Rank 2 local systems and abelian varieties II (with Ambrus Pál)
Let X/Fq be a smooth, geometrically connected, quasi-projective variety. Let E be a semi-simple overconvergent F-isocrystal on X such that for each closed point x of X, the polynomial Px(E,t) has coefficients in Q. Suppose that each summand has rank 2, determinant Qp(-1), and infinite monodromy around infty. We show that E comes from a family of abelian varieties.
This is in analogy to the celebrated work of Corlette-Simpson on rigid rank 2 local systems on smooth complex varieties. We use as input Drinfeld's first work on the Langlands correspondence for GL2, Abe's work on the p-adic Langlands correspondence, the theory of (both l-adic and p-adic) companions, Poonen's Bertini theorem over finite fields, and crucially, the pigeonhole principle!
Rank 2 local systems and abelian varieties (with Ambrus Pál).
Let X/F_q be a smooth, geometrically connected, quasi-projective variety. We formulate a conjecture that absolutely irreducible rank 2 local systems on X "come from families of abelian varieties". We prove that when for projective varieties, a p-adic variant of our conjecture reduces to the case of projective curves. This "p-adic variant" assumes that there exists a "complete set of p-adic companions", a strong form of Deligne's petits camarades cristallin conjecture [Weil II, Conj 1.2.10 (vi)]. Again assuming the existence of a complete set of p-adic companions, it follows that when X is projective variety, the l-adic version of the conjecture also reduces to the case of curves on X. Along the way, we prove Lefschetz theorems on homomorphisms of abelian varieties and p-divisible groups. We also answer a question of Grothendieck on extending abelian schemes via their p-divisible groups.
Here is a talk I gave on an early version of this work.
Rank 2 local systems, Barsotti-Tate groups, and Shimura curves
This paper subsumes the second half of my thesis. Let X be a smooth projective curve over a finite field whose order is a square. We construct a "natural" bijection between (certain) rank 2 local systems on X and (certain) Barsotti-Tate groups on X. We conjecture that both of these "come from" families of abelian varieties; more precisely, we believe they come from families of fake elliptic curves. We use this bijection to give a criterion for being a Shimura curve over Fq. We also study the field-of-coefficients of compatible systems. Other than Abe's recent work, the main new ingredient is a descent-of-scalars criterion for general K-linear Tannakian categories.
Correspondences without a core
This paper mostly subsumes the first half of my thesis and improves the results on dynamics. To a "correspondence without a core", we associate a graph G together with a large group of "algebraic" automorphisms A. This graph reflects the "generic dynamics" of the correspondence, i.e. the dynamics of the generic point. In the case of a Hecke correspondence of modular curves, this is related to the action of PSL2(Ql) on its building. When the graph G is a tree, we show that (G,A) in fact shares properties with the action of PSL2(Ql) on its building.
The underlying goal of the article is to show that the "formal" structure of an (étale) correspondence without a core shares many properties with Hecke correspondences. However, there are interesting examples of étale correspondences without a core that are not directly related to Shimura curves in characteristic p: both central leaves in Shimura varieties and Drinfeld modular curves furnish examples. Nonetheless, the only examples we know are "modular". This poses the following natural question: given an étale correspondence of curves X←Z→Y without a core, are there infinitely many other minimal étale correspondences without a core between X and Y?
The last two sections are not in my PhD thesis. They introduce the notion of an "invariant line bundle" and "invariant sections" on a correspondence without a core. Using these concepts we obtain results on the number of "bounded étale orbits" on a correspondence without a core, generalizing recent work of Hallouin-Perret. Their argument uses spectral graph theory (including the Perron-Frobenius theorem!) and applies to correspondences over the algebraic closure of a finite field whose "graph of generic dynamics is a tree". (This is true, e.g., for Hecke correspondences of modular curves.) Our method is purely algebro-geometric, works for any correspondence without a core (relaxing the "tree" condition), and drops the assumption on the base field.
Some simple applications: we obtain a "non-computational" proof that every pair of supersingular elliptic curves over the algebraic closure of the prime field are related by an l-primary isogeny for any l≠p. In characteristic 0, we prove that an étale correspondence of projective curves without a core has no bounded orbit. This yields a "non-computational" proof that every Hecke correspondence of compactified modular curves is ramified at at least one cusp. It also shows that for any compact Shimura curve over the complex numbers, the iterated orbit of a point under any single Hecke correspondence is unbounded.
Maximal class numbers of CM number fields
(with Ryan Daileda and Anton Malyshev).
Conditional on GRH and Artin's conjecture on L-functions, we prove an upper bound on the maximal class number of CM number fields, fixing the totally real index-2 subfield. We show that this bound is optimal by realizing a family of such number fields with as-large-as-possible class number. The construction uses the following trick. Let N be a positive number. Then there exists a bound B such that for all primes p bigger than B, there are N consecutive quadratic resiues mod p. In fact, given any sequence of N gaps, one can find such a B such that for all primes p bigger than B, there are quadratic residues with precisely that gap sequence.
Frobenius trace fields of cohomologically rigid local systems
(with Joshua Lam).
We study the following question: do rigid local systems L admit a spreading out with bounded Frobenius trace field? This question is motivated by Simpson's conjecture that rigid local systems are of geometric origin (though not implied by it). We provide various partial results in the direction of this question for cohomologically rigid local systems on projective varieties. For instance, we prove there is a spreading out of L such that for a positive proportion of primes, the Frobenius trace field is bounded. We also prove that L^h has a spreading out with bounded Frobenius traces. In the quasi-projective results, for certain monodromy profiles we are able to partial results that are in a sense stronger.
The Manin-Mumford conjecture in genus 2 and rational curves on K3 surfaces
(with Philip Engel and Daniel Litt).
Let A be a simple Abelian surface over an algebraically closed field k. Let S⊂A(k) be the set of torsion points of A contained in the image of a map from a genus 2 curve C to A, sending a Weierstrass point of C to the origin. The purpose of this note is to show that if k has characteristic zero, then S is finite -- this is in contrast to the situation where k is the algebraic closure of a finite field, where S is all of A(k), as shown by Bogomolov and Tschinkel. We deduce that if k= Q, the Kummer surface associated to A has infinitely many k-points not contained in a rational curve arising from a genus 2 curve in A, again in contrast to the situation over the algebraic closure of a finite field.
Constructing abelian varieties from rank 2 Galois representations
(with Jinbang Yang and Kang Zuo).
Let X/K be a smooth curve over a number field. Let L be an irreducible rank 2 l-adic local system on X with cyclotomic determinant, that has infinite monodromy around infinity and such that there exists a number field E such that all Frobenius traces of an integral model of L lie in E. Then we show that L occurs inside the cohomology of a family of abelian varieties over X. Our argument broadly follows a recent theorem of Snowden-Tsimerman, who consider the case when E=Q. This theorem perhaps provides a bit of evidence for the relative Fontaine-Mazur conjecture, recently posed by several authors.
- Periodic de Rham bundles over curves (with Mao Sheng).
- Periodic Higgs bundles over curves (with Mao Sheng).
We study Higgs bundles on smooth projective curves over C that, after a spreading out, satisfy periodicity for a twisted Frobenius for almost all primes p. We conjecture such Higgs bundles are motivic, and provide some evidence for this. Interestingly, when the base curve has genus 1, our conjecture is equivalent to the existence of infinitely many primes of supersingular reduction.
- Finiteness theorems for logarithmic crystalline representations I, II (with Jinbang Yang and Kang Zuo)
We prove a finiteness theorem for absolutely irreducible logarithmic crystalline representations. This is meant in analogy with Deligne's finiteness theorem for local systems underlying a PVHS.
- A Lefschetz theorem for crystalline representations (with Jinbang Yang and Kang Zuo).
As a simple corollary of his nonabelian Hodge theorem, Simpson proves a Lefschetz theorem for polarized variations of Hodge structures: let X/C be a smooth projective variety of dimension at least 2 and let D be a smooth ample divisor. Let LX be a complex local system on X such that the restriction LD underlies a PVHS on D. Then LX underlies a PVHS on X. A natural arithmetic analog of a polarized variation of Hodge structure is a crystalline representation. We prove a Lefschetz theorem for crystalline representations, subject to several mild restrictions. This theorem suggests a strong Lefschetz theorem for families of motives. The main technique is p-adic nonabelian Hodge theory and vanishing theorems. Our techniques in fact allow us to treat the case of logarithmic crystalline representations.
Deformation theory of periodic Higgs-de Rham flows
We study the deformation theory of periodic Higgs-de Rham flows. To do this, we explicitly construct the tangent-obstruction theory to deforming filtered de Rham bundles and graded Higgs bundles. We introduce the notion of a periodic Higgs-de Rham flow being ordinary. When applied to the (logarithmic) uniformizing Higgs bundle of a hyperbolic curve over a finite field, this specializes to Mochizuki's notion of an ordinary curve. We give an application to the deformation theory of torsion crystalline representations.
Notes not intended for publication
- Gonality growth of Galois covers