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).Comment: 8 page

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.Comment: 10 pages; to appear, Canad. Math. Bull. v2: error corrected (see abstract

Convergence rates for ordinal embedding

We prove optimal bounds for the convergence rate of ordinal embedding (also known as non-metric multidimensional scaling) in the 1-dimensional case. The examples witnessing optimality of our bounds arise from a result in additive number theory on sets of integers with no three-term arithmetic progressions. We also carry out some computational experiments aimed at developing a sense of what the convergence rate for ordinal embedding might look like in higher dimensions

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

Homology of FI-modules

We prove an explicit and sharp upper bound for the Castelnuovo-Mumford regularity of an FI-module V in terms of the degrees of its generators and relations. We use this to refine a result of Putman on the stability of homology of congruence subgroups, extending his theorem to previously excluded small characteristics and to integral homology while maintaining explicit bounds for the stable range.Comment: 34 pages. v2: major reorganization; to appear in Geometry and Topolog

The number of extensions of a number field with fixed degree and bounded discriminant

We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant less than X; these bounds improve on work of Schmidt. We also prove various related results, such as lower bounds for the number of extensions and upper bounds for Galois extensions

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)Comment: 17 page

On large subsets of FqnF_q^n with no three-term arithmetic progression

In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of $F_q^n$ with no three terms in arithmetic progression by $c^n$ with $c < q$. For $q=3$, the problem of finding the largest subset with no three terms in arithmetic progression is called the `cap problem'. Previously the best known upper bound for the cap problem, due to Bateman and Katz, was $O(3^n / n^{1+\epsilon})$.Comment: 4 pages. This paper supersedes arXiv:1605.05492 and combines the solutions to the cap set problem independently obtained by the two author

Detection of Planted Solutions for Flat Satisfiability Problems

We study the detection problem of finding planted solutions in random instances of flat satisfiability problems, a generalization of boolean satisfiability formulas. We describe the properties of random instances of flat satisfiability, as well of the optimal rates of detection of the associated hypothesis testing problem. We also study the performance of an algorithmically efficient testing procedure. We introduce a modification of our model, the light planting of solutions, and show that it is as hard as the problem of learning parity with noise. This hints strongly at the difficulty of detecting planted flat satisfiability for a wide class of tests

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.Comment: 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 Revisions, this version: some typos fixed and references added, minor revisions per suggestions of Igor Pak, Emmanuel Breuillard, and referee; this version will be the one that appears in the conference proceeding