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.Comment: 54 pages. v4: new title, paper completely reorganized; final version, to appear in Duke Math Journa

In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of F n q with no three terms in arithmetic progression by c n with c < q. For q = 3, the problem of finding the largest subset of F n 3 with no three terms in arithmetic progression is called the cap problem. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz [BK12], was on order n −1−ǫ 3 n .The problem of finding large subsets of an abelian group G with no three-term arithmetic progression, or of finding upper bounds for the size of such a subset, has a long history in number theory. The most intense attention has centered on the cases where G is a cyclic group Z/NZ or a vector space (Z/3Z) n , which are in some sense the extreme situations. We denote by r 3 (G) the maximal size of a subset of G with no three-term arithmetic progression. The fact that r) was first proved by Brown and Buhler [BB82], which was improved to O(3 n /n) by Meshulam [Mes95]. The best known upper bound, O(3 n /n 1+ǫ ), is due to Bateman and Katz [BK12]. The best lower bound, by contrast, is around 2.2 n [Ede04].The problem of arithmetic progressions in (Z/3Z) n has sometimes been seen as a model for the corresponding problem in Z/NZ. We know (for instance, by a construction of Behrend [Beh46]) that r 3 (Z/NZ) grows more quickly than N 1−ǫ for every ǫ > 0. Thus it is natural to ask whether r 3 ((Z/3Z) n ) grows more quickly than (3 − ǫ) n for every ǫ > 0. In general, there has been no consensus on what the answer to this question should be.In the present paper we settle the question, proving that for all odd primes p, r 3 ((Z/pZ) n ) 1/n is bounded away from p as n grows.The main tool used here is the polynomial method, in particular the use of the polynomial method developed in the breakthrough paper of Croot, Lev, and Pach [CLP16], which drastically improved the best known upper bounds for r 3 ((Z/4Z) n ). In this case, they show that a subset of G with no three-term arithmetic progression has size at most c n for some c < 4. In the present paper, we show that the ideas of their paper can be extended to vector spaces over a general finite field.

Abstract. 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, a positive fraction of quadratic extensions of Fq(t) have the ℓ-part of their class group isomorphic to A.

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 nrepresentations. 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.

We introduce a new method to bound -torsion in class groups, combining analytic ideas with reflection principles. This gives, in particular, new bounds for the 3-torsion part of class groups in quadratic, cubic and quartic number fields, as well as bounds for certain families of higher degree fields and for higher . Conditionally on GRH, we obtain a nontrivial bound for -torsion in the class group of a general number field.