Department of


Seminar Calendar
for Number Theory Seminar events the year of Saturday, February 29, 2020.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
     January 2020          February 2020            March 2020     
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
           1  2  3  4                      1    1  2  3  4  5  6  7
  5  6  7  8  9 10 11    2  3  4  5  6  7  8    8  9 10 11 12 13 14
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Tuesday, January 14, 2020

11:00 am in 241 Altgeld Hall,Tuesday, January 14, 2020

Superimposing theta structure on a generalized modular relation

Atul Dixit (Indian Institute of Technology in Gandhinagar)

Abstract: By a modular relation for a certain function $F$, we mean that which is governed by the map $z\to -1/z$ but not necessarily by $z\to z+1$. Equivalently, the relation can be written in the form $F(\alpha)=F(\beta)$, where $\alpha\beta=1$. There are many generalized modular relations in the literature such as the general theta transformation $F(w,\alpha)=F(iw, \beta)$ or the Ramanujan-Guinand formula $F(z, \alpha)=F(z, \beta)$ etc. The latter, equivalent to the functional equation of the non-holomorphic Eisenstein series on $\mathrm{SL}_{2}(\mathbb{Z})$, admits a beautiful generalization of the form $F(z, w,\alpha)=F(z, iw, \beta)$, that is, one can superimpose theta structure on it.

Recently, a modular relation involving infinite series of the Hurwitz zeta function $\zeta(z, a)$ was obtained. It generalizes a result of Ramanujan from the Lost Notebook. Can one superimpose theta structure on it? While answering this question affirmatively, we were led to a surprising new generalization of $\zeta(z, a)$. We show that this new zeta function, $\zeta_w(z, a)$, satisfies a beautiful theory. In particular, it is shown that $\zeta_w(z, a)$ can be analytically continued to the whole complex plane except $z=1$. Hurwitz's formula for $\zeta(z, a)$ is also generalized in this setting. We also prove a generalized modular relation involving infinite series of $\zeta_w(z, a)$, which is of the form $F(z, w,\alpha)=F(z, iw, \beta)$. This is joint work with Rahul Kumar.

Thursday, January 23, 2020

11:00 am in 241 Altgeld Hall,Thursday, January 23, 2020

Heights and p-adic Hodge Theory

Lucia Mocz (University of Chicago)

Abstract: We discuss connections between p-adic Hodge theory and the Faltings height. Most namely, we show how new tools in p-adic Hodge theory can be used to prove new Northcott properties satisfied by the Faltings height, and demonstrate phenomenon which are otherwise predicted by various height conjectures. We will focus primarily on the Faltings height of CM abelian varieties where the theory can be made to be computational and explicit.

Thursday, January 30, 2020

11:00 am in 241 Altgeld Hall,Thursday, January 30, 2020

Modularity of some $\mathrm{PGL}_2(\mathbb{F}_5)$ representations

Patrick Allen (Illinois)

Abstract: Serre's conjecture, proved by Khare and Wintenberger, states that every odd two dimensional mod p representation of the absolute Galois group of the rationals comes from a modular form. This admits a natural generalization to totally real fields, but even the real quadratic case seems completely out of reach. I'll discuss some of the difficulties one encounters and then discuss some new cases that can be proved when p = 5. This is joint work with Chandrashekhar Khare and Jack Thorne.

Thursday, February 6, 2020

11:00 am in 241 Altgeld Hall,Thursday, February 6, 2020

The Kuznetsov formulas for GL(3)

Jack Buttcane (University of Maine)

Abstract: The Kuznetsov formulas for GL(2) connect the study of automorphic forms to the study of exponential sums. They are useful in a wide variety of seemingly unrelated problems in analytic number theory, and I will (briefly) illustrate this with a pair of examples: First, if we consider the roots v of a quadratic polynomial modulo a prime p, then the sequence of fractions v/p is uniformly distributed modulo 1; this is the “mod p equidistribution” theorem of Duke, Friedlander, Iwaniec and Toth. Second, the Random Wave Conjecture states that a sequence of automorphic forms should exhibit features of a random wave as their Laplacian eigenvalues tend to infinity. I will discuss their generalization to GL(3) and applications.

Thursday, February 13, 2020

11:00 am in 241 Altgeld Hall,Thursday, February 13, 2020

Divisors of integers, permutations and polynomials

Kevin Ford (Illinois Math)

Abstract: We describe a probabilistic model that describes the statistical behavior of the divisors of integers, divisors of permutations and divisors of polynomials over a finite field. We will discuss how this can be used to obtain new bounds on the concentration of divisors of integers, improving a result of Maier and Tenenbaum. This is joint work with Ben Green and Dimitris Koukoulopoulos.

Thursday, February 20, 2020

11:00 am in 241 Altgeld Hall,Thursday, February 20, 2020

The Third Moment of Quadratic L-Functions

Ian Whitehead (Swarthmore College)

Abstract: I will present.a smoothed asymptotic formula for the third moment of Dirichlet L-functions associated to real characters. Beyond the main term, which was known, the formula has an unexpected secondary term of size $X^{3/4}$ and an error of size $X^{2/3}$. I will give background on the multiple Dirichlet series techniques that motivated this result. And I will describe the new ideas about local and global multiple Dirichlet series that made the final, sieving step in the proof possible. This is joint work with Adrian Diaconu.

Thursday, February 27, 2020

11:00 am in 241 Altgeld Hall,Thursday, February 27, 2020

The shape of low degree number fields

Bob Hough (Stony Brook University)

Abstract: In his thesis, M. Bhargava proved parameterizations and identified local conditions which he used to give asymptotic counts for $S_4$ quartic and quintic number fields, ordered by discriminant. This talk will discuss results in an ongoing project to add detail to Bhargava's work by considering in addition to the field discriminant, the lattice shape of the ring of integers in the canonical embedding, and by giving strong rates with lower order terms in the asymptotics. These results build on earlier work of Taniguchi-Thorne, Bhargava-Shankar-Tsimerman and Bhargava Harron.

Thursday, March 5, 2020

11:00 am in 241 Altgeld Hall,Thursday, March 5, 2020

Potential automorphy of Galois representations into general spin groups

Shiang Tang (Illinois Math)

Abstract: Given a connected reductive group $G$ defined over a number field $F$, the Langlands program predicts a connection between suitable automorphic representations of $G(\mathbb A_F)$ and geometric $p$-adic Galois representations $\mathrm{Gal}(\overline{F}/F) \to {}^LG$ into the L-group of $G$. Striking work of Arno Kret and Sug Woo Shin constructs the automorphic-to-Galois direction when $G$ is the group $\mathrm{GSp}_{2n}$ over a totally real field $F$, and $\pi$ is a cuspidal automorphic representation of $\mathrm{GSp}_{2n}(\mathbb A_F)$ that is discrete series at all infinite places and is a twist of the Steinberg representation at some finite place: To such a $\pi$, they attach geometric $p$-adic Galois representations $\rho_{\pi}: \mathrm{Gal}(\overline{F}/F) \to \mathrm{GSpin}_{2n+1}$. In this work we establish a partial converse, proving a potential automorphy theorem, and some applications, for suitable $\mathrm{GSpin}_{2n+1}$-valued Galois representations. In this talk, I will explain the background materials and the known results in this direction before touching upon the main theorems of this work.

Thursday, April 2, 2020

11:00 am in 241 Altgeld Hall,Thursday, April 2, 2020

To Be Announced

Zarathustra Brady (MIT)

Tuesday, April 7, 2020

11:00 am in 241 Altgeld Hall,Tuesday, April 7, 2020

To Be Announced

Amita Malik (Rutgers University)

Thursday, April 16, 2020

11:00 am in 241 Altgeld Hall,Thursday, April 16, 2020

To Be Announced

Christelle Vincent (University of Vermont)

Thursday, April 30, 2020

11:00 am in 241 Altgeld Hall,Thursday, April 30, 2020

To Be Announced

Eric Stubley (University of Chicago)