University of Illinois, Urbana-Champaign
Number Theory Seminar
Fall 2024

This is the homepage for the UIUC Number Theory Seminar.

We will meet in Gregory Hall, Room 205 on Tuesdays at 11:00am.

Seminars from 2012-2022 are archived here.

• September 10:   Bruce Reznick (UIUC)
  Title:  Equal sums of two cubes of quadratic forms 
  Abstract:  Number theorists since Euler have been interested in finding parameterizations of equal sums of two cubes of rational numbers. We present a complete solution to the equation f_1^3 + f_2^3 = f_3^3 + f_4^3 over C[x,y] in quadratic forms f_i, as well as finding when there is a change of variables leading to forms in Q[x,y]. There are two “flips” of this equation: f_1^3 + (-f_3)^3 = (-f_2)^3 + f_4^3 and  f_1^3 + (-f_4)^3 = (-f_2)^3 + f_3^3.  One novelty of this solution is showing that in two of these three cases, there is a third representation of the sum as f_5^3 + f_6^3, where these are also binary quadratic forms. This work began as an attempt to understand an example of Ramanujan.
  
  
  

• September 24:   Scott Ahlgren (UIUC)
  Title:  Automorphic forms and the partition function
  Abstract:   The partition function p(n) counts the number of ways to break a positive integer into parts. Its values are the coefficients of a modular form of weight -1/2, and this opens the door to study properties of p(n) using the theory of automorphic forms.  There are two branches to this study; the analytic side involves Maass forms and spectral theory and the arithmetic side involves holomorphic modular forms and Galois representations.  In all cases the study can be viewed as a "testing ground"for more general theorems about modular forms.    I will discuss a number of results which have been proved with various collaborators (most of whom have an Illinois connection) in the last few years.
  
  
  

• October 1:   William Chen (UIUC)
  Title:  Geometry and arithmetic of noncongruence subgroups of SL(2,Z)
  Abstract:  Famously, SL(2,Z) does not have the congruence subgroup property. This means that it admits finite index subgroups which do not contain the kernel of the reduction map to SL(2,Z/n) for any n. Given the outstanding success of the congruence theory over the last century, it is natural to wonder if noncongruence subgroups may eventually aspire to the same success. In this talk I'll explain how a couple of exceptional isomorphisms in "low degree" leads to an understanding of noncongruence subgroups as capturing the geometry and arithmetic of punctured elliptic curves. While elliptic curves are abelian varieties, punctured elliptic curves are "anabelian" varieties in the sense of Grothendieck. A key observation is that noncongruence modular curves are moduli spaces for branched (possibly nonabelian) covers of elliptic curves, at most branched above the origin. This leads to a host of new questions, in topics ranging from discrete groups, Teichmuller dynamics, representation theory, algebraic geometry, and of course number theory. As time allows I'll describe some results and open problems.  
  
  
  

• October 8:  Maksym Radziwiłł (Northwestern)
  Title:  Trigonometric polynomials with multiplicative coefficients and small L^1 norms
  Abstract:  I will discuss joint work with Mayank Pandey in which we show that if a trigonometric polynomial has multiplicative coefficients and small L^1 norm then the coefficients have to be characters. One can view this as a variant of Littlewood's conjecture restricted to trigonometric polynomials with multiplicative coefficients.
  
  
  

• October 15:  Debmalya Basak and Cruz Castillo (UIUC)
  Title:  Surfaces Associated to Zeros of Automorphic L-functions
  Abstract:   Assuming the Riemann Hypothesis, Montgomery established results concerning pair correlation of zeros of the Riemann zeta function. Rudnick and Sarnak extended these results to automorphic L-functions and all level correlations. We show that automorphic L-functions exhibit additional geometric structures related to the correlation of their zeros. This is joint work with Alexandru Zaharescu.
  
  
  

• October 24:  Robert Lemke Oliver (Tufts)  (Thursday Talk!)
  Title:  The least prime whose Frobenius is in a rational equivalence class
  Abstract:  The Chebotarev density theorem is a powerful tool in number theory, in part because it guarantees the existence of primes whose Frobenius lies in a given conjugacy class in a fixed Galois extension of number fields.  However, for some applications, it is necessary to know not just that such primes exist, but to additionally know something about their size, say in terms of the degree and discriminant of the extension.  In this talk, I'll discuss forthcoming work with Cho and Zaman on the least prime with a specified Frobenius in a fixed Galois S_n extension.  Our approach is comparatively elementary, but when combined with existing results based on the zeros of L-functions, it leads to the strongest known bounds in this setting.
  
  
  

• November 12: Ziquan Zhuang (Johns Hopkins) (This is an Algebraic Geometry Seminar talk)
  Title:  Boundedness of singularities and discreteness of local volumes
  Abstract: The local volume of a Kawamata log terminal (klt) singularity is an invariant that plays a central role in the local theory of K-stability. By the stable degeneration theorem, every klt singularity has a volume preserving degeneration to a K-semistable Fano cone singularity. I will talk about a joint work with Chenyang Xu on the boundedness of Fano cone singularities when the volume is bounded away from zero. This implies that local volumes only accumulate around zero in any given dimension.
  
  
  

• November 19:  Carlo Pagano (Concordia University)
  Title:  Hilbert 10 via additive combinatorics
  Abstract: In 1970 Matiyasevich, building on earlier work of Davis--Putnam--Robinson, proved that every enumerable subset of Z is Diophantine, thus showing that Hilbert's 10th problem is undecidable for Z. The problem of extending this result to the ring of integers of number fields (and more generally to finitely generated infinite rings) has attracted significant attention and, thanks to the efforts of many mathematicians, the task has been reduced to the problem of constructing, for certain quadratic extension of number fields L/K, an elliptic curve E/K with rk(E(L))=rk(E(K))>0. In this talk I will explain joint work with Peter Koymans, where we use Green--Tao to construct the desired elliptic curves, settling Hilbert 10 for every finitely generated infinite ring.
  
  
  

• December 3:  Daniel Le (Purdue)
  Title:  The weight part of Serre's conjecture for GL_n and GSp_4
  Abstract:  The phenomenon of congruences between q-expansions of modular forms played a large role in the development of the theory of Galois representations and eventually to modularity. The weight part of Serre's conjecture asks when two cuspidal eigenforms of different weights can be congruent. A rather complete answer has been given in terms of Galois representations for classical and Hilbert modular forms. We will discuss recent generalizations to higher rank in several joint works with B.V. Le Hung, H. Lee, B. Levin, and S. Morra.