Organizers.
Nathan Grieve: nathan.m.grieve@gmail.com (Contact organizer)
Antonio Lei: alantoniolei@gmail.com
Webpage of the Ottawa-Carleton Number Theory group:
Fall 2025
We will schedule a collection of virtual and in person talks. The defalut seminar time slot is Monday @ 09:00-10:00 Ottawa time (however this may vary from time to time). The Zoom link is: here. Email any of the organizers for the Zoom password.
List of confirmed speakers:
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22 September, 2025
In person lecture (with Zoom live stream)
Brandon Hanson
Title: Arithmetic combinatorics from high-dimensional probability
Abstract: The use of probability in combinatorics was pioneered by Erdos, rather famously. Recently, techniques from high-dimensional probability have proved fruitful in attacking problems where sparsity is a prominent feature. I will highlight its role in work joint with Rudnev, Shkredov and Zhelezov on the sum-product problem, and if time permits, newer results joint with Waterhouse convolutions in the boolean cube.
Location: STEM-664
Time: 09:00-10:00
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29 September, 2025
In person lecture (with Zoom live stream)
Omer Avci (University of Ottawa)
Title: Torsion of Rational Elliptic Curves over the Galois Extensions of $\mathbb{Q}$
Abstract: Mazur's celebrated theorem gives a complete classification of the torsion subgroups $E(\mathbb{Q})_{\mathrm{tors}}$ for elliptic curves $E/\mathbb{Q}$. This result inspired the broader problem of classifying $E(L)_{\mathrm{tors}}$ for elliptic curves $E/L$, where $L$ is a field of characteristic zero. In this talk, I will first review results from the literature and some variants of this problem. I will then focus on the case where $L/\mathbb{Q}$ is a Galois extension, outlining our methods and presenting two families of results: when $L = \mathbb{Q}(\zeta_p)$ for a prime $p$, and when $L$ is a $\mathbb{Z}_p$-extension of a quadratic field $K$.
Location: STEM-664
Time: 09:00-10:00
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6 October, 2025
Virtual lecture (Zoom live stream)
Nic Banks (University of Waterloo)
Title: Classification results for intersective polynomials with no integral roots
Abstract: In this talk, I describe the contents of my recently-defended PhD thesis on strongly intersective polynomials. These are polynomials with no integer roots but with a root modulo every positive integer, thereby constituting a failure of the local-global principle. We start by describing their relation to Hilbert's 10th Problem and an algorithm of James Ax. These are fascinating objects which make contact with many areas of math, including permutation group theory, splitting behaviour of prime ideals in number fields, and Frobenius elements from class field theory.
In particular, we discuss constraints on the splitting behaviour of ramified primes in splitting fields of intersective polynomials, building on the work of Berend-Bilu (1996) and Sonn (2008). We also explain the computation of a list of possible Galois groups of such polynomials, which includes many examples and which supports some recent conjectures of Ellis & Harper (2024).
Time permitting, we end by discussing future work, including results from permutation group theory and from character theory.
Location: Zoom live stream
Time: 09:00-10:00
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20 October, 2025
In person lecture (with Zoom live stream)
Sara Sajadi (University of Toronto)
Title: A Unified Finiteness Theorem For Curves
Abstract: This talk presents a unified framework for finiteness results concerning arithmetic points on algebraic curves, exploring the analogy between number fields and function fields. The number field setting, joint work with F. Janbazi, generalizes and extends classical results of Birch–Merriman, Siegel, and Faltings. We prove that the set of Galois-conjugate points on a smooth projective curve with good reduction outside a fixed finite set of places is finite, when considered up to the action of the automorphism group of a proper integral model. Motivated by this, we consider the function field analogue, involving a smooth and proper family of curves over an affine curve defined over a finite field. In this setting, we show that for a fixed degree, there are only finitely many étale relative divisors over the base, up to the action of the family's automorphism group (and including the Frobenius in the isotrivial case). Together, these results illustrate both the parallels and distinctions between the two arithmetic settings, contributing to a broader unifying perspective on finiteness.
Location: STEM-464
Time: 09:00-10:00
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27 October, 2025
In person lecture (with Zoom live stream)
Charlie Wu (University of Toronto)
Title: Compactness and character varieties
Abstract: Let $X$ be an orientable genus $g$ surface with $n$ punctures. Relative character varieties are spaces parametrizing isomorphism classes of representations of $\pi_1(X)$ satisfying some local conditions around the punctures. When this space is a single point, some remarkable work of Katz shows that the unique representation in this space has interesting arithmetic and complex geometric properties - namely that it is defined over the ring of integers of a number field and it ``comes from geometry". We discuss the geometry of this space when it is larger than a point, and we give a classification of their compact components. This is joint work with Daniel Litt.
Location: STEM-664
Time: 09:00-10:00
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3 November, 2025
In person lecture (with Zoom live stream)
Felix Baril Boudreau (CICMA & Université du Luxembourg)
Title: Abelian varieties with homotheties
Abstract: 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 $G_\ell$ of the $\ell$-adic representation of the absolute Galois group of $K$ contains all $c$-th power homotheties, where $c$ is a positive constant. If $K$ is a global function field, the analogous statement fails in general, since Zahrin has shown the existence of ordinary Abelian varieties of positive dimensions defined over $K$, for which $G_\ell$ only contains finitely many homotheties. In this talk, I will discuss my ongoing joint work with Sebastian Petersen (University of Kassel), in which we prove, under suitable additional assumptions, an analogue of Bogomolov--Serre Theorem when $K$ is a finitely generated field of positive characteristic.
Location: STEM-664
Time: 09:00-10:00
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10 November, 2025
In person lecture (with Zoom live stream)
Luochen Zhao (Morningside Center of Mathematics)
Title: On the arithmetic of Bernoulli--Hurwitz periods
Abstract: Let E be an elliptic curve having good ordinary reduction at a prime p. The values of the classical Eisenstein series at E are algebraic and are called Bernoulli--Hurwitz numbers, and they admit a p-adic interpolation by specializing Katz's one-variable Eisenstein measure at E. We will explain that the periods of this p-adic measure are modular, i.e., are special values of certain weight one higher level Eisenstein series. Furthermore, we explain a new proof of the interpolation by this modularity, as well as how one can get a p-adic Kronecker's first limit formula.
Location: STEM-664
Time: 09:00-10:00
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14 November, 2025
In person lecture (with Zoom live stream)
Luochen Zhao (Morningside Center of Mathematics)
Title: On the structure of anticyclotomic Selmer groups of a supersingular elliptic curve
Abstract: Let p be a fixed prime. The study of the variation of Selmer groups attached to a given Galois representation over a Z_p-extension is a central topic in Iwasawa theory. When the Galois representation is the Tate module of a rational elliptic curve having good supersingular reduction at p, the information of the Selmer groups becomes rather elusive due to the failure of Mazur's control theorem. In this expository talk I'll explain a strategy (due to Matar) to study the growth of Selmer groups of a supersingular elliptic curve over the anticyclotomic Z_p extension, assuming the indivisibility of the Heegner point. Along the way, we will introduce Kobayashi's modification of Selmer groups (i.e., plus/minus Selmer groups), and explain how they could be fitted together to yield information on the usual Selmer groups.
Location: STEM-664
Time: 09:00-10:00
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17 November, 2025
In person lecture (with Zoom live stream)
Luochen Zhao (Morningside Center of Mathematics)
Title: On the structure of anticyclotomic Selmer groups of modular forms
Abstract: I will report the recent work with Antonio Lei and Luca Mastella, in which we determine the structure of the Selmer group of a modular form over the anticyclotomic Zp extension, assuming the imaginary quadratic field satisfies the Heegner hypothesis, that p splits in it and at which the form has good reduction, and that the bottom generalized Heegner class is primitive. Here the last assumption springs from Gross's treatment of Kolyvagin's bound on Shafarevich--Tate groups, and was put in the Iwasawa theoretic context by Matar--Nekovář and Matar for elliptic curves. This talk will focus on our use of the vanishing of BDP Selmer groups in proving the result, which allows us to treat both ordinary and supersingular reduction types uniformly.
Location: STEM-664
Time: 09:00-10:00
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24 November, 2025
In person lecture (with Zoom live stream)
Ben Forras (University of Ottawa)
Title: Graduated orders in equivariant Iwasawa theory
Abstract: We describe graduated orders over regular local rings of dimension at most two, and explain how this can be used to prove integrality results in equivariant Iwasawa theory.
Location: STEM-664
Time: 09:00-10:00
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8 December, 2025
In person lecture (with Zoom live stream)
Nathan Grieve (Carleton University)
Title: Complexity thresholds for divisors and explicit effective Diophantine approximation of rational points
Abstract: I will survey recent results which surround complexity thresholds for divisors, including measures of positivity and singularities thereof, and explain how they interplay with Diophantine approximation of algebraic points. A portion of these results include recent joint work C. Noytaptim. Further, I will place emphasis on explicit and effective results.
Location: Carleton University (Room TBD)
Time: 10:00-11:00
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TBD
In person lecture (with Zoom live stream)
Romain Branchereau (McGill University)
Title: TBD
Abstract: TBD
Location: STEM-664
Time: TBD (Winter 2026?)
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TBD
Virtual lecture (Zoom live stream)
Flora Poon (NCTS, Taiwan)
Title: TBD
Abstract: TBD
Location: Zoom live stream
Time: TBD (Winter 2026)
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TBD
In person lecture (with Zoom live stream)
Sunil Lakshmana Naik (Queen's University)
Title: TBD
Abstract: TBD
Location: Zoom live stream
Time: TBD (Winter 2026)
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TBD
Virtual lecture (Zoom live stream)
Alessandro Fazzari (University of Montreal)
Title: TBD
Abstract: TBD
Location: Zoom live stream
Time: TBD (Winter 2026)
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