# 2023-2024 -- Ottawa/Carleton Number Theory Seminar

Organizers.

Nathan Grieve: nathan.m.grieve@gmail.com (Contact organizer)

Antonio Lei: alantoniolei@gmail.com

Erman Isik: eisik@uottawa.ca

Webpage of the Ottawa-Carleton Number Theory group:

Spring 2024

We plan to have a day of talks scheduled for Monday 13 May 2024. The talks will take place in person at the University of Ottawa STEM-664.

List of confirmed speakers and schedule of talks.

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Time: 9:00 - 10:00

Speaker: Gary Walsh (Tutte Institute and University of Ottawa)

Title: Solving problems of Erdos using elliptic curves and an elliptic curve analogue of the Ankeny-Artin-Chowla Conjecture

Abstract: We describe how the Mordell-Weil group of rational points on a certain families of elliptic curves give rise to solutions to conjectures of Erdos on powerful numbers, and state a related conjecture, which can be viewed as an elliptic curve analogue of the Ankeny-Artin-Chowla Conjecture.

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Time: 10:30-11:30

Speaker: Arul Shankar (University of Toronto)

Title: Conditional bounds on the 2, 3, 4, and 5 torsion of the class groups of number fields

Abstract: Let n be a positive integer, and let K be a degree n number field. It is believed that the class group of K should be a cyclic group, up to factors that are negligible compared to the size of the discriminant of K. Another way of phrasing this is to say that for any fixed m, the m torsion subgroup of the class group of K is negligible in size. This is only known for the 2 torsion subgroups of quadratic fields by work of Gauss.

For other pairs m and n, it is a natural question to obtain nontrivial bounds for the sizes of the m torsion in the class groups of degree n fields K.

In this talk, I will discuss joint work with Jacob Tsimerman, in which we prove such bounds, conditional on some standard elliptic curve conjectures, for the cases m=2, 3, 4, and 5 (and where n is allowed to be any positive integer).

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Time: 13:30-14:30

Speaker: Mohammadreza Mohajer (University of Ottawa)

Title: Exploring p-adic periods of 1-motive

Abstract: Period numbers and p-adic periods are crucial in number theory, offering insights into transcendence theory and arithmetic geometry. Classical period numbers, arising from integrals of algebraic differential forms, serve as transcendental numbers, encoding deep arithmetic information. Studying classical periods is well-explored in curtain cases however, extending these concepts to their p-adic counterparts present greater complexity. In this work, we develop an integration theory for 1-motives with good reduction, serving as a generalization of Fontaine-Messing p-adic integration. For 1-motive M with good reduction, the p-adic numbers resulting from this integration are called Fontaine-Messing p-adic periods of M. We identify a suitable p-adic Betti-like Q-structure inside the crystalline realisation and we show that a p-adic version Kontsevich-Zagier conjecture holds for M, if one takes the Fontaine-Messing p-adic periods of M relative to its p-adic Betti lattice. This theorem is the p-adic version of analytic subgroup theorem for 1-motives with good reduction.

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Time: 15:00-16:00

Speaker: Mathilde Gerbelli-Gauthier (McGill U.)

Title: Statistics of automorphic forms using endoscopy

Abstract: Classical questions about modular forms on SL_2 have direct analogues on higher-rank groups: What is the dimension of spaces of forms of a given weight and level? How are the Hecke eigenvalues distributed? What is the sign of the functional equation of the associated L-function? Though exact answers can be hard to obtain in general for groups of higher rank, I’ll describe some statistical results towards these questions, and outline how we obtain them using the stable trace formula. This is joint work, some of it in progress, with Rahul Dalal.

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Additional details about this day of talks will be posted here in due time.

Another Ottawa based event this spring that may be of interest is the June workshop on Galois Cohomolgy and Massey Products. There is also the CNTA XVI which will takes place in Toronto at the Fields Institute.

Winter 2024

Schedule of Talks. The details for the seminar talks that have been scheduled at various times throughout the winter semester are posted below.

All lectures are intended to be about 50 minutes with 5-10 minutes for questions and/or discussions. We intend to have a collection of both in person and virtual lectures. If you are interested in joining the seminar mailing list and/or giving a talk, then please email the contact organizer. The seminar schedule is also available on researchseminars.org. The static Zoom link for the seminar is: here. Please be in touch in case that you require a password to join.

List of confirmed speakers:

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Tuesday 5 March 2024

In person lecture (with Zoom live stream)

Felix Baril Boudreau (U. Lethbridge)

Title: The Distribution of Logarithmic Derivatives of Quadratic L-functions in Positive Characteristic

Abstract: To each square-free monic polynomial D in a fixed polynomial ring Fq[t], we can associate a real quadratic character 𝜒D, and then a Dirichlet L-function L(s,𝜒D). We compute the limiting distribution of the family of values L'(1,𝜒D)/L(1,𝜒D) as D runs through the square-free monic polynomials of Fq[t] and establish that this distribution has a smooth density function. Time permitting, we discuss connections of this result with Euler-Kronecker constants and ideal class groups of quadratic extensions. This is joint work with Amir Akbary.

Time: 16:00-17:00 EDT

Location: uOttawa STEM-664 and Zoom live stream

Slides: .pdf

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Tuesday 12 March 2024 (Postponed)

In person lecture

Gary Walsh (Tutte Institute and University of Ottawa)

Title: Integral points on elliptic curves - an excursion into speculative number theory

Abstract: We continue work of ours and of Bennett concerning the existence of primitive integral points on elliptic curves with rational 2 torsion, and relate the existence to various problems in Analytic Number Theory and Diophantine Analysis.

Numerous open problems will be discussed, along with a status report using recent theorems that follow from the work of Wiles and others on ternary Diophantine equations.

Time: 16:00-17:00 EDT

Location: uOttawa STEM-664 and Zoom live stream

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Tuesday 19 March 2024

Virtual Lecture

Fırtına Küçük (University College Dublin)

Title: Factorization of algebraic p-adic L-functions of Rankin-Selberg products

Abstract: In the first part of the talk, I will give a brief review of Artin formalism and its p-adic variant. Artin formalism gives a factorization of L-functions whenever the associated Galois representation decomposes. I will explain why the p-adic Artin formalism is a non-trivial problem when there are no critical L-values. In particular, I will focus on the case where the Galois representation arises from a self-Rankin-Selberg product of a newform, and present the results in this direction including the one I obtained in my PhD thesis.

In the last part of the talk, I will discuss the case where the newform f in question has a theta-critical p-stabilization, i.e. if f is in the image of the theta operator. Unlike the ordinary and the non-critical slope cases, one cannot simply define the p-adic L-function of f in terms of its interpolative properties. I will discuss technical difficulties paralleling this and explain the degenerate properties of the theta-critical forms in terms of the algebro-geometric properties of the eigencurve.

Time: 16:00-17:00 EDT

Location: Zoom live stream (Link to Zoom recording)

Slides: .pdf

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Tuesday 9 April 2024

Virtual Lecture

Abhishek Bharadwaj (Queen's U.)

Title: Sufficient conditions for a problem of Polya

Abstratct: There is an old result attributed to Polya on identifying algebraic integers by studying the power traces; and a finite version of this result was proved by Bart de Smit. We study the generalisation of these questions, namely determining algebraic integers by imposing certain constraints on the power sums. This is a joint work with V Kumar, A Pal and R Thangadurai. Time permitting, we will also describe related results in an ongoing project.

Time: 16:00-17:00 EDT

Location: Zoom live stream

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Fall 2023

Schedule of Talks. All lectures are intended to be about 50 minutes with 5-10 minutes for questions and/or discussions. We intend to have a collection of both in person and virtual lectures. If you are interested in joining the seminar mailing list and/or giving a talk, then please email the contact organizer. The seminar schedule is also available on researchseminars.org. The static Zoom link for the seminar is: here. Please be in touch in case that you require a password to join.

List of confirmed speakers:

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Tuesday 10 October 2023

In person lecture

Muhammad Manji (University of Warwick)

Title: Iwasawa Theory for GU(2,1) at inert primes

Time: 16:00-17:00 EDT

Location: uOttawa STEM-464

Notes: .pdf

Abstract: The Iwasawa main conjecture was stated by Iwasawa in the 1960s, linking the Riemann Zeta function to certain ideals coming from class field theory, and proved in 1984 by Mazur and Wiles. This work was generalised to the setting of modular forms, predicting that analytic and algebraic constructions of the p-adic L-function of a modular form agree, proved by Kato (’04) and Skinner--Urban (’06) for ordinary modular forms. For the non-ordinary case there are some modern approaches which use p-adic Hodge theory and rigid geometry to formulate and prove cases of the conjecture. I will review these cases and discuss my work in the setting of automorphic representations of unitary groups at non-split primes, where a new approach uses the L-analytic regulator map of Schneider—Venjakob. My aim is to state a version of the conjecture which was previously unknown, and discuss what is still needed to prove the conjecture in full.

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Tuesday 17 October 2023

In person lecture

Erman Isik (University of Ottawa)

Title: Modular approach to Diophantine equation $x^p+y^p=z^3$ over some number fields

Time: 16:00-17:00 EDT

Location: uOttawa STEM-464 and Zoom live stream

Notes: .pdf

Abstract: Solving Diophantine equations, in particular, Fermat-type equations is one of the oldest and most widely studied topics in mathematics. After Wiles’ proof of Fermat’s Last Theorem using his celebrated modularity theorem, several mathematicians have attempted to extend this approach to various Diophantine equations and number fields over several number fields.

The method used in the proof of this theorem is now called “modular approach”, which makes use of the relation between modular forms and elliptic curves. I will first briefly mention the main steps of the modular approach, and then report our asymptotic result (joint work with {\"O}zman and Kara) on the solutions of the Fermat-type equation $x^p+y^p=z^3$ over various number fields.

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Thursday 26 October 2023

Special in person lecture (Fields-Carleton Distinguished Lecture)

Chelsea Walton (Rice U.)

Title: Modernizing Modern Algebra, I: Category Theory is coming, whether we like it or not

Time: 19:00-20:00 EDT

Location: 274, 275 Teraanga Commons, Carleton U.

https://science.carleton.ca/events/fields-carleton-distinguished-lecture/

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Friday 27 October 2023

Special in person lecture (Fields-Carleton Distinguished Lecture)

Chelsea Walton (Rice U.)

Title: Modernizing Modern Algebra, II: Category Theory is coming, whether we like it or not

Time: 13:30-14:30 EDT (Coffee/tea starting at 1 p.m.)

Location: 4351 Herzberg Building, Macphail Room, Carleton U.

https://science.carleton.ca/events/fields-carleton-distinguished-lecture/

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Tuesday 7 November 2023

Virtual lecture

Chatchai Noytaptim (University of Waterloo)

Title: Arithmetic Dynamical Questions with Local Rationality

Time: 16:00-17:00 EDT

Location: Zoom

Abstract: In this talk, we first introduce a numerical criterion which bounds the degree of any algebraic integer in short intervals (i.e., intervals of length less than 4). As an application, we classify all unicritical polynomials defined over the maximal totally real extension of the field of rational numbers. Using tools from complex and p-adic potential theory, we also classify all quadratic unicritical polynomials defined over the field of rational numbers in which they have only finitely many totally real preperiodic points. In particular, we are able to explicitly compute totally real preperiodic points of some quadratic unicritical polynomials by applying the numerical tool and p-adic dynamics. This is based on joint work with Clay Petsche.

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Tuesday 14 November 2023

Virtual lecture

David Nguyen (Queen's U.)

Title: Variance over Z and moments of L-functions

Time: 16:00-17:00 EDT

Location: Zoom

Notes: .pdf

Abstract: One of the central problems in analytic number theory has been to evaluate moments of the absolute value of L-functions on the critical line. Bounds on these moments are approximations to the Lindelöf hypothesis and, thus, subconvexity bounds for these L-functions. Besides a few low moments where rigorous results are known, sharp bounds on higher moments are wide open. Recently, in 2018, it has been discovered that there is a certain connection between asymptotics of moments of L-functions and variance over the integers (the Keating--Rodgers--Roditty-Gershon--Rudnick--Soundararajan conjecture in arithmetic progressions). Certain analogues of this conjecture are completely known, i.e., are theorems, in the function field setting. In this lecture, I plan to explain this new connection between asymptotics of variance over Z and those of moments, and discuss my work on confirming a smoothed version of this conjecture in a restricted range.

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Tuesday 21 November 2023

Virtual lecture

Akash Sengupta (University of Waterloo)

Title: Radical Sylvester-Gallai configurations

Time: 16:00-17:00 EDT

Location: Zoom

Notes: .pdf

Abstract: In 1893, Sylvester asked a basic question in combinatorial geometry: given a finite set of distinct points v_1,..., v_m in R^n such that the line joining any pair of distinct points v_i,v_j contains a third point v_k in the set, must all points in the set be collinear?

The classical Sylvester-Gallai (SG) theorem says that the answer to Sylvester’s question is yes, i.e. such finite sets of points are all collinear. Generalizations of Sylvester's problem, which are known as Sylvester-Gallai type problems have been widely studied by mathematicians, have found remarkable applications in algebraic complexity theory and coding theory. The underlying theme in all Sylvester-Gallai type questions is the following:

Are Sylvester-Gallai type configurations always low-dimensional?

In this talk, we will discuss a non-linear generalization of Sylvester's problem, and its connections with the Stillman uniformity phenomenon in Commutative Algebra. I’ll talk about an algebraic-geometric approach towards studying such SG-configurations and a result showing that radical SG-configurations are indeed low dimensional as conjectured by Gupta in 2014. This is based on joint work with Rafael Oliveira.

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5 December 2023

In person lecture

Harun Kir (Queen's U.)

Title: The refined Humbert invariant as an ingredient

Time: 16:00-17:00

Location: STEM 664 EDT and Zoom

Notes: .pdf

Abstract: In this talk, I will advertise the refined Humbert invariant, which is the main ingredient of my research. It was introduced by Ernst Kani (1994) upon observing that every curve $C$ comes equipped with a canonically defined positive definite quadratic form $q_C$. This result can be used to define algebraically the (usual) Humbert invariant (1899) and Humbert surfaces.

The beauty of the refined Humbert invariant is that it translates the geometric questions into the arithmetic questions. Therefore, it allows us to solve many interesting geometric problems regarding the nature of curves of genus $2$ including the automorphism groups and the elliptic subcovers of these curves, the intersection of the Humbert surfaces, and the CM points on the Shimuracurves in this intersection.

I will also give the classification of this invariant in the CM case as these illustrations reveal how interesting the refined Humbert invariant is.

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