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

Constructing abelian varieties from rank 2 Galois representations

Let U be a smooth affine curve over a number field K with a compactification X and let L be a rank 2, geometrically irreducible lisse Ql-sheaf on U with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field E ⊂ Ql, and has bad, infinite reduction at some closed point x of X\U. We show that L occurs as a summand of the cohomology of a family of abelian varieties over U. The argument follows the structure of the proof of a recent theorem of Snowden and Tsimerman, who show that when E=Q, then L is isomorphic to the cohomology of an elliptic curve EU -> U.Peer Reviewe

Constructing abelian varieties from rank 3 Galois representations with real trace field

Let $U/K$ be a smooth affine curve over a number field and let $L$ be an irreducible rank 3 $\overline{\mathbb Q}_{\ell}$-local system on $U$ with trivial determinant and infinite geometric monodromy around a cusp. Suppose further that $L$ extends to an integral model such that the Frobenius traces are contained in a fixed totally real number field. Then, after potentially shrinking $U$, there exists an abelian scheme $f\colon B_U\rightarrow U$ such that $L$ is a summand of $R^2f_*\overline{\mathbb Q}_{\ell}(1)$. The key ingredients are: (1) the totally real assumption implies $L$ admits a square root $M$; (2) the trace field of $M$ is sufficiently bounded, allowing us to use recent work of Krishnamoorthy-Yang-Zuo to construct an abelian scheme over $U_{\bar K}$ geometrically realizing $L$; and (3) Deligne's weight-monodromy theorem and the Rapoport-Zink spectral sequence, which allow us to pin down the arithmetizations using the total degeneration.Comment: 3 pages, comments welcome

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