Department of

Mathematics


Seminar Calendar
for Number Theory Seminar events the year of Thursday, July 2, 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.
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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 amThursday, April 2, 2020

Poisson imitators and sieve theory

Zarathustra Brady (MIT)

Abstract: I'll describe how sieve theory is actually a question about probability distributions whose low moments agree with the low moments of Poisson distributions. In particular, we can derive Selberg’s “parity problem” without using properties of the Möbius function or the Liouville function - instead, we use the fact that the alternating group forms a subgroup of the symmetric group.

Thursday, April 16, 2020

11:00 am in Zoom (email Patrick Allen for the meeting ID and password),Thursday, April 16, 2020

The Wiles defect for Hecke algebras that are not complete intersections

Jeff Manning (University of California at Los Angeles)

Abstract: In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings R->T to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod p Galois representation at non-minimal level were isomorphic and complete intersections, provided the same was true at minimal level. In addition to proving modularity theorems, this numerical criterion also implies a connection between the order of a certain Selmer group and a special value of an L-function. In this talk I will consider the case of a Hecke algebra acting on the cohomology a Shimura curve associated to a quaternion algebra. In this case, one has an analogous map of ring R->T which is known to be an isomorphism, but in many cases the rings R and T fail to be complete intersections. This means that Wiles' numerical criterion will fail to hold. I will describe a method for precisely computing the extent to which the numerical criterion fails (i.e. the 'Wiles defect"), which will turn out to be determined entirely by local information at the primes dividing the discriminant of the quaternion algebra. This is joint work with Gebhard Bockle and Chandrashekhar Khare.

Thursday, April 23, 2020

11:00 am in Zoom (email Patrick Allen for the meeting ID and password),Thursday, April 23, 2020

Log-free zero density estimates for automorphic L-functions

Chen An (Duke University)

Abstract: One of the most important topics in number theory is the study of zeros of L-functions. Near the edge of the critical strip, one may show that the number of zeros for certain L-functions is small; such a result is called a zero density estimate. For Dirichlet L-functions, this topic is well understood by the work of Gallagher, Selberg, Jutila, etc. For families of automorphic L-functions, Kowalski and Michel show that the number of zeros near the edge of the critical strip is small on average. The proof uses a large sieve inequality with key objects called pseudo-characters. I will present my recent progress on the refinement of Kowalski-Michel's large sieve inequality, which gives rise to a better zero density estimate for automorphic L-functions.

Thursday, April 30, 2020

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

Locally Split Galois Representations and Hilbert Modular Forms of Partial Weight One

Eric Stubley (University of Chicago)

Abstract: The Galois representation attached to a p-ordinary eigenform is upper triangular when restricted to a decomposition group at p. A natural question to ask is under what conditions this upper triangular decomposition splits as a direct sum. Ghate and Vatsal have shown that for Galois representations coming from families of p-ordinary eigenforms, the restriction to a decomposition group at p is split if and only if the family has complex multiplication; in their proof, the weight one members of the family play a key role. I'll talk about work in progress which aims to answer similar questions in the case of Galois representations for a totally real field which are split at only some of the primes above p. In this work Hilbert modular forms of partial weight one play a central role; I'll discuss what is known about them and to what extent the techniques of Ghate and Vatsal can be adapted to this situation.