**Raju Krishnamoorthy**

NSF Postdoctoral Fellow

Arithmetic Geometry Group

Freie Universität Berlin

FB Mathematik und Informatik

Arnimallee 3

14195 Berlin

**Email:**

firstname[at]math[dot]columbia[dot]edu OR

firstname[at]math[dot]fu-berlin[dot]de

**About Me**

I am an NSF postdoctoral fellow under the supervision of Hélène Esnault at FU Berlin. Before that, I was a graduate student in the Mathematics Department at Columbia University, where my advisor was Johan de Jong.

**Research Interests**

These days I mostly think about l-adic local systems and overconvergent F-isocrystals on smooth varieties over finite fields. I am also especially interested in p-divisible groups on complete varieties over finite fields and central leaves on Shimura varieties.

These interests essentially arose from my PhD Thesis. Motivated by results of Mochizuki, I tried to "characterize" Shimura curves over a finite field using purely group theoretic data: the notion of an étale correspondence without a core. These exhibit many formal similarities with Hecke correspondences of Shimura curves: for instance, given a correspondence without a core, one can construct an infinite graph with a large group of "algebraic" automorphisms. In the case of a Hecke correspondence of Shimura curves, this specializes to the action of PSL_2(Q_p) on it's building. Relatedly, given a correspondence without a core, one may construct an infinite tower of covers; these specialize to adding progressively higher (full) level structure in the familiar case of Hecke correspondences of Shimura/modular curves.

A more elaborate group-theoretic hypothesis: one may make assumptions about the Galois groups of this infinite tower of curves. By assuming that certain Galois groups are related to linear groups over local fields, one is led to the following question. Let X←Z→X be an étale correspondence without a core and suppose there is an SL_2(Q_l) local system on X such that the two pullbacks to Z are isomorphic as local systems. Then is the whole package "related to" a Hecke correspondence of Shimura curves? The example of modular curves with *Igusa level structures* show that the phrase "related to" is absolutely essential: the correspondence may not simply deform to characteristic 0.

Recent work of Tomoyuki Abe completes the so-called *companions conjecture* of Deligne in the case of curves by proving a p-adic Langlands correspondence for curves over a finite field. Using Abe's results combined with foundational work of de Jong, we translated the condition on local systems to a condition on associated p-divisible groups. Under sufficiently auspicious circumstances, the correspondence together with the p-divisible groups deforms to characteristic 0 and Mochizuki's theorem then implies that everything in sight is at least "related to" a Hecke correspondence of Shimura curves.

The work of Lafforgue (resp. Abe) shows that in general the local systems (resp. overconvergent F-isocrystals) that occur in this story are "motivic"; this was indeed the original motivation for the companions conjecture. More recently, Deligne, Drinfeld, and Abe-Esnault have proven almost all of the companions conjecture for higher dimensional varieties: loosely speaking, we don't know how to go from "l to p" (for the experts: we also don't know how to go from "p to p".) These questions would all be resolved if one could prove that absolutely irreducible local systems (resp. overconvergent F-isocrystals) were of *geometric origin*. Since finishing my PhD, I have been mostly thinking about this "geometricity" problem and its corollaries.

**Preprints/Papers**

**3**. Local Systems and Barsotti-Tate groups. *in preparation*

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*. 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 consequence: let X be a compact Shimura curve 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 resolves 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.

**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!