**Brief summary of papers**

**4**. Rank 2 Local Systems and Abelian varieties. *in preparation*

Here is a talk I gave on this work at BIRS.

**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 *submitted*.

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.

Another expected consequence: let X be a compact Shimura curve of Hodge type over the complex numbers with good reduction X at p. Then there are exactly two Newton polygons that occur in universal abelian scheme over X. This would resolve part of Conjecture 1.3 of a paper of Sheng-Zhang-Zuo. The key property of Shimura curves is that they admit Hecke correspondences, i.e. *étale correspondences without a core.*

**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: