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*Annals of Mathematics*, vol 185 (2017). arXiv version. - "Averages of l-torsion in class groups of number fields," with L. Pierce and M.M. Wood, submitted. arXiv version.
- "Algebraic structures on cohomology of configuration spaces of manifolds with flows," with J. Wiltshire-Gordon, submitted. arXiv version.
- "Homology of FI-modules," with T. Church, to appear,
*Geometry and Topology*. arXiv version. - "Detection of planted solutions for flat satisfiability problems," with Q. Berthet, submitted. arXiv version.
- "Furstenberg sets and Furstenberg schemes over finite fields," with D. Erman, to appear,
*Algebra and Number Theory*. arXiv version. - "An incidence conjecture of Bourgain over fields of positive characteristic," with M. Hablicsek,
*Forum of Math, Sigma*, vol 4 (2016). arXiv version. (Blog post) - "Representation stability in cohomology and asymptotics for families of varieties over finite fields," with T. Church and B. Farb,
*Contemporary Mathematics*620 (2014) 1-54. arXiv version. - "Homological stability for Hurwitz spaces and the Cohen Lenstra conjecture over function fields, II," with A. Venkatesh and C. Westerland, submitted. arXiv version. (Blog post)
**WARNING**(posted 18 Sept 2013): We have been made aware that there is a gap in Section 12 of the above paper (Homological stability ... II) namely, the application to Cohen-Lenstra. This also affects Section 6 and some theorems stated in the introduction. More precisely, the homological stabilization maps used in this paper and the previous paper are not exactly the same. At present, we have been able to fix this under some additional restrictions on boundary monodromy, but not in general. For a detailed descripton of the error in the paper and a guide to which parts of the paper are correct, see this blog post. A modified version will be uploaded to the arXiv at some point. Until then, please contact the authors if you need more information. - "FI-modules over Noetherian rings," with T. Church, B. Farb, and R. Nagpal,
*Geometry and Topology*18-5, 2951-2984 (2014). arXiv version. (Blog post) - "Superstrong approximation for monodromy groups," in
*Thin Groups and Superstrong Approximation*, MSRI Publication series 61 (2014) arXiv version. (Blog post) - "FI-modules: a new approach to stability for S_n-representations," with T. Church and B. Farb,
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*Duke Math J.*161 (3) (2012) (former title: "Every curve is a Teichmuller curve.") (Blog post) - "Understanding persistent homology and Plex using a networking dataset," with L. Balzano, preprint.
- "Modeling lambda invariants by p-adic random
matrices," with S. Jain and A. Venkatesh,
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*Group Theory (Granville, OH, 1992)*, 250--261, World Sci. Publishing, River Edge, NJ (1993)

We report on a pilot study of the topological structure of data sets using Plex, a software package for computational homology. Plex assigns to a set of points in R^n a "barcode," which is intended to reveal topological structure in data. In particular, if the data are drawn from a low-dimensional manifold M, the barcode is meant to capture the homology of M. We compared barcodes coming from three sources: bytecount data from the UW-Madison core computer network, Gaussian noise centered at a point, and Gaussian noise convolved with a circle. One of our goals is to understand the result of Plex needs to be in order to distinguish a data set with topological structure from one consisting solely of noise.

We model the variation of lambda-invariants of imaginary quadratic fields by statistics of p-adic random matrices, and test the resulting predictions against numerical data.

We discuss some problems of arithmetic distribution, including conjectures of Cohen-Lenstra, Malle, and Bhargava; we explain how such conjectures can be heuristically understood for function fields over finite fields, and discuss a general approach to their proof in the function field context based on the topology of Hurwitz spaces. This approach also suggests that the Schur multiplier plays a role in such questions over number fields.

We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This topological theorem has the following arithmetic consequence: let l > 2 be prime and A a finite abelian l-group. Then there exists Q = Q(A) such that, for q greater than Q and not congruent to 1 modulo l, a positive fraction of quadratic extensions of Fq(t) have the l-part of their class group isomorphic to A.

We study the variation of Mordell-Weil ranks in the Jacobians of curves in a pro-p tower over a fixed number field. In particular, we show that under mild conditions the Mordell-Weil rank of a Jacobian in the tower is bounded above by a constant multiple of its dimension. In the case of the tower of Fermat curves, we show that the constant can be taken arbitrarily close to 1. The main result is used in the forthcoming paper of Guillermo Mantilla-Soler on the Mordell-Weil rank of the modular Jacobian J(Np^m).

We discuss Linnik's work on the distribution of integral solutions to x^2+y^2+z^2 =d, as d goes to infinity. We give an exposition of Linnik's ergodic method; indeed, by using large-deviation results for random walks on expander graphs, we establish a refinement of his equidistribution theorem. We discuss the connection of these ideas with modern developments (ergodic theory on homogeneous spaces, L-functions).

Write G_S(p) for the Galois group of the maximal pro-p extension of Q unramified away from a set of primes S not including p. We describe a conjectural probability distribution on the isomorphism class of G_S(p) when S is a random set of g primes. This can be thought of as a non-abelian pro-p version of the Cohen-Lenstra heuristic. We present two heuristic arguments leading to the same conjecture, one in the spirit of Cohen-Lenstra and the other based on analogies between number fields and function fields.

**"Every curve is a Teichmuller curve,"** with D. B. McReynolds, submitted.

In a previous paper, we proved that the equation A^4 + B^2 = C^p has no nontrivial solutions when p is a prime no less than 211. In the present paper, we show that the equations A^4 + B^2 = C^n and A^4 + 2 B^2 = C^n have no nontrivial solutions for n >= 4 apart from 1^4 + 2*11^2 = 3^5. The proof involves a mixture of geometric, analytic, and computational techniques.

In a previous paper with A. Venkatesh, we gave an upper bound on GRH for the order of the p-torsion subgroup of the ideal class group of a number field K, which could be made unconditional in certain cases (for instance, the 3-torsion in the class group of a quadratic field.) In this note, we observe that those bounds could be (slightly) sharpened if one had some control over the number of points on P^1(K) whose height is bounded by some small positive power of the discriminant of K. This problem has apparently not been seriously investigated, apart from some numerical experiments carried out by D. Lecoanet.

.pdf version, as appearing in published volume

log N(X) < C log X exp(D sqrt(log log X))

**"Asymptotics of coinvariants of Iwasawa modules under non-normal subgroups,"** with A. Logan, *Math. Res. Lett.* 14, no.5, 769-773 (2007)

Let G be a subgroup of S_n. Malle has conjectured that the number of number fields which have degree n over Q, Galois group G, and discriminant between -X and X is asymptotic to C X^a (log X)^b where a and b are constants depending on G. We study the problem of counting extensions of F_q(T) with the above properties; this is essentially a problem of counting F_q-rational points on Hurwitz varieties. We show that the heuristic "an irreducible d-fold over F_q has q^d rational points" yields the analogue of Malle's conjecture, with identical values of a and b. Moreover, the function field setting suggests more general heuristics about the distribution of discriminants of number fields. (Remark: in this paper we count only extensions of F_q(T) containing no constant field extension. There are some situations in which the fields with nontrivial constant subextensions actually dominate the ones we count, contrary to the remark on the top of p.154 in our paper.)

**"Serre's conjecture over
F _{9},"**

**"On the average number of octahedral modular forms,"** *Math. Res. Lett.* 10, 269--273 (2003)

PUBLISHED VERSION .dvi version .pdf version

**"Endomorphism algebras of Jacobians,"** * Adv. Math.* 162, 243--271 (2001)

Van der Geer and Oort have written:

Previous works of Brumer, Mestre, Ekedahl-Serre, and others have justified this expectation by providing examples of families of curves whose Jacobians have large endomorphism rings. We give a general procedure for producing families of branched covers of the line whose Jacobians have extra endomorphisms. We show that many of the examples produced by the above authors are "explained" in this way, and produce some new examples. For instance, we obtain curves whose Jacobians have real multiplication by the index-n subfield of"...one expects excess intersection of the Torelli locus and the loci corresponding to abelian varieties with very large endomorphism rings; that is, one expects that they intersect much more than their dimensions suggest."

**"On the modularity of Q-curves,"**,with C. Skinner, *Duke Math. J.* 109, no. 1, 97--122 (2001)

**"Congruence ABC implies ABC,"** *Indag. Math.*, N.S., 11 (2), 197--200 (2000)

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Jordan Ellenberg * ellenber@math.wisc.edu * revised 27 May 2010