Department of


Seminar Calendar
for Number Theory Seminar events the year of Sunday, December 8, 2019.

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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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Thursday, January 17, 2019

11:00 am in 241 Altgeld Hall,Thursday, January 17, 2019

What is Carmichael's totient conjecture?

Kevin Ford (Illinois Math)

Abstract: A recent DriveTime commercial features a mathematician at a blackboard supposedly solving "Carmichael's totient conjecture". This is a real problem concerning Euler's $\phi$-function, and remains unsolved, despite the claim made in the ad. We will describe the history of the conjecture and what has been done to try to solve it.

Thursday, January 24, 2019

11:00 am in 241 Altgeld Hall,Thursday, January 24, 2019

Some statistics of the Euler phi function

Harold Diamond (Illinois Math)

Abstract: Questions about the distribution of value of the Euler phi function date to work of Schoenberg and Erdos. This talk will survey this theme and include a result of mine in which two applications of the Perron inversion formula are applied to count the number of points (n, phi(n)) lying in a specified rectangle.

Thursday, January 31, 2019

11:00 am in 241 Altgeld Hall,Thursday, January 31, 2019

Monodromy for some rank two Galois representations over CM fields

Patrick Allen (Illinois Math)

Abstract: In the automorphic-to-Galois direction of Langlands reciprocity, one aims to construct a Galois representation whose Frobenius eigenvalues are determined by the Hecke eigenvalues at unramified places. It is natural to ask what happens at the ramified places, a problem known a local-global compatibility. Varma proved that the p-adic Galois representations constructed by Harris-Lan-Taylor-Thorne satisfy local-global compatibility at all places away from p, up to the so-called monodromy operator. Using recently developed automorphy lifting theorems and a strategy of Luu, we prove the existence of the monodromy operator for some of these Galois representations in rank two. This is joint work with James Newton.

Thursday, February 21, 2019

11:00 am in 241 Altgeld Hall,Thursday, February 21, 2019

Prime number models, large gaps, prime tuples and the square-root sieve

Kevin Ford (Illinois Math)

Abstract: We introduce a new probabilistic model for primes, which we believe is a better predictor for large gaps than the models of Cramer and Granville. We also make strong connections between our model, prime k-tuple counts, large gaps and the "square-root sieve". In particular, our model makes a prediction about large prime gaps that may contradict the models of Cramer and Granville, depending on the tightness of a certain sieve estimate. This is joint work with Bill Banks and Terence Tao.

Thursday, February 28, 2019

11:00 am in 241 Altgeld Hall,Thursday, February 28, 2019

Using q-analogues to transform singularities

Kenneth Stolarsky (Illinois Math)

Abstract: This is a mostly elementary talk about polynomials and their q-analogues, filled with conjectures based on numerical evidence. For example, if ( x - 1 ) ^ 4 is replaced by a q-analogue, what happens to the root at x = 1 ? These investigations accidentally answer a question posed by J. Browkin about products of roots that was also answered by Schinzel some decades ago. We also look at how certain q-analogues are related to each other.

Thursday, March 7, 2019

11:00 am in 241 Altgeld Hall,Thursday, March 7, 2019

Diophantine problems and a p-adic period map

Brian Lawrence (University of Chicago)

Abstract: I will outline a proof of Mordell's conjecture / Faltings's theorem using p-adic Hodge theory. I'll start with a discussion of cohomology theories in algebraic geometry, and build from there. The paper is joint with Akshay Venkatesh.

Thursday, March 14, 2019

11:00 am in 241 Altgeld Hall,Thursday, March 14, 2019

Extremal primes for elliptic curves without complex multiplication

Ayla Gafni (Rochester Math)

Abstract: Fix an elliptic curve $E$ over $\mathbb{Q}$. An ''extremal prime'' for $E$ is a prime $p$ of good reduction such that the number of rational points on $E$ modulo $p$ is maximal or minimal in relation to the Hasse bound. In this talk, I will discuss what is known and conjectured about the number of extremal primes $p\le X$, and give the first non-trivial upper bound for the number of such primes when $E$ is a curve without complex multiplication. The result is conditional on the hypothesis that all the symmetric power $L$-functions associated to $E$ are automorphic and satisfy the Generalized Riemann Hypothesis. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in recent work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.

Thursday, March 28, 2019

11:00 am in 241 Altgeld Hall,Thursday, March 28, 2019

Core partitions, Numerical semigroups, and Polytopes

Hayan Nam (University of California at Irvine)

Abstract: A partition is an $a$-core partition if none of its hook lengths are divisible by $a$. It is well known that the number of $a$-core partitions is infinite and the number of simultaneous $(a, b)$-core partitions is a generalized Catalan number if $a$ and $b$ are relatively prime. Numerical semigroups are additive monoids that have finite complements, and they are closely related to core partitions. The first half of the talk, we will talk about an expression for the number of simultaneous $(a_1,a_2,\dots, a_k)$-core partitions. In the second half, we discuss the relationship between numerical semigroups and core partitions, along with how to count numerical semigroups with certain restrictions.

2:00 pm in 241 Altgeld Hall,Thursday, March 28, 2019

Joint Shapes of Quartic Fields and Their Cubic Resolvents

Piper Harron (University of Hawaii)

Abstract: In studying the (equi)distribution of shapes of quartic number fields, one relies heavily on Bhargava's parametrizations which brings with it a notion of resolvent ring. Maximal rings have unique resolvent rings so it is possible to live a long and healthy life without understanding what they are. The authors have decided, however, to forsake such bliss and look into what ever are these rings and what happens if we consider their shapes along with our initial number fields. What indeed! Please stay tuned. (Joint with Christelle Vincent)

Thursday, April 4, 2019

11:00 am in 241 Altgeld Hall,Thursday, April 4, 2019

Low-lying zeros of Dirichlet L-functions

Kyle Pratt (Illinois Math)

Abstract: I will present work in progress with Sary Drappeau and Maksym Radziwill on low-lying zeros of Dirichlet L-functions. By way of motivation I will discuss some results on the spacings of zeros of the Riemann zeta function, and the conjectures of Katz and Sarnak relating the distribution of low-lying zeros of L-functions to eigenvalues of random matrices. I will then describe some ideas behind the proof of our theorem.

Thursday, April 11, 2019

11:00 am in 241 Altgeld Hall,Thursday, April 11, 2019

Vanishing of Hyperelliptic L-functions at the Central Point

Wanlin Li (Wisconsin Math)

Abstract: We study the number of quadratic Dirichlet L-functions over the rational function field which vanish at the central point s=1/2. In the first half of my talk, I will give a lower bound on the number of such characters through a geometric interpretation. This is in contrast with the situation over the rational numbers, where a conjecture of Chowla predicts there should be no such L-functions. In the second half of the talk, I will discuss joint work with Ellenberg and Shusterman proving as the size of the constant field grows to infinity, the set of L-functions vanishing at the central point has 0 density.

2:00 pm in 241 Altgeld Hall,Thursday, April 11, 2019

Conversations on the exceptional character

Abstract: We will spend the last few weeks of the semester discussing Landau-Siegel zeros. In particular, we will be discussing Henryk Iwaniec's survey article "Conversations on the exceptional character."

Thursday, April 25, 2019

11:00 am in 241 Altgeld Hall,Thursday, April 25, 2019

Local models for potentially crystalline deformation rings and the Breuil-Mézard conjecture

Stefano Morra (Paris 8)

Abstract: Available at

Thursday, May 2, 2019

11:00 am in 241 Altgeld Hall,Thursday, May 2, 2019

The Distribution of log ζ(s) Near the Zeros of ζ

Fatma Cicek (Rochester Math)

Friday, August 30, 2019

3:00 pm in 343 Altgeld Hall,Friday, August 30, 2019

Organizational Meeting

Aubrey Laskowski (UIUC)

Abstract: This will be the organizational meeting for the graduate student number theory seminar. We will discuss the schedule for weekly meetings, as well as begin sign-up for speakers.

Thursday, September 5, 2019

11:00 am in 241 Altgeld Hall,Thursday, September 5, 2019

On the modularity of elliptic curves over imaginary quadratic fields

Patrick Allen (Illinois)

Abstract: Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point, implying that the mod 3 Galois representation attached to the elliptic curve arises from a modular form of weight one. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch" that gives a criterion for when a given mod 6 representation arises from an elliptic curve. As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.

Friday, September 6, 2019

3:00 pm in Illini Hall 1,Friday, September 6, 2019

Series and Polytopes

Vivek Kaushik (Illinois Math)

Abstract: Consider the series $S(k)=\sum_{n \geq 0} \frac{(-1)^{nk}}{(2n+1)^k}$ for $k \in \mathbb{N}.$ It is well-known that $S(k)$ is a rational multiple of $\pi^k$ using standard techniques from either Fourier Analysis or Complex Variables. But in this talk, we evaluate $S(k)$ through multiple integration. On one hand, we start with a $k$-dimensional integral that is equal to the series in question. On the other hand, a trigonometric change of variables shows the series is equal to the volume of a convex polytope in $\mathbb{R}^k.$ This volume is proportional to a probability involving certain pairwise sums of $k$ independent uniform random variables on $(0,1).$ We obtain this probability using combinatorial analysis and multiple integration, which ultimately leads to us finding an alternative, novel closed formula of $S(k).$

Thursday, September 12, 2019

11:00 am in 241 Altgeld Hall,Thursday, September 12, 2019

Moments of half integral weight modular L–functions, bilinear forms and applications

Alexander Dunn (Illinois Math)

Abstract: Given a half-integral weight holomorphic newform $f$, we prove an asymptotic formula for the second moment of the twisted L-function over all primitive characters modulo a prime. In particular, we obtain a power saving error term and our result is unconditional; it does not rely on the Ramanujan-Petersson conjecture for the form $f$. This gives a very sharp Lindelöf on average result for L-series attached to Hecke eigenforms without an Euler product. The Lindelöf hypothesis for such series was originally conjectured by Hoffstein. In the course of the proof, one must treat a bilinear form in Salié sums. It turns out that such a bilinear form also has several arithmetic applications to equidistribution. These are a series of joint works with Zaharescu and Shparlinski-Zaharescu.

Thursday, September 19, 2019

11:00 am in 241 Altgeld Hall,Thursday, September 19, 2019

Indivisibility and divisibility of class numbers of imaginary quadratic fields

Olivia Beckwith (Illinois)

Abstract: For any prime p > 3, the strongest lower bounds for the number of imaginary quadratic fields with discriminant down to -X for which the class group has trivial (resp. non-trivial) p-torsion are due to Kohnen and Ono (Soundararajan). I will discuss refinements of these classic results in which we consider the imaginary quadratic fields for which the class number is indivisible (divisible) by p and which satisfy the property that a given finite set of rational primes split in a prescribed way. We prove a lower bound for the number of such fields with discriminant down to -X which is of the same order of magnitude as in Kohnen and Ono's (Soundararajan's) results. For the indivisibility case, we rely on a result of Wiles establishing the existence of imaginary quadratic fields with trivial p-torsion in their class groups which satisfy a finite set of local conditions, and a result of Zagier which says that the Hurwitz class numbers are the Fourier coefficients of a mock modular form.

Friday, September 20, 2019

3:00 pm in 1 Illini Hall,Friday, September 20, 2019

The prime number theorem through the Ingham-Karamata Tauberian theorem

Gregory Debruyne (Illinois Math)

Abstract: It is well-known that the prime number theorem can be deduced from certain Tauberian theorems. In this talk, we shall present a Tauberian approach that is perhaps not that well-known through the Ingham-Karamata theorem. Moreover, we will give a recently discovered "simple" proof of a so-called one-sided version of this theorem. We will also discuss some recent developments related to the Ingham-Karamata theorem. The talk is based on work in collaboration with Jasson Vindas.

Thursday, September 26, 2019

11:00 am in 241 Altgeld Hall,Thursday, September 26, 2019

Large prime gaps and Siegel zeros

Kevin Ford (Illinois Math)

Abstract: We show that the existence of zeros of Dirichlet L-functions very close to 1 ("Siegel zeros") implies larger prime gaps than are currently known. We also present a heuristic argument that the existence of Siegel zeros implies gaps of larger order than $\log^2 x$, that is, larger than the Cramer conjecture.

Thursday, October 3, 2019

11:00 am in 241 Altgeld Hall,Thursday, October 3, 2019

Optimality of Tauberian theorems

Gregory Debruyne (Illinois & Ghent University)

Abstract: The Wiener-Ikehara and Ingham-Karamata theorems are two celebrated Tauberian theorems which are known to lead to short proofs of the prime number theorem. In this talk, we shall investigate quantified versions of these theorems and show that these are optimal. For the optimality, rather than constructing counterexamples, we shall use an attractive functional analysis argument based on the open mapping theorem. The talk is based on work in collaboration with David Seifert and Jasson Vindas.

Thursday, October 17, 2019

11:00 am in 241 Altgeld Hall,Thursday, October 17, 2019

A new approach to bounds for L-functions

Jesse Thorner (University of Florida)

Abstract: Let $L(s)$ be the $L$-function of a cuspidal automorphic representation of $GL(n)$ with analytic conductor $C$. The Phragmen-Lindelof principle implies the convexity bound $|L(1/2)| \ll C^{1/4+\epsilon}$ for all fixed $\epsilon>0$, while the generalized Riemann hypothesis for $L(s)$ implies that $|L(1/2)|\ll C^{\epsilon}$. A major theme in modern number theory is the pursuit of subconvexity bounds of the shape $|L(1/2)| \ll C^{1/4-\delta}$ for some fixed constant $\delta>0$. I will describe how to achieve (i) an unconditional nontrivial improvement over the convexity bound for all automorphic $L$-functions (joint work with Kannan Soundararajan), and (ii) an unconditional subconvexity bound for almost all automorphic $L$-functions (joint work with Asif Zaman).

Thursday, October 24, 2019

11:00 am in 241 Altgeld Hall,Thursday, October 24, 2019

Non-vanishing of Dirichlet L-functions

Rizwanur Khan (University of Mississippi)

Abstract: $L$-functions are fundamental objects in number theory. At the central point $s = 1/2$, an $L$-function $L(s)$ is expected to vanish only if there is some deep arithmetic reason for it to do so (such as in the Birch and Swinnerton-Dyer conjecture), or if its functional equation specialized to $s = 1/2$ implies that it must. Thus when the central value of an $L$-function is not a "special value", and when it does not vanish for trivial reasons, it is conjectured to be nonzero. In general it is very difficult to prove such non-vanishing conjectures. For example, nobody knows how to prove that $L(1/2, \chi)$ is nonzero for all primitive Dirichlet characters $\chi$. In such situations, analytic number theorists would like to prove 100% non-vanishing in the sense of density, but achieving any positive percentage is still valuable and can have important applications. In this talk, I will discuss work on establishing such positive proportions of non-vanishing for Dirichlet $L$-functions.

Thursday, October 31, 2019

11:00 am in 241 Altgeld Hall,Thursday, October 31, 2019

Eisenstein ideal with squarefree level

Carl Wang-Erickson (University of Pittsburgh)

Abstract: In his landmark paper "Modular forms and the Eisenstein ideal," Mazur studied congruences modulo a prime p between the Hecke eigenvalues of an Eisenstein series and the Hecke eigenvalues of cusp forms, assuming these modular forms have weight 2 and prime level N. He asked about generalizations to squarefree levels N. I will present some work on such generalizations, which is joint with Preston Wake and Catherine Hsu.

Thursday, November 7, 2019

11:00 am in 241 Altgeld Hall,Thursday, November 7, 2019

An even parity instance of the Goldfeld conjecture

Ashay Burungale (Caltech)

Abstract: We show that the even parity case of the Goldfeld conjecture holds for the congruent number elliptic curve. We plan to outline setup and strategy (joint with Ye Tian).

Friday, November 8, 2019

3:00 pm in Illini Hall 1,Friday, November 8, 2019

On two central binomial series related to $\zeta(4)$

Vivek Kaushik (UIUC)

Abstract: In this expository talk, we prove two related central binomial series identities: $\sum_{n \geq 0} \frac{{2n}\choose{n}}{2^{4n}(2n+1)^3}=\frac{7 \pi^3}{216}$ and $\sum_{n \in \mathbb{N}} \frac{1}{{n^4}{{2n}\choose{n}}}=\frac{17 \pi^4}{3240}.$ These series resist all standard approaches used to evaluate other well-known series such as the Dirichlet $L$ series. Our method to prove these central binomial series identities in question will be to evaluate two log-sine integrals that are equal to the series representations. The evaluation of these log-sine integrals will lead to computing closed forms of polylogarithms evaluated at certain complex exponentials. After proving our main identities, we discuss some polylogarithmic integrals that can be readily evaluated using the knowledge of these central binomial series.

Thursday, November 14, 2019

11:00 am in 241 Altgeld Hall,Thursday, November 14, 2019

Bounds on $S(t)$

Ghaith Hiary (Ohio State University)

Abstract: I survey some upper and lower bounds in the theory of the Riemann zeta function, in particular lower bounds on $S(t)$, the fluctuating part of the zeros counting function for the Riemann zeta function. I outline a new unconditional lower bound on $S(t)$, which is work in progress.

Thursday, November 21, 2019

11:00 am in 241 Altgeld Hall,Thursday, November 21, 2019

Sums with the Mobius function twisted by characters with powerful moduli

William Banks (University of Missouri)

Abstract: In the talk, I will describe some recent joint work with Igor Shparlinski, in which we have combined classical ideas of Postnikov and Korobov to derive new bounds on short character sums for certain nonprincipal characters of powerful moduli. Our results are used to bound sums with the Mobius function twisted by such characters, and we obtain new results on the size and zero-free region of Dirichlet L-functions attached to the same class of moduli.

Thursday, December 5, 2019

11:00 am in 241 Altgeld Hall,Thursday, December 5, 2019

Fourth Moments of Modular Forms on Arithmetic Surfaces

Ilya Khayutin (Northwestern University)

Abstract: I will describe a method to study the fourth moment of periods of Hecke eigenforms using a second moment pre-trace formula. The second moment pre-trace formula is constructed out of the usual pre-trace formula using non-standard test functions involving all Hecke operators. It can also be understood using the theta correspondence. Our main application is to the sup-norm problem for modular forms on arithmetic surfaces. Joint work with Raphael Steiner.