ large, where the largeness condition is uniform and does not depend on $p$, and also that $F_p$ is unramified at $p$ assuming a further condition on $p$. We also speculate on the relation between the uniform boundedness of the $F_p$'s, and the local system $L$ being strongly of geometric origin, a notion due to Langer-Simpson.Comment: Comments welcome