Add Headings and they will appear in your table of contents.
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The webpage for the 2022-2023 seminar is here.
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Organizers.
Saban Alaca: SabanAlaca@cunet.carleton.ca
Nathan Grieve: nathan.m.grieve@gmail.com (Contact organizer)
Damien Roy: damien.roy@uottawa.ca
Abdellah Sebbar: asebbar@uottawa.ca
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Wednesday 29 June 2022
Special day of in person lectures
In person meeting Location: Room 201 of the STEM building (U. of Ottawa)
Virtual Zoom live stream details (please contact nathan.m.grieve@gmail.com for the Zoom link details).
Schedule, talk titles and abstracts:
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The intent is that the hour long talks will be about 45 mintues to allow for questions and a breif break. Similarly, the intent for the 30 minute talks is that they will be about 25 mintues.
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9:00--10:00
Nathan Grieve (RMC/Carleton/UQAM)
Title: On geometry of numbers and Diophantine approximation for linear series
Abstract: In some sense, this lecture will be a follow-up to the talk that I gave in Banff last week. First, I will provide context and motivation for the manner in which Diophantine arithmetic questions for projective varieties lead naturally to the study of filtered linear series. Then, I will report on some recent results which pertain to a form of Central Limit Theorem for a concept that I call Harder and Narasimhan data. Finally, I will discuss some recent ongoing work with Ruiran Sun. It deals with a numerical concept of slope stability, for big and nef birational divisor classes, with respect to prime birational divisors. In particular, I intend to mention some illustrative examples that we have been studying over the past few months.
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10:00--10:30
Mohammadreza Mohajer (U. of Ottawa)
Title: Geometry of Arithmetic Periods
Abstract: Period numbers have been studied in a wide rage of mathematical areas. Questions about algebraic relation and linear relation of a period space have had always substantial attention in number theory. I will talk about arithmetic and geometric source of periods and their relations. We will see how the behaviour of non-trivial vanishing periods and L-values relates to Grothendieck period conjecture. For 1-periods which be viewed also as periods of Deligne 1-motives, I will mention new results such as Kontsevich period Conjecture for such periods. Like original periods, we have p-adic periods in the sense of crystalline Frobenius which have both arithmetic and geometric source similarly. We are looking for analogous results and connections for such p-adic periods.
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10:30--11:30
Allysa Lumley (McGill U.) (Virtual lecture)
Title: Distribution of Values of L-functions in the critical strip
Abstract: In this talk, we will discuss the distribution of values for various L-functions in the critical strip (complex numbers with real part between 0 and 1). We will start with some key motivations for studying L(1+it,f), describe the state of the art for results and techniques. If there is time , we will also discuss what happens in the cases for $L(\sigma+it,f)$ where $1/2\le \sigma <1$.
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11:30--13:30 Lunch
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13:30--14:30
Ruiran Sun (McGill U.)
Title: One-pointed Shafarevich's conjecture for moduli spaces of canonically polarized manifolds
Abstract: Motivated by Shafarevich's conjecture, Arakelov-Parshin proved the following finiteness result: for every curve C, the set of isomorphism classes of nonconstant morphisms C \to M_g is finite (g \geq 2). For moduli stacks parametrizing higher dimensional varieties Arakelov-Parshin's finiteness theorem fails for trivial reason, i.e. the existence of product families. In this talk we will explain that this is somehow the only obstruction: the finiteness theorem holds true for the Hom set of "pointed" curves (in which the product families are excluded). We also discuss some application of this result. This is a joint work with Ariyan Javanpeykar, Steven Lu and Kang Zuo.
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14:30--15:00
Gary Walsh (U. of Ottawa and CSE)
Title: An application of Runge's and Baker's theorems to an effective version of a theorem of Shioda on ranks of elliptic curves
Abstract: Numerous papers have appeared in the literature on ranks of elliptic curves of particular types. An effective generalization containing all of these curves is proved, and we show how this generalization is closely related to work of Shioda. We further this investigation to an interesting subfamily having curves of considerably higher rank.
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15:00--16:00
Damien Roy (U. of Ottawa)
Title: Diophantine approximation with constraints
Abstract: Consider an arbitrary n-tuple of real numbers which are linearly independent over the field of rational numbers. A result of Dirichlet from 1842 tells us that there are infinitely many integral linear combinations of these numbers whose absolute value is bounded above by an explicit function of their norm, namely the largest absolute value of their coefficients. Moreover, this result is essentially best possible, in the sense that, for some n-tuples, one cannot do better except for a multiplicative constant. Following work of Schmidt in 1976 and of Thurnheer in 1990, we study what happens when one asks the first n-m coefficients of the linear combinations to be positive for some positive integer m at most equal to n-2. To show that our estimate is essentially best possible, we first extend the theory of parametric geometry of numbers to allow angular constraints. This is joint work with my former MSc student Jérémy Champagne.
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16:00--17:00
Adam Logan (Carleton U. and Govt. of Canada)
Title: Lattice neighbours and modular forms (joint work with Eran Assaf, Daniel Fretwell, Colin Ingalls, Spencer Secord, and John Voight)
Abstract: Kneser introduced the relation of neighbourhood of lattices, which has been very important both for understanding lattices themselves and in applications to algebraic geometry and elsewhere. Chenevier and Lannes proved some beautiful formulas involving modular forms for the number of ways in which one lattice is a neighbour of another; however, the nature of their work restricted them to a very small number of lattices. In this talk we will define the terms above and describe our attempts to find related equalities in a more general setting, as well as some connections with special values of L-functions and congruences of modular forms. We will also present a mysterious conjecture about lattices of rank 6 that emerged from our calculations.
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Past seminar talks from 2021-2022 academic year.
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Schedule of Talks. All lectures are intended to be about 50 minutes. They are scheduled as virtual talks unless explicitly noted otherwise. In case that the travel/visitor situation improves, then additional person lectures may be possible. For virtual talks, the Zoom meeting information will be sent to the seminar mailing list and posted here at a time closer to the seminar date. If you are interested in joining the seminar mailing list and/or giving a talk, then please email the contact organizer.
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Tuesday 21 September 2021
Special in person lecture
Title: Omega results for cubic field counts via the Katz-Sarnak philosophy
Speaker: Daniel Fiorilli (Université Paris-Saclay, CNRS)
Time: 15:30
Room: STEM 664
Abstract: I will discuss recent joint work with P. Cho, Y. Lee and A. Södergren. Since the work of Davenport-Heilbronn, much work has been done to obtain a precise estimate for the number of cubic fields of discriminant at most X. This includes work of Belabas-Bhargava-Pomerance, Bhargava-Shankar-Tsimerman and Taniguchi-Thorne. In this talk I will present a negative result, which states that the GRH implies that the error term in this estimate cannot be too small. Our approach involves low-lying zeros of Dedekind zeta functions of cubic fields (first studied by Yang), and is strongly related to the Katz-Sarnak conjectures and the ratios conjecture of Conrey, Farmer and Zirnbauer.
Note: To access U.O. campus, first upload your proof of complete vaccination on
https://www.uottawa.ca/coronavirus/en/mandatory-covid-19-vaccination
Then, fill the COVID-19 Daily Health Check-In form online before coming to campus. You will get an automatic confirmation.
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Wednesday 20 October 2021
Virtual Zoom Lecture
Title: A generalization to Elkies's theorem
Speaker: Wanlin Li (CRM)
Time: 14:30
Recorded Talk: mp4
Abstract: Elkies proved that for a fixed elliptic curve over Q, there exist infinitely many primes at which its reductions are supersingular. In this talk, we give the first generalization to Elkies’s theorem for some curves of genus >2. We consider families of cyclic covers of the projective line ramified at 4 points whose moduli space is embedded in a Shimura curve.This is joint work in progress with Elena Mantovan, Rachel Pries, and Yunqing Tang.
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Wednesday 27 October 2021
Virtual Zoom Lecture
Title: Class groups, congruences, and cup products
Speaker: Eric Stubley (McGill)
Time: 14:30
Slides: slides.pdf
Abstract: The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the p-part of the class group of the p-th cyclotomic field to congruences of Bernoulli numbers mod p. For p and N prime with N=1 mod p, a similar result of Calegari and Emerton relates the rank of the p-part of the class group of Q(N^(1/p)) to whether or not a certain quantity (Merel's number) is a p-th power mod N.
I'll speak about joint work with Karl Schaefer in which we study this rank, refining the result of Calegari and Emerton and proving exact characterizations of the rank for small p in terms of similar p-th power congruence conditions. Our main tactic is to relate elements of the class group to the vanishing of cup products in Galois cohomology. I'll aim to give some gentle exposition of how to think about these cup products, and I'll highlight the ways in which our work was informed by computation.
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Wednesday 3 November 2021
Virtual Zoom Lecture
Title: Primes in short intervals - Heuristics and calculations
Speaker: Allysa Lumley (McGill)
Time: 14:30
Abstract: We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length y around $x$, where $y\ll(\log x)^2$. In particular, we conjecture that the maximum grows surprisingly slowly as $y$ ranges from $\log x$ to $(\log x)^2$. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification. This is joint work with Andrew Granville.
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Wednesday 10 November 2021
Virtual Zoom Lecture
Title: Squarefrees (and B-frees) in short intervals
Time: 14:30
Speaker: Brad Rodgers (Queen's)
Abstract: In this talk I will discuss the distribution of squarefree integers in random short intervals. In particular I hope to explain forthcoming work with O. Gorodetsky and A. Mangerel demonstrating that counts of squarefrees in short intervals exhibit gaussian behavior, answering an old question of R. R. Hall. In fact, this result can be proved for B-free integers, which are a generalization of squarefrees to be discussed in the talk. I will also explain a curious connection between these counts and fractional Brownian motion.
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Tuesday 16 November 2021
Virtual Zoom Lecture
Title: Explicit Rational Equivalences of Points on Surfaces
Time: 14:30
Slides: slides.pdf
Speaker: Jonathan Love (McGill)
Abstract: The Chow group of zero-cycles on a smooth projective surface X is obtained by taking the free abelian group generated by closed points on X, and declaring two elements ("zero-cycles") to be equal if their difference can be written as a sum of divisors of rational functions on curves in X; in this case we say the zero-cycles are "rationally equivalent." These Chow groups are notoriously difficult to compute; while a set of conjectures due to Bloch and Beilinson predict the existence of certain relations in these groups when X is defined over a number field, there are very few non-trivial cases in which these relations have been proven to hold. In this talk, I will discuss techniques that can be used to compute rational equivalences exhibiting some of the expected relations, in the case that X is a product of two elliptic curves over Q.
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Tuesday 23 November 2021
Virtual Zoom Lecture
Title: Characters of VOAs and modular forms
Time: 14:30
Slides: slides.pdf
Speaker: Cameron Franc (McMaster)
Abstract: Since their first appearance in mathematics and physics, the theory of vertex operator algebras has been intimately connected with the theory of modular forms. In this talk we will review some of this story, emphasizing aspects of Zhu's theorem and related results that make this connection precise. We will then discuss how the theory of modular forms can be used to help classify nice classes of VOAs, akin to using restrictions on character tables to classify representations of finite groups. Both old and new cases where this program has been carried out successfully will be discussed, as well as some open problems. No background on VOA theory will be assumed.
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Wednesday 1 December 2021
Virtual Zoom Lecture
Title: Computing an L-function modulo a prime
Time: 14:30
Slides: slides.pdf
Speaker: Felix Baril Boudreau (Western)
Abstract: Let K be a function field with a constant field of size q. If E is an elliptic curve over K with nonconstant j-invariant then its L-function L(T,E/K) is a polynomial in 1 + T Z[T]. Inspired by the algorithms of Schoof and Pila for computing zeta functions of curves over finite fields, we consider the problem of computing the reduction of L(T,E/K) modulo an integer without first computing the whole L-function. Doing so for sufficiently many integers coprime with q completely determines L(T,E/K). The existing literature on this problem could be summarized as follows: Under the assumption that the Mordell-Weil group E(K) has a subgroup of order N ≥ 2, with N coprime with q, Chris Hall gave an explicit formula for the reduction L(T,E/K) mod N. We present novel theorems going beyond Hall's. https://arxiv.org/abs/2110.12156
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Wednesday 8 December 2021
Virtual Zoom Lecture
Title: Greatest common divisors near $S$-units and applications
Time: 14:30
Slides: slides.pdf
Speaker: Zheng Xiao (Michigan State)
Abstract: The notion of GCD between integers was generalized to algebraic numbers and further to blow-ups along a closed subscheme. In this talk, we will discuss the results of GCD on integers and polynomials by Begeaud-Corvaja-Zannier and Levin. Then followed with my generalization of $S$-units to almost $S$-units and the improved version of polynomial GCD relating the Vojta's conjecture. This leads to applications to linear recurrences. If time allows, we will also discuss the GCD problem over abelian surfaces and its relation between Vojta's proven conjecture of arithmetic discriminant.
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Wednesday April 6 2022
Special in person lecture
Title: Integers as Sums of Three Cubes
Time: 14:30
Location: Room 464 of the STEM building (U. of Ottawa)
Speaker: Gary Walsh (U. of Ottawa and CSE)
Abstract: In this lecture we discuss a variety of topics on the diophantine equation $x^3+y^3+z^3=k$, from a historical look at the computational attempts to solve the case $k=3$, to recent extraordinary discoveries. We also develop an algorithmic approach that appears to have been overlooked, and shows how solutions with many decimal digits can often arise naturally from the existence of certain algebraic numbers of remarkably small height.
Note: To access U.O. campus, first upload your proof of complete vaccination on
https://www.uottawa.ca/coronavirus/en/mandatory-covid-19-vaccination
Then, fill the COVID-19 Daily Health Check-In form online before coming to campus. You will get an automatic confirmation.