Philadelphia Area Number Theory Seminar

Underlying a lot of modern number theory is the philosophy that arithmetic quantities for which no obvious reason for correlation exists should indeed be uncorrelated in a precise quantitative sense. A classical example is provided by the square-root cancellation in exponential sums such as the quadratic Gauss sums (which feature in the proof of quadratic reciprocity) or Kloosterman sums. Polygonal paths traced by their normalized incomplete sums give a fascinating insight into their chaotic formation. In this talk, we will present our recent results describing the limiting shape distribution in two ensembles of Gauss and Kloosterman sum paths as well as related results on sums of products of Kloosterman sums.

Dubi Kelmer (Boston College and Princeton)

Following Margulis's proof of the Oppenheim conjecture we know that integer values of an irrational indefinite quadratic form in n >= 3 variables are dense on the real line. The same is true for an inhomogeneous form obtained by shifting values by a fixed vector if either the form or the shift is irrational.  In this talk, I will describe several approaches to this problem that give effective results that hold for a fixed rational form Q and almost all shifts, by reducing it to the density of certain orbits of a discrete group acting on the torus. I will then describe different approaches using dynamics, representation theory, and estimates on exponential sums for this problem.

Sachi Hashimoto (Brown University)

Modular curves are, roughly speaking, curves whose points parameterize elliptic curves with extra structure on their torsion points. An open question in number theory is to find all rational points on all modular curves: this is known as "Mazur's program B". We will discuss some questions related to and inspired by Mazur's program B. In particular, we will discuss how to parameterize the set of all rational points on modular curves, conditional on a conjecture of Zywina. Using our parameterization we try to answer the following question: to what extent do the rational points on modular curves come from the intrinsic geometry of the curves? This is joint work with Maarten Derickx, Filip Najman, and Ari Shnidman.

We investigate subsets A of the natural numbers having the property that, for some positive number p < 2, one has

   int_0^1 | sum_{n in A\cap [1,N]} e(n alpha) |^p d alpha

   << | A\cap [1,N] |^p N^{eps-1}.

Examples of such sets include (but are not restricted to) the square-free, or more generally, the r-free numbers. We show that there are many other examples of such sets. For polynomials 

   psi(x; a) = a _kx^k + … + a_1x,

having coefficients a_i satisfying suitable irrationality conditions, we obtain Weyl-type estimates for associated exponential sums restricted to subconvex L^p sets, and we show that the sequence (psi(n; a))_{n in A} is equidistributed modulo 1. We also discuss applications to averages of arithmetic functions.

Fernando Trejos, Princeton University