2025 CMS Winter Meeting
Toronto, Dec 5 - 8, 2025
Number Theory by Early Career Researchers
Org: Jérémy Champagne, AJ Fong and Zhenchao Ge (University of Waterloo)
[PDF]
Org: Jérémy Champagne, AJ Fong and Zhenchao Ge (University of Waterloo)
[PDF]
- ALI ALSETRI, University of Kentucky
Burgess-type character sum estimates over generalized arithmetic progressions of rank 2. [PDF]
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We extend the classical Burgess estimates to character sums over proper generalized arithmetic progressions (GAPs) of rank 2 in prime fields. The core of our proof is a sharp upper bound for the multiplicative energy of these sets, established by adapting an argument of Konyagin and leveraging tools from the geometry of numbers. This is joint work with Xuancheng Shao.
- FÉLIX BARIL BOUDREAU, Université du Luxembourg / Concordia University
Abelian varieties with homotheties [PDF]
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Let $A$ be an Abelian variety defined over a number field $K$. The celebrated Bogomolov--Serre theorem states that, for any prime $\ell$, the image of the $\ell$-adic representation of the absolute Galois group of $K$ contains all $c$-th power homotheties, where $c$ is a positive integer. When $K$ is a global function field, Zarhin has shown that the corresponding statement fails in general. In this talk, I will present an analogue to Bogomolov--Serre's theorem when $K$ is a finitely generated field of positive characteristic. This is part of an ongoing joint work with Sebastian Petersen (University of Kassel).
- HYMN CHAN, University of Toronto
- ALEX COWAN, University of Waterloo
- JOSE CRUZ, University of Calgary
- NIC FELLINI, Queen’s University
- KEIRA GUNN, Mt Royal University
- HAZEM HASSAN, McGill University
- FATEMEH JALALVAND, University of Calgary
- FATEMEHZAHRA JANBAZI, University of Toronto
- NICOL LEONG, University of Lethbridge
- ISABELLA NEGRINI, University of Toronto
Rigid Cocycles and the p-adic Kudla Program [PDF]
- ALEX COWAN, University of Waterloo
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Rigid cocycles, introduced by Darmon and Vonk in 2017, offer a promising framework to extend complex multiplication theory to real quadratic fields, suggesting a theory of “real multiplication.” They exhibit striking parallels with modular forms and are central to the emerging p-adic Kudla program. While the classical Kudla program studies the theta correspondence between automorphic forms on different groups, the p-adic version appears to replace automorphic forms with rigid cocycles. Although a theory for a p-adic theta correspondence has yet to be developed, recent results suggest its existence. In this talk, I present some of these p-adic results, draw comparisons to the classical setting, and discuss the evidence for an underlying p-adic theta correspondence.
- PAUL PÉRINGUEY
- EMILY QUESADA-HERRERA, University of Lethbridge
- FATEME SAJADI, University of Toronto
- GIAN CORDANA SANJAYA, University of Waterloo
Squarefree density of discriminant of polynomials with restricted coefficients [PDF]
- EMILY QUESADA-HERRERA, University of Lethbridge
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The squarefree density problem, which asks to determine the probability that a multivariate integer polynomial $F(x_1, \ldots, x_n)$ attains a squarefree value, is a classical problem. Some recent progress has been made in the case where $F$ is the discriminant of a monic integer polynomial. Namely, Bhargava, Shankar, and Wang proved that the density of monic integer polynomials with squarefree discriminant exists and is given by the product of the local densities, which were previously computed by Yamamura.
In this talk, I will discuss the case where $F$ is the discriminant of a monic integer polynomial with restricted coefficients, with the emphasis on local density computations. This talk is partially based on a joint work with Valentio Iverson and Xiaoheng Wang.
- KYLE YIP, Georgia Institute of Technology
Diophantine tuples and Diophantine powersets [PDF]
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Let $k,n$ be integers with $k\geq 2$ and $n \neq 0$. A set $A$ of positive integers is a Diophantine tuple with property $D_{k}(n)$ if the product of $ab+n$ is a perfect $k$-th power for every $a,b\in A$ with $a\neq b$. These Diophantine tuples have been studied extensively. In this talk, I will discuss some recent progress on ``Diophantine powersets" (first studied by Gyarmati, S\'{a}rk\"{o}zy, and Stewart), where we allow $ab+n$ to be a perfect power instead of a perfect $k$-th power for some fixed $k$. Joint work with Ernie Croot.
- XIAO ZHONG, University of Waterloo