University of Illinois, Urbana-Champaign
Number Theory Seminar
Fall 2023

This is the homepage for the UIUC Number Theory Seminar.

We will meet in Literature, Cultures, and Linguistics Building, Room G-32 (Directions from Altgeld Hall).

Seminars from 2012-2022 are archived here.

• August 29:  Jesse Thorner (UIUC)
  Title:  A new zero-free region for Rankin-Selberg L-functions
  Abstract:  I will present a new zero-free region for GL(1)-twists of GL(m)xGL(n) Rankin-Selberg L-functions.  The proof is inspired by Siegel's celebrated lower bound for Dirichlet L-functions at s = 1.  This is joint work with Gergely Harcos.
  
  
  

• September 5:  Valeriya Kovaleva (Université de Montréal)  (This talk will take place over Zoom!)
  Title:  Traces of random matrices over finite fields and cancellation in character sums
  Abstract:  Let X be a matrix drawn uniformly at random from GL_n(F_q), then one may conjecture that traces of powers of such matrices Tr(X^k) should have an asymptotically uniform distribution on Fq. Further, one may wonder how robust this phenomenon is and how large can k be. On the one hand, this question is an analogue of a classic problem from random matrix theory, and, on the other hand, it is intimately related to short character sums over function fields with the power k serving as the conductor. In our work, we prove that the distribution of Tr(X^k) is indeed asymptotically uniform and that the respective short interval character sums exhibit cancellation for k = q^{o(n^2)}. This is a much wider range than one could hope to obtain for general characters, and in fact, this phenomenon seems to have no analogue over the integers. This is joint work with Ofir Gorodetsky.
  
  
  

• September 12:  Alexandru Zaharescu (UIUC) and  Jack  Anderson (UIUC)
  Title:  Distribution of angles to lattice points seen from a fast moving observer
  Abstract:  We consider a square expanding with constant speed seen from an observer moving away with constant acceleration, and study the distribution of angles between rays from the observer towards the lattice points in the square. We prove the existence of the gap distribution as time tends to infinity and provide explicit formulas for the corresponding density function. This is joint work with Florin Boca and Cristian Cobeli.
  
  
  

• September 19:  Yuan Liu (UIUC)
  Title:  Recent progress in the nonabelian Cohen--Lenstra heuristics
  Abstract:  The nonabelian Cohen—Lenstra program studies the distribution of the Galois group of maximal unramified extensions of a family of global fields. In this talk, we will first introduce the history of this area, which includes the algebraic number theory background knowledge and the original (abelian) Cohen—Lenstra heuristics on the distribution of class groups. Then we will review the progress made in the recent decade, and discuss the future questions and directions.
  
  
  

• September 26:  Kunjakanan Nath (UIUC)
  Title:  On binary problems in analytic number theory
  Abstract:  Given any two sequences {a(n)}_{n=1}^\infty and {b(n)}_{n=1}^\infty of  ''number-theoretic'' interest, consider a binary problem of the form S(N):=\sum_{n=1}^{N} a(n)b(n). In general, it is an extremely difficult question to evaluate the sum S(N). In this talk, we will give a few examples to demonstrate the application of Fourier analysis in conjunction with the arithmetic structure of the given sequence and the bilinear form method to estimate the sum S(N).
  
  
  

• October 3:  Peter Zenz (Stanford and Brown)
  Title:  On real zeros of holomorphic Hecke cusp forms
  Abstract:  In this talk we are going to explore so-called “real” zeros of holomorphic Hecke cusp of large weight on the modular surface. Gosh and Sarnak established that the number of real zeros tends to infinity as the weight k goes to infinity. To do so, they studied the behavior of holomorphic Hecke cusp forms close to the cusp, i.e. for z = x + iy with y > k^{1/2}. In this region the cusp form is well approximated by a single Fourier coefficient and the problem of finding real zeros boils down to studying sign changes of Fourier coefficients of holomorphic Hecke cusp forms. Low in the fundamental domain, say for 1 ≤ y ≤ 2, investigating sign changes of Fourier coefficients is not sufficient and the problem is less understood. In this talk we explain ongoing work on detecting real zeros low in the fundamental domain on average over a large Hecke basis. The averaging allows us to establish an almost sharp quantitative bound for the number of real zeros, compared to what is predicted by a random model for holomorphic Hecke cusp forms.
  
  
  

• October 10:  William Chen (Rutgers)
  Title:  The dark side of the modular group SL(2,Z)
  Abstract:  The group SL(2,Z) has a number of dual natures. In group theory, it is the (special) automorphism group of Z^2, as well as the special outer automorphism group of a free group of rank 2. In surface topology, it is the mapping class group of both a torus as well as a punctured torus. In the theory of arithmetic groups, it is also special in that it has both congruence, and noncongruence subgroups. While the abelian/unpunctured/congruence side is now well understood, and has led to the profoundly successful theory of congruence modular curves and modular forms, the nonabelian/punctured/noncongruence occupies a rather mysterious spot that manages to simultaneously, and just barely, escape the reach of a number of seemingly relevant results. In this talk we will explain some interesting questions and conjectures in the area, and describe some connections with topics such as combinatorial group theory, representation theory, and number theory.
  
  
  

• October 17:  Qihang Sun (UIUC)
  Title:  Asymptotics in partitions: circle method and Kloosterman sums
  Abstract:   In 1917, Hardy and Ramanujan established an asymptotic formula for the integer partition function p(n). Rademacher later proved that this formula converges to p(n) when summed to infinity. The concept of rank for partitions was introduced by Dyson to explain Ramanujan's congruences. Since then, the asymptotic properties of these rank functions and their connections with modular forms have become a vast area of research. In this talk, we will explore the recent progresses related to the rank of partitions, delving into both the circle method and Kloosterman sums.
  
  
  

• October 24:   No talk.
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• October 31:  Lasse Grimmelt (Oxford)  (This talk will take place over Zoom.)
  Title:  Primes in large arithmetic progressions
  Abstract:  One important topic in analytic prime number theory is the study of the distribution of primes in arithmetic progressions. In the case of large arithmetic progressions, sums of Kloosterman sums and the spectral theory of automorphic forms are the main tools. In this talk I will give the background behind this connection and explain how my coauthors V. Blomer, J. Li, S. Rydin Myerson and I applied it when considering additive problems with almost prime squares. The aim will be to not dive into the technical details, but instead to get an intuitive understanding of why the seemingly unrelated area of automorphic forms is so effective in this application.
  
  
  

• November 7:  Nicole Looper (UIC)
  Title:   Dynamical Arakelov-Green functions in higher dimensions and arithmetic applications
  Abstract:  This talk will discuss the connection between Arakelov-Green functions and the arithmetic of dynamical systems. For curves, this connection is best known in terms of arithmetic on their Jacobian varieties and the equidistribution of points of small canonical height with respect to rational functions on P^1. For higher dimensional projective spaces, the situation is much more complex and remains relatively uncharted. In this talk, I will discuss how the Fekete-Leja transfinite diameter is a promising avenue for future progress on this issue, allowing for a generalization of Arakelov-Green functions to the higher-dimensional setting that admits similar quantitative bounds.
  
  
  

• November 14:  Dimitris Koukoulopoulos (Université de Montréal)
  Title:   The mean value of the Erdos-Hooley Delta function
  Abstract:  The Erdos-Hooley Delta function is defined for positive integers n as Delta(n) = sup_{u\in\mathbb{R}} #{d|n : e^u < d ≤  e^{u+1}}. In a seminal 1979 paper, Hooley proved that estimates of its mean value can be exploited to count solutions to certain Diophantine equations. In this talk, I will present recent work, joint with Kevin Ford and Terence Tao, that establishes improved upper and lower bounds on the mean value of Delta.
  
  
  

• November 21:  Thanksgiving Break
  
  
  

• November 28:  Caroline Turnage-Butterbaugh (Carleton College)
  Title:  Moments of Dirichlet L-functions  
  Abstract:  In recent decades there has been much interest and measured progress in the study of moments of the Riemann zeta-function and, more generally, of various L-functions. Despite a great deal of effort spanning over a century, asymptotic formulas for moments of L-functions remain stubbornly out of reach in all but a few cases. In this talk, we consider the problem for the family of all Dirichlet L-functions of even primitive characters of bounded conductor. I will outline how to harness the asymptotic large sieve to prove an asymptotic formula for the general 2kth moment of an approximation to this family. The result, which assumes the generalized Lindelöf hypothesis for large values of k, agrees with the prediction of Conrey, Farmer, Keating, Rubenstein, and Snaith. Moreover, it provides the first rigorous evidence beyond the so-called “diagonal terms” in their conjectured asymptotic formula for this family of L-functions.