**Brief summary of papers**

**7**. A Lefschetz theorem for crystalline representations - with Jinbang Yang and Kang Zuo, arXiv 2003.08906, *submitted*.

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 **L_X** be a complex local system on **X** such that the restriction **L_D** underlies a PVHS on **D**. Then **L_X** 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. 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.

**6**. Deformation theory of Periodic Higgs-de Rham flows - with Jinbang Yang and Kang Zuo, arXiv 2005.00579.

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.

**5**. Rank 2 local systems and abelian varieties II - with Ambrus Pál, *submitted*.

Let X/**F**_q 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 P_{x}(*E*,t) has coefficients in **Q**. Suppose that each summand has rank 2, determinant Q_{p}(-1), and infinite monodromy around infty. We show that *E* comes from a family of abelian varieties.

We use as input Drinfeld's first work on the Langlands correspondence for GL_{2}, Abe's work on the *p*-adic Langlands correspondence, and Poonen's Bertini theorem over finite fields.

**4**. Rank 2 Local Systems and abelian varieties - with Ambrus Pál, *submitted.*

Here is a talk I gave on an early work at BIRS.

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.

**3**. Rank 2 Local Systems, Barsotti-Tate groups, and Shimura Curves. *submitted*

This paper will subsume 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 **F**_{q}. 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.

**2**. Correspondences without a Core - Algebra and Number Theory 12:5 (2018) 1173-1214.

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 PSL_{2}(Q_{l}) on its building. When the graph G is a tree, we show that (G,A) in fact shares properties with the action of PSL_{2}(Q_{l}) 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 new: 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.

**1**. Maximal Class Numbers of CM Number Fields - with R. Daileda and A. Malyshev, J. Number Theory 130 (2010), no. 4, 936-943.

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.

**Notes not intended for publication**

Gonality Growth of Galois Covers

Let X be a smooth projective curve that does not map nontrivially to a genus 1 curve. In this note, I prove a (tight) lower bound for the gonality growth of (unramified) Galois covers X. I suspect that if the Jacobian of X is (geometrically) simple, then the growth of gonality is in fact *linear*, but I have no idea how to prove this. More precisely, I suspect that if one has a tower of (unramified) Galois extensions whose Galois groups are sufficiently non-abelian, then gonality growth should be linear. Examples would include semi-simple linear algebraic groups over Z/l^{n }Z. This is compatible with work of Abramovich on towers of modular curves (for the case of SL_{2}.) If you have any ideas, please let me know!

Some friends and I at Banff:

Mao Sheng and I in front of the cherry blossoms