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Jordan S. Ellenberg

Publications and source records attributed to Jordan S. Ellenberg.

At least 37 records · Page 2Linked to original sources
arXiv2015

Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This 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 F_q(t) have the l-part of their class group isomorphic to A.

math.NT↗
arXiv2015

Furstenberg sets and Furstenberg schemes over finite fields

We give a lower bound for the size of a subset of $\mathbb F_q^n$ containing a rich k-plane in every direction, a k-plane Furstenberg set. The chief novelty of our method is that we use arguments on non-reduced subschemes and flat families to derive combinatorial facts about incidences between points and k-planes in space.

math.AG↗
arXiv2014

FI-modules and stability for representations of symmetric groups

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tautological ring of the moduli space of n-pointed curves - the space of polynomials on rank varieties of n x n matrices - the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module.

math.RT↗
arXiv2014

FI-modules over Noetherian rings

FI-modules were introduced by the first three authors in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of S_n-representations. In this paper we prove the Noetherian property for FI-modules over arbitrary Noetherian rings: any sub-FI-module of a finitely generated FI-module is finitely generated. This lets us extend many of the results of [CEF] to representations in positive characteristic, and even to integral coefficients. We focus on three major applications of the main theorem: on the integral and mod p cohomology of configuration spaces; on diagonal coinvariant algebras in positive characteristic; and on an integral version of Putman's central stability for homology of congruence subgroups.

math.RT↗
arXiv2013

Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II

We prove a version of the Cohen--Lenstra conjecture over function fields (completing the results of our prior paper). This is deduced from two more general theorems, one topological, one arithmetic: We compute the direct limit of homology, over puncture-stabilization, of spaces of maps from a punctured manifold to a fixed target; and we compute the Galois action on the set of stable components of Hurwitz schemes.

math.NT↗
arXiv2013

An incidence conjecture of Bourgain over fields of positive characteristic

In this note we generalize a recent theorem of Guth and Katz on incidences between points and lines in $3$-space from characteristic $0$ to characteristic $p$, and we explain how some of the special features of algebraic geometry in characteristic $p$ manifest themselves in problems of incidence geometry.

math.CO↗
arXiv2013

Representation stability in cohomology and asymptotics for families of varieties over finite fields

We consider two families X_n of varieties on which the symmetric group S_n acts: the configuration space of n points in C and the space of n linearly independent lines in C^n. Given an irreducible S_n-representation V, one can ask how the multiplicity of V in the cohomology groups H*(X_n;Q) varies with n. We explain how the Grothendieck-Lefschetz Fixed Point Theorem converts a formula for this multiplicity to a formula for the number of polynomials over F_q (or maximal tori in GL_n(F_q), respectively) with specified properties related to V. In particular, we explain how representation stability in cohomology, in the sense of [CF, arXiv:1008.1368] and [CEF, arXiv:1204.4533], corresponds to asymptotic stability of various point counts as n goes to infinity.

math.GT↗
arXiv2013

Superstrong approximation for monodromy groups

This document is an expanded version of a lecture presented at a conference on "Thin Groups and Superstrong Approximation" held at the Mathematical Sciences Research Institute in February 2012. Superstrong approximation is a criterion on a finitely generated group, saying that certain Cayley graphs associated to finite quotients of the group form an expander family. In recent years, our knowledge about superstrong approximation for infinite-index Zariski-dense subgroups of arithmetic lattices ("thin groups") has drastically improved. We briefly survey the construction of monodromy groups, discuss our (limited) knowledge about whether such groups are thin, and discuss an application to arithmetic geometry (see the paper "Expander graphs, gonality, and variation of Galois representations") deriving from recent advances in superstrong approximation. We conclude by indulging in some speculations about more general contexts, asking: what are the interesting questions about "nonabelian superstrong approximation" and "superstrong approximation for Galois groups?" We discuss the relation of these notions with the Product Replacement Algorithm and the Bogomolov property for infinite algebraic extensions of number fields.

math.NT↗
arXiv2012

Random pro-p groups, braid groups, and random tame Galois groups

We introduce a heuristic prediction for the distribution of the isomorphism class of the Galois group of the maximal pro-p extension of Q unramified outside a "random" set of primes. This is guided by reasoning similar to that governing the Cohen-Lenstra conjectures. We conclude by describing theoretical and experimental evidence for our heuristic.

math.GR↗
arXiv2012

Random Dieudonne modules, random p-divisible groups, and random curves over finite fields

We describe a probability distribution on isomorphism classes of principally quasi-polarized p-divisible groups over a finite field k of characteristic p which can reasonably be thought of as "uniform distribution," and we compute the distribution of various statistics (p-corank, a-number, etc.) of p-divisible groups drawn from this distribution. It is then natural to ask to what extent the p-divisible groups attached to a randomly chosen hyperelliptic curve (resp. curve, resp. abelian variety) over k are uniformly distributed in this sense. For instance, one can ask whether the proportion of genus-g curves over F_p whose Jacobian is ordinary approaches the limit that such a heuristic would predict. This heuristic is analogous to conjectures of Cohen-Lenstra type for fields k of characteristic other than p, in which case the random p-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over F_3 appear substantially less likely to be ordinary than hyperelliptic curves over F_3.

math.NT↗
arXiv2010

Upper bounds for the growth of Mordell-Weil ranks in pro-p towers of Jacobians

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).

math.NT↗
arXiv2010

Linnik's ergodic method and the distribution of integer points on spheres

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).

math.NT↗
arXiv2009

Positive motivic measures are counting measures

Let K be a field. A positive motivic measure on the Grothendieck ring K_0(Var_K) is a homomorphism from K_0(Var_K) to the real numbers assigning a nonnegative value to every variety. In this note we show that the only positive motivic measures are the counting measures: measures on K_0(Var_{F_q}) which send a variety to its number of rational points over some fixed finite extension of F_q.

math.AG↗
arXiv2005

A sharp diameter bound for unipotent groups of classical type over Z/pZ

The unipotent subgroup of a finite group of Lie type over a prime field Z/pZ comes equipped with a natural set of generators; the properties of the Cayley graph associated to this set of generators have been much studied. In the present paper, we show that the diameter of this Cayley graph is bounded above and below by constant multiples of np + n^2 log p, where n is the rank of the associated Lie group. This generalizes a result of the first author, which treated the case of SL_n(Z/pZ). (Keywords: diameter, Cayley graph, finite groups of Lie type. AMS classification: 20G40, 05C25)

math.GR↗
arXiv2005

On the error term in Duke's estimate for the average special value of L-functions

Let F be an orthonormal basis of weight 2 cusp forms on Gamma_0(N). We show that various weighted averages of special values L(f \tensor chi, 1) over f in F are equal to 4 pi + O(N^{-1 + epsilon}). A previous result of Duke gives an error term of O(N^{-1/2} log N). The bound here is used in the author's paper "Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p," (to appear, Amer. J. Math.) to show that certain spaces of cuspforms arising there contain forms whose L-functions have nonvanishing special value. Version of May 2005: Nathan Ng found an error in the earlier version which yielded a bound too strong by a factor of log N; this is the corrected version, as it will appear in Canad. Math. Bull. The change does not affect the application to the Amer. J. Math. paper.

math.NT↗
arXiv2003

Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p

We show that the mod p Galois representations attached to a Q-curve E of degree d over an imaginary quadratic number field K are surjective for all p larger than some constant M_{K,d}, if E has potentially multiplicative reduction at any prime not dividing 6. The proof uses Mazur's formal immersion method, a result of Darmon and Merel on non-split modular curves, and an analytic argument showing that Jacobians of certain twisted modular curves admit quotients with Mordell-Weil rank 0. In combination with a previous theorem of the author and Skinner on modularity of Q-curves, the surjectivity result allows one to show that the generalized Fermat equation A^4 + B^2 = C^p has no nontrivial primitive solutions for p >= 211.

math.NT↗