Location: PIMS 4133 

Tentative schedule: Thursdays 2-3pm (occasionally we will meet at 1-2pm on Thursday)

Organizers:

Paul Péringuey peringuey@math.ubc.ca 

Kyle Yip kyleyip@math.ubc.ca 

Kin Ming Tsang kmtsang@math.ubc.ca

Please contact one of the organizers if you would like to give a talk. In person talks are particularly encouraged, but occasionally we will also have zoom talks. 

The talk will be simultaneously broadcasted on zoom, please contact one of the organizers for the zoom link.

Title and abstracts of talks:

Apr 4, Paul Péringuey (UBC)

Title: Refinements of Artin's primitive root conjecture

Abstract: Let ordₚ(a) be the order of a in (ℤ/pℤ)^*. In 1927 Artin conjectured that the set of primes p for which a given integer a (that is neither a square number nor −1) is a primitive root (i.e. ordₚ(a)=p−1) has a positive asymptotic density among all primes. In 1967 Hooley proved this conjecture assuming the Generalized Riemann Hypothesis.

In this talk we will study the behaviour of ordₚ(a) as p varies over primes, in particular we will show, under GRH, that the set of primes p for which ordₚ(a) is ``k prime factors away'' from p−1 has a positive asymptotic density among all primes except for particular values of a and k. We will interpret being ``k prime factors away'' in three different ways, namely k=𝜔((p−1)/ ordₚ(a)), k=𝛺((p−1)/ordₚ(a)) and k=𝜔(p−1)−𝜔(ordₚ(a)), and present conditional results analogous to Hooley's in all three cases and for all integer k. This is joint work with Leo Goldmakher and Greg Martin.

Apr 11 Ling Long (Louisiana State University)

Title: Hypergeometric functions through the arithmetic kaleidoscope

Abstract: The classical theory of hypergeometric functions, developed by generations of mathematicians including Gauss, Kummer, and Riemann, has been used substantially in the ensuing years within number theory, geometry, and the intersection thereof. In more recent decades, these classical ideas have been translated from the complex setting into the finite field and $p$-adic settings as well.

In this talk, we will give a friendly introduction to hypergeometric functions, especially in the context of number theory.

Apr 18 Sarah Dijols (UBC) 

Title: Generic representations and ABV-packets for $p$-adic groups

Abstract: After a brief introduction on the theory of $p$-adic groups complex representations, I will explain why tempered and generic Langlands parameters are open. I will further derive a number of consequences, in particular for the enhanced genericity conjecture of Shahidi and its analogue in terms of ABV packets. This is a joint work with Clifton Cunningham, Andrew Fiori, and Qing Zhang.

Previous talks:

Sep 14, Yifeng Huang (UBC)

Title: Counting matrix solutions of Diophantine equations

Abstract: A central subject in number theory is counting solutions of polynomial equations in a finite field. In the talk, I will recall the famous Sato--Tate conjecture in this theme and several of its variants. Can matrices be involved in this story, and what extra things do they bring? To answer this question, I will define the notion of n x n matrix solutions, and present results about their counts, where both arithmetic (e.g., Sato--Tate) and combinatorics (e.g., partitions of n) play a role.

Part of this talk is based on joint work with Ono and Saad and joint work with Jiang.

Sep 21, Kin Ming Tsang (UBC)

Title: Explicit Zero-free region for Automorphic L-functions

Abstract: Number Theorists are interested in the zeros of L-functions, for example, zeros of the Riemann zeta function. In this talk, we discuss the explicit zero-free region for automorphic L-functions and how the use of Stechkin’s trick leads to a better explicit bound in the zero-free region.

This talk is based on an ongoing joint work with Steven Creech, Alia Hamieh, Simran Khunger, Kaneenika Sinha and Jakob Streipel initiated in Inclusive Paths in Explicit Number Theory Summer School.

Sep 28, Stephen Choi (Simon Fraser University) [Abstract in pdf] 

Title: Gap Principle of Divisibility Sequences of Polynomials

Abstract: Let $f \in \mathbb{Z}[x]$ and $\ell \in \mathbb{N}$. Consider the set of all $(a_{0},a_{1},\dots,a_{\ell}) \in \mathbb{N}^{\ell+1}$ with $a_{i} < a_{i+1}$ and $f(a_{i}) \mid f(a_{i+1})$ for all $0 \leq i \leq \ell-1$. We say that $f$ satisfies the gap principle of order $\ell$ if $\lim a_{\ell}/a_{0} = \infty$ as $a_{0} \to \infty$ for any such $(a_{0},a_{1},\dots,a_{\ell})$. We also define the gap order of $f(x)$ to be the smallest positive integer $\ell$ such that $f(x)$ satisfies the gap principle of order $\ell$. If such $\ell$ does not exist, we say that $f(x)$ does not satisfy the gap principle. In this talk, we will discuss a conjecture by Chan, Choi and Lam that $f(x)$ does not satisfy the gap principle  if and only if $f(x)$ is in the form of $f(x)=A(Bx+C)^n$ for some $A, B, C \in \mathbb{Z}$. Moreover, we completely determine the gap order of any polynomial  that if $f(x)$ is not in the form of $A(Bx+C)^n$, then $f(x)$ has gap order $2$ if $f(x)$ is a quadratic polynomial or a power of a quadratic polynomial; and has gap order $1$ otherwise.  Related to the proof of above results, the multiplicative order of the fundamental solution of Pell's equation $X^2-DY^2=1$ in $\mathbb{Z} [\sqrt{D}]/<D>$ will also be discussed. These are joint work with Tsz Ho Chan, Peter Cho-Ho Lam and Daniel Tarnu.

Oct 5 (Special time: 1pm), Elchin Hasanalizade (ADA University)

Title: On some explicit results for the sum of unitary divisor function

Abstract: Let $\sigma^*(n)$ be the sum of all unitary (i.e. coprime) divisors of $n$. As an analogue of Lehmer’s totient problem, Subbarao proposed the following conjecture. The congruence $\sigma^*(n)\equiv 1\pmod{n}$ is possible iff $n$ is a prime power. This problem is still open. We strengthen considerably the lower estimations for the potential counterexamples to Subbarao’s conjecture.

In the second part of our talk, we discuss the growth of the function $\sigma^*(n)$. We establish a new explicit upper bound, namely $\sigma^*(n)<1.2678n\log\log{n}$ for all $n\ge223092870$. For this purpose, we use explicit estimates for Chebyshev’s $\theta$-function and for some product defined over prime numbers.

Oct 17 (Special time: Tuesday at 4pm) Kyle Yip (UBC)

Title: Diophantine tuples over integers and finite fields

Abstract: A set $\{a_{1}, a_{2},\ldots, a_{m}\}$ of distinct positive integers is a Diophantine $m$-tuple if the product of any two distinct elements in the set is one less than a square. There is a long history and extensive literature on the study of Diophantine tuples and their generalizations in various settings. In this talk, we focus on the following generalization: for each $n \ge 1$ and $k \ge 2$, we call a set of positive integers a Diophantine tuple with property $D_{k}(n)$ if the product of any two distinct elements is $n$ less than a $k$-th power, and we denote $M_k(n)$ be the largest size of a Diophantine tuple with property $D_{k}(n)$. Using various tools from number theory, we show that there is $k=k(n)$ such that $k,n \to \infty$ and $M_k(n)=o(\log n)$, breaking the $\log n$ barrier. A key ingredient is to study the finite field model of the same problem. Joint work with Seoyoung Kim and Semin Yoo.

Oct 19, Raghu Pantangi (University of Regina)

Title: On sums of coefficients of polynomials related to the Borwein conjectures

Abstract: Peter Borewein empirically discovered quite a number of mysteries involving sign patterns of coefficients of polynomials of the form $f_{p,s,n}(q):=\prod_{j=0}^{n} \prod_{k=1}^{p-1} (1-q^{pj+k})^{s}$ ($p$ a prime and $s,n \in \mathbb{N}$). In the case $(p,s) \in \{(3,1), (3,2)\}$, he conjectured that the coefficients follow  a repeating + - - pattern, and in the case $(p,s)=(5,1)$, it was conjectured that the coefficients follow a repeating + - - - - sign pattern. We consider a weaker problem of finding the signs of partial sums of coefficients along some arithmetic progressions. We use a combinatorial sieving principle by Li-Wan and elementary character theory to asymptotically estimate and find the signs of these partial sums. We find that the signs of these partial sums are compatible with the sign pattern in Borewein's conjectures. This is based on joint work with Ankush Goswami. 

Oct 26, Lior Silberman (UBC)

Title: Arithmetic Quantum Unique Ergodicity for $SL_2$ over number fields

Abstract: Let $M = SL_2(\mathbb{Z}[i])\backslash \mathbb{H}^{(3)}$ be the Bianchi orbifold, and let $\{f_n\}_{n=1}^\infty \subset L^2(M)$ be a sequence of Hecke--Maass forms on it. We should that the probability measures on $M$ with densities $|f_n(x)|^2$ with respect to the Riemannian volume become equidistributed on $M$. This generalizes the case $SL_2(\mathbb{Z})\backslash \mathbb{H}^{(2)}$ due to Lindenstrauss.

In the first part of the talk I will introduce the quantum unique ergodicity problem and the setup necessary to state the theorem. In the second half I will try to give ideas of the proof, including of the full result. More generally the result holds when we replace $\mathbb{Q}(i)$ with a general number field $F$, $SL_2$ with the group $G$ of norm-1 elements of a quaternion algebra over $F$, $\mathbb{H}^{(3)}$ with $G(F_\infty)/K_\infty$ where $F_\infty = F\otimes_\mathbb{Q} \mathbb{R}$ and $K_\infty$ is a maximal compact subgrouop, and $SL_2(\mathbb{Z}[i])$ with a congruence lattice. Joint work with Z. Shem-Tov, also relying on work with A. Zaman.

Nov 2, Greg Martin (UBC)

Title: Statistics of the multipicative group

Abstract: For every positive integer n, the quotient ring Z/nZ is the natural ring whose additive group is cyclic. The "multiplicative group modulo n" is the group of invertible elements of this ring, with the multiplication operation. As it turns out, many quantities of interest to number theorists can be interpreted as "statistics" of these multiplicative groups. For example, the cardinality of the multiplicative group modulo n is simply the Euler phi function of n; also, the number of terms in the invariant factor composition of this group is closely related to the number of primes dividing n. Many of these statistics have known distributions when the integer n is chosen at random (the Euler phi function has a singular cumulative distribution, while the Erdös–Kac theorem tells us that the number of prime divisors follows an asymptotically normal distribution). Therefore this family of groups provides a convenient excuse for examining several famous number theory results and open problems. We shall describe how we know, given the factorization of n, the exact structure of the multiplicative group modulo n, and go on to outline the connections to these classical statistical problems in multiplicative number theory.

Nov 9, Fatma Çiçek (University of Northern British Columbia)

Title: Uniform distribution modulo one of ordinates of certain nontrivial zeros 

Abstract: A result of Rademacher is that on the Riemann hypothesis, the positive ordinates of the nontrivial zeros of the Riemann zeta function, that is, the $ \gamma$ with $\gamma>0$, are uniformly distributed modulo one. This talk will be on a recent conditional result of the speaker with Steve Gonek, which proves that a special subsequence of this sequence is also uniformly distributed modulo one. The elements of the subsequence are obtained from the original sequence by imposing a size restriction on the values $\log|\zeta'(\frac12+i\gamma)|$ conditionally on the Riemann hypothesis. 

Nov 16, Seoyoung Kim (University of Göttingen)

Title: Birch and Swinnerton-Dyer conjecture and Nagao's conjecture

Abstract:  In 1965, Birch and Swinnerton-Dyer formulated a conjecture on the Mordell-Weil rank $r$ of elliptic curves which also implies the convergence of the Nagao-Mestre sum. We show that if the Nagao-Mestre sum converges, then the limit equals $-r+1/2$, and study the connections to the Riemann hypothesis for E. We also relate this to Nagao’s conjecture. Furthermore, we discuss a generalization of the above results for the Selberg classes and hence (conjecturally) for larger classes of  $L$-functions. 

Nov 23, Shubhrajit Bhattacharya (UBC)

Title: Integer Polynomials and Toric Geometry

Abstract: Arithmetic Statistics is an emerging subbranch of Number Theory where we count arithmetic objects when bounded by some quantitative invariant, like height. One example is counting polynomials with integer coefficients having a fixed Galois group. Bhargava’s proof of B. L. van der Waerden’s conjecture about the count of integer polynomials of degree n and bounded largest coefficient with Galois group not equal to the full symmetry group of order n is a recent breakthrough. We will introduce a new way of parametrizing monic integer cubic polynomials with Galois group C3 using rational points on a toric variety. 

We also introduce a new height function on polynomials arising from height functions in toric geometry. This results in a nice relation between the height zeta function of the toric variety, defined by Batyrev–Tschinkel, and the Dirichlet series attached to the counting sequence of monic abelian cubics! Using this we prove explicit and asymptotic formulas for the number of monic abelian cubics of a given height.

We also discuss future research avenues using our method. This is based on joint work with Andrew O’Desky. Preprint available here https://arxiv.org/abs/2310.17831.

Nov 30, Severin Schraven (UBC)

Title: Local to global principle for higher moments of the natural density

Abstract: In this talk I will explain how to obtain a local to global principle for expected values over free Z-modules of finite rank. We use the same philosophy as Ekedhal’s Sieve for densities, later extended and improved by Poonen and Stoll in their local to global principle for densities. This strategy can also be extended to higher moments and to holomorphy rings of any global function field. These results were obtained in collaboration with A. Hsiao, J. Ma, G. Micheli, S. Tinani, V. Weger, Y.Q. Wen.

Dec 7, Frederik Broucke (Ghent University)

Title: Examples of well-behaved Beurling number systems

Abstract: A Beurling number system consists a non-decreasing unbounded sequence of reals larger than 1, which are called generalized primes, and the sequence of all possible products of these generalized primes, which are called generalized integers. With both sequences one associates counting functions. Of particular interest is the case when both counting functions are close to their classical counter parts: namely when the prime-counting function is close to Li(x), and when the integer-counting function is close to ax for some positive constant a.

A Beurling number systems is well-behaved if it admits a power saving in the error terms for both these counting functions. In this talk, I will discuss some general theory of these well-behaved systems, and present some recent work about examples of such well-behaved number systems. This talk is based on joint work with Gregory Debruyne and Szilárd Révész.

Dec 14, Shamil Asgarli (Santa Clara University)

Title: Linear system of hypersurfaces passing through a Galois orbit

Abstract: Consider the vector space (parameter space) of all homogeneous forms of degree d in n+1 variables defined over some field K. Geometrically, the vanishing set of such a form corresponds to a hypersurface of degree d in the projective space P^{n}. The dimension of this parameter space is $m = \binom{n+d}{d}$. If P_1, ..., P_m are in "general position", then no hypersurface of degree d can pass through all these m points, because passing through each additional point imposes 1 new linearly independent condition. In this talk, we address the following variant: for a given K, d, and n, can we always find m points P_1, ..., P_m so that:

(a) P_1, P_2 ..., P_m form a Gal(L/K)-orbit of a single point P defined over a Galois extension L / K with [L:K] = m,  and

(b) No hypersurface of degree m defined over K passes through P_1, P_2, ..., P_m.

We show that the answer is "Yes" if the base field K has at least 3 elements. In other words, the concept of "general position" for points can be modelled by Galois orbits. As an application, we compute the maximum dimension of a linear system of hypersurfaces over a finite field F_q where each F_q-member of the system is irreducible over F_q. This is joint work with Dragos Ghioca and Zinovy Reichstein. 

Jan 18, (Special time: 1pm) Théo Untrau (ENS Rennes)

Title: Equidistribution of some families of short exponential sums

Abstract: Exponential sums play a role in many different problems in number theory. For instance, Gauss sums are at the heart of some early proofs of the quadratic reciprocity law, while Kloosterman sums are involved in the study of modular and automorphic forms. Another example of application of exponential sums is the circle method, an analytic approach to problems involving the enumeration of integer solutions to certain equations. In many cases, obtaining upper bounds on the modulus of these sums allow us to draw conclusions, but once the modulus has been bounded, it is natural to ask the question of the distribution of exponential sums in the region of the complex plane in which they live. After a brief overview of the motivations mentioned above, I will present some results obtained with Emmanuel Kowalski on the equidistribution of exponential sums indexed by the roots modulo p of a polynomial with integer coefficients.

Jan 25, Kübra Benli (University of Lethbridge)

TItle: Sums of proper divisors with missing digits

Abstract: Let $s(n)$ denote the sum of proper divisors of a positive integer $n$. In 1992, Erd\H{o}s, Granville, Pomerance, and Spiro conjectured that if $\mathcal{A}$ is a set of integers with asymptotic density zero then the preimage set $s^{-1}(\mathcal{A})$ also has asymptotic density zero. In this talk, we will discuss the verification of this conjecture when $\mathcal{A}$ is the set of integers with missing digits (also known as ellipsephic integers) by giving a quantitative estimate on the size of the set $s^{-1}(\mathcal{A})$. This talk is based on the joint work with Giulia Cesana, C\'{e}cile Dartyge, Charlotte Dombrowsky and Lola Thompson. 

Feb 1, Dragos Ghioca (UBC)

Title: Collision of orbits under the action of a Drinfeld module

Abstract: We present various results and conjectures regarding unlikely intersections of orbits for families of Drinfeld modules. Our questions are motivated by the groundbreaking result of Masser and Zannier (from 15 years ago) regarding torsion points in algebraic families of elliptic curves.

Feb 8, Debanjana Kundu (University of Texas Rio Grande Valley) 

Title: A Conjecture of Mazur predicting the growth of Mordell--Weil ranks in $\mathbb{Z}_p$-extensions

Abstract: Let $p$ be an odd prime. We study Mazur's conjecture on the growth of the Mordell--Weil ranks of an elliptic curve $E/\mathbb{Q}$ over an imaginary quadratic field in which $p$ splits and $E$ has good reduction at $p$. In particular, we obtain criteria that may be checked through explicit calculation, thus allowing for the verification of Mazur's conjecture in specific examples. This is joint work with Rylan Gajek-Leonard, Jeffrey Hatley, and Antonio Lei.

Feb 15, Ertan Elma (University of Lethbridge)

Title: A Discrete Mean Value of the Riemann Zeta Function and its Derivatives

Abstract: In this talk, we will discuss an estimate for a discrete mean value of the Riemann zeta function and its derivatives multiplied by Dirichlet polynomials. Assuming the Riemann Hypothesis, we obtain a lower bound for the 2kth moment of all the derivatives of the Riemann zeta function evaluated at its nontrivial zeros. This is based on a joint work with Kübra Benli and Nathan Ng.

Feb 29, Kim Klinger-Logan (Kansas State University)

Title: A shifted convolution problem arising from physics

Abstract: Certain eigenvalue problems involving the invariant Laplacian on moduli spaces have potential applications to scattering problems in physics. Green, Russo, Vanhove, et al., discovered the behavior of gravitons (hypothetical particles of gravity represented by massless string states) is also closely related to eigenvalue problems for the Laplace-Beltrami operator on various moduli spaces. In this talk we will examine applications and results related to solutions $(\Delta - \lambda) f = E_aE_b$ on $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})/SO_2(\mathbb{R})$, where $E_s$ is a non-holomorphic Eisenstein series on $GL(2)$ and $\Delta = y^2(\partial_x^2+\partial_y^2)$. One such interesting finding from this work is a family of identities relating convolution sums of divisor functions to Fourier coefficients on modular forms. This work is in collaboration with Ksenia Fedosova, Stephen D. Miller, Danylo Radchenko, and Don Zagier. 

Mar 7,  Michael Bennett  (UBC)

Title: Arithmetic progressions in sumsets of geometric progressions

Abstract : If A and B are two geometric progressions, we characterize all 3-term arithmetic progressions in the sumset A+B. Somewhat surprisingly, while mostly elementary, this appears to require quite deep machinery from Diophantine Approximation.

Mar 14, Simone Coccia  (UBC)

Title: Analogues of the Hilbert Irreducibility Theorem for integral points on surfaces

Abstract: We will discuss conjectures and results regarding the Hilbert Property, a generalization of Hilbert's irreducibility theorem to arbitrary algebraic varieties. In particular, we will explain how to use conic fibrations to prove the Hilbert Property for the integral points on certain surfaces, such as affine cubic surfaces.

Mar 21, Emanuele Bodon (UBC)

Title: Pro-p Iwahori Invariants

Abstract: Let $F$ be the field of $p$-adic numbers (or, more generally, a non-archimedean local field) and let $G$ be $\mathrm{GL}_n(F)$ (or, more generally, the group of $F$-points of a split connected reductive group). In the framework of the local Langlands program, one is interested in studying certain classes of representations of $G$ (and hopefully in trying to match them with certain classes of representations of local Galois groups).

In this talk, we are going to focus on the category of smooth representations of $G$ over a field $k$. An important tool to investigate this category is given by the functor that, to each smooth representation $V$, attaches its subspace of invariant vectors $V^I$ with respect to a fixed compact open subgroup $I$ of $G$. The output of this functor is actually not just a $k$-vector space, but a module over a certain Hecke algebra. The question we are going to attempt to answer is: how much information does this functor preserve or, in other words, how far is it from being an equivalence of categories? We are going to focus, in particular, on the case that the characteristic of $k$ is equal to the residue characteristic of $F$ and $I$ is a specific subgroup called "pro-$p$ Iwahori subgroup".