Queen's Number Theory Seminar
Queen's University, Kingston, ON
Organizer: Seoyoung Kim and Brad Rodgers.
Please contact me know if you are interested in giving a talk in our seminar!
Winter Semester 2021 (Virtually via Zoom)
Tariq Osman
Date: Friday, March the 26th
Time: 4:00-5:00pm
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Title: Tails of Generalised Quadratic Weyl Sums.
Abstract: We define generalized quadratic Weyl sums as exponential sums of the form S_N(x, c_1, c_0, \alpha) := \sum_{n=1}^{N} e((\frac{1}{2}n^2 + c_1 n + c_0) x + n \alpha). We prove that for x chosen uniformly from the unit interval and specific rational choices of c_1 and \alpha the limiting distribution of \frac{1}{\sqrt N} S_N has compact support. Outside of these specific choices of c_1 and \alpha the limiting distribution of \frac{1}{\sqrt N} S_N can be shown to have heavy tails. This is Joint work with Francesco Cellarosi.
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Julie Desjardins
Date: Friday, March the 19th
Time: 4:00-5:00pm
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Title: Root number in families of elliptic curves
Abstract: The root number $W(E)$ of $E$ an elliptic curve over $\Q$ is a useful substitute to the geometric rank $rk(E)$. By a weak version of the Birch and Swinnerton-Dyer conjecture (the « parity conjecture »), it is equal to $(-1)^rk(E)$. In particular, under the parity conjecture, $W(E)=-1$ implies that $rk(E)$ is non-zero. In addition to having its own interest from a arithmetic statistics point of view, studying the behavior of the root number in a family of elliptic curves has applications in arithmetic geometry to prove the density of rational points on elliptic surfaces.
As proven in my thesis, the root number takes both value $+1,-1$ for infinitely many fibers of a non-isotrivial family of elliptic curves. When we restrict our attention to the « integer fibers » of the families, however, the situation changes. As illustrated by Washington example $y^2=x^3+tx^2-(t-3)x+1$, the root number can take the same value for all values of $t\in\mathbb{Z}$ (on this example it is always $-1$). In a project with Rena Chu, we fully identify the non-isotrivial families of elliptic curves with coefficients of small degree (\geq2) such that the root number is the same for all integer fibers.
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Jyothsnaa Sivaraman
Date: Friday, March the 12nd
Time: 4:00-5:00pm
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Title: Euclidean ideal classes in Abelian extensions of small degree.
Abstract: In 1979, H. W. Lenstra showed that if the ring of integers of a number field has infinitely many units, then it has a Euclidean ideal class if and only if the class group is cyclic provided the generalised Riemann hypothesis is true. For a number field K, we call the rank of the free part of O∗_K the unit rank of O_K. In 2013, Graves and Murty introduced techniques from sieve theory to remove the dependence on the generalised Riemann hypothesis in this theorem for a family of fields. For a family of abelian number fields, they showed unconditionally that if the unit rank of OK is at least 4, then the class group is cyclic if and only if it has a Euclidean ideal class. This result was extended to a family of number fields for which unit rank O_K is 3 by Deshouillers, Gun and Sivaraman. In this talk, we will look at some unconditional results towards this problem in the case of real quadratic and Galois cubic fields. One is joint work with Prof. Sanoli Gun and the other with Prof. Kumar Murty.
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Victoria Cantoral-Farfan
Date: Friday, March the 5th
Time: 4:00-5:00pm
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Title: Around Mumford–Tate and Sato–Tate conjectures.
Abstract: The famous Mumford-Tate conjecture asserts that, for every prime number, Hodge cycles are Q-linear combinations of Tate cycles, through Artin's comparisons theorems between Betti and etale cohomology. On the other hand, the generalized Sato-Tate conjecture predicts the equidistribution of the normalized L-functions associated to an abelian variety. The goal of this talk is to introduce and present the state-of-the-art of the aforementioned conjectures as well as to describe the strong relations between those conjectures.
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Ram Murty
Date: Friday, February the 26th
Time: 4:00-5:00pm
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Title: An introduction to discrete Markov chains.
Abstract: We give a quick introduction to the theory of discrete Markov chains. We also give a new proof of the ergodic theorem for markov chains.
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Seoyoung Kim
Date: Friday, February the 12th
Time: 4:00-5:00pm
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Title: The Markov triples and the Markov tree
Abstract: In this talk we look at various (conjectural) properties of the Markov triples and their tree generated by Vieta involutions. We present results related to its growth (Zagier) and introduce some (conjectural) results, for instance, by Bourgain-Gamburd-Sarnak.
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David Nguyen
Date: Friday, February the 5th
Time: 4:00-5:00pm
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Title: The k-fold divisor function in arithmetic progressions
Abstract: In this talk we will look at various sums of the k-fold divisor function $\tau_k(n)$, which has the Dirichlet generating function $\sum_n \tau_k(n) n^{-s} = \zeta^k(s)$, in arithmetic progressions. The distribution of $\tau_k(n)$ in arithmetic progressions is closely related to the distribution of prime numbers and to moments of Dirichlet L-functions. For k larger than 3, we show that $\tau_k(n)$ is well distributed to special large moduli exceeding $X^{1/2}$ with an effective power-saving error term. This extends the method of Zhang (2013) to $\tau_k(n)$. For $k=3$, we show further that the ternary divisor function $\tau_3(n)$ is equidistributed on average for moduli up to $X^{2/3}$, extending the individual estimate of Friedlander and Iwaniec (1985). If time permits, we will also discuss a work in progress of an averaged variance of $\tau_3(n)$ in arithmetic progressions related to a recent conjecture of Rodgers and Soundararajan (2018) about asymptotic of this variance.
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Brad Rodgers
Date: Friday, January 29nd
Time: 4:00-5:00pm
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Title: A look at some proofs of the prime number theorem
Abstract: We will give a modern look at some old proofs of the prime number theorem, including the role (or lack of role) of the zeros of the zeta function in these proofs.
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Manami Roy
Date: Friday, January 22nd
Time: 4:00-5:00pm
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Title: An equidistribution result for cuspidal automorphic representations of GSp(4)
Abstract:
There are some well-known classical equidistribution results for elliptic curves and elliptic modular forms. The so-called vertical Sato-Tate conjecture for GL(2) states that the normalized eigenvalues of elliptic modular forms are equidistributed with respect to some measure. In this talk, we will discuss a similar equidistribution result for a family of cuspidal automorphic representations of GSp(4). We formulate our theorem explicitly in terms of the number of cuspidal automorphic representations in this family. To count the number of these cuspidal automorphic representations, we will explore the connection between Siegel modular forms and automorphic representations of GSp(4). The talk is based on a recent work with Ralf Schmidt and Shaoyun Yi.
Fall Semester 2020 (Virtually via Zoom)
Siddhi Pathak
Date: Tuesday, Nov. 24th
Time: 4:00-5:00pm
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Title: Special values of L-functions with periodic coefficients
Abstract: It is well known that the values of the Riemann zeta-function at even positive integers are rational multiples of powers of $\pi$. Hence these values are transcendental and algebraically dependent. On the contrary, the odd zeta-values are not only expected to be transcendental but also algebraically independent. In this talk, we discuss the analogous problem of determining the arithmetic nature of special values of Dirichlet L-functions, and more generally, Dirichlet series with periodic coefficients. Under the assumption of standard conjectures, we derive a vanishing criterion for the values L(k,f), generalizing the work of Okada. These results are obtained jointly with A. Bharadwaj.
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John Charles Saunders
Date: Tuesday, Nov. 17th.
Time: 4:00-5:00pm
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Title: Random Fibonacci Sequences From Balancing Words
Abstract: We study growth rates of random Fibonacci sequences of a particular structure. A random Fibonacci sequence is an integer sequence starting with 1, 1 where the next term is determined to be either the sum or the difference of the two preceeding terms where the choice of taking either the sum or the difference is chosen randomly at each step. There has been much study on such sequences. For instance, in 2012, McLellan proved that if the pluses and minuses follow a periodic pattern and Gn is the nth term of the resulting random Fibonacci sequence, then
limn→∞|Gn|1/n
exists. We extend her results by showing that this limit also exist if the choices of pluses and minuses follow a balancing word pattern. This is joint work with Dr. Kevin Hare.
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Tariq Osman
Date: Tuesday, Nov. 10th
Time: 4:00-5:00pm
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Title: Limit Theorems for Quadratic Weyl Sums.
Abstract: Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)
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Daniel Cloutier
Date: Tuesday, Nov. 3rd.
Time: 4:00-5:00pm
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The Ito-Kawada Theorem
Abstract: I will present the proof of the Ito-Kawada Theorem in the case of cyclic groups. The theorem gives a necessary and sufficient condition for a sequence of random elements on a group to tend to equidistribution. The proof makes use of finite Fourier analysis and the representation theory of finite groups. Time permitting, I will also talk about the non-abelian and compact cases.
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Sasha Mangerel
Date: Tuesday, October 27th.
Time: 4:00-5:00pm
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Title: Monotone Chains of Fourier Coefficients of Cusp FormsAbstract: Given a multiplicative function f: \mathbb{N} \rightarrow \mathbb{R} it is generally a difficult problem to determine the frequency with which chains of inequalities like
f(n+1) \leq f(n+2) \leq \cdots \leq f(n+k), k \geq 2
occur. I will explain how recent developments in the theory of automorphic forms and in multiplicative number theory enable us to show, among other things, that when e.g., f is the Ramanujan \tau function the set \{n \in \mathbb{N} : \tau(n+1) < \tau(n+2) < \tau(n+3)\} has positive upper density (Joint work w/ Oleksiy Klurman).
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Matilde Lalin
Date: Tuesday, October 20th.
Time: 4:00-5:00pm
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Title: Moments and non-vanishing for cubic L-functions.
Abstract: We will start by discussing the problem of finding moments of Dirichlet L-functions, and how this implies non-vanishing results. We will then discuss the problem for function fields. Finally, we will present two results: one on the first moment of L-functions associated to cubic characters over F_q(t), and another one for non-vanishing of such functions. The proofs of these results involve estimations of cubic Gauss sums, and bounding the second mollified moment, which uses techniques of Soundararajan, Harper, Lester and Radziwill for the Riemann zeta function and quadratic twists of modular forms. This is joint work with C. David and A. Florea.
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Ahmet Muhtar Güloğlu
Date: Tuesday, October 13th.
Time: 4:00-5:00pm
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Title: One level density for cubic Dirichlet L-functions
Abstract: We study the distribution of non-trivial zeros of Hecke L-functions attached to cubic Dirichlet characters over Eisenstein field that are near the critical point s=1/2 and thereby get a non-trivial non-vanishing result for these functions at s=1/2.
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Ram Murty
Date: Tuesday, October 6th.
Time: 4:00-5:00pm
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Title: NORMAL NUMBER OF PRIME FACTORS OF SUMS OF FOURIER COEFFICIENTS OF MODULAR FORMS
Abstract: We will study the normal order of the number of prime factors of sums of Fourier coefficients of modular forms and obtain some conditional results. This is joint work with Kumar Murty and Sudhir Pujahari.
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Brad Rodgers
Date: Tuesday, September 29th.
Time: 4:00-5:00pm
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Title: On the distribution of polynomials with random multiplicative coefficients
Abstract: In this talk we consider random polynomials P(z) with multiplicative coefficients, and discuss the distribution of P(z) in the complex plane where z is chosen at random uniformly from the unit circle. We will show that almost surely P(z) tends to a gaussian distribution -- an analogue of a classic theorem of Salem and Zygmund. Our approach is via moments and depends on a point-counting method pioneered by Vaughan and Wooley. This is ongoing joint work with Jacques Benatar and Alon Nishry.
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