
Date 
Speaker 
Title 


Jan 9 
Youness Lamzouri 
A walk on Legendre paths 

at noon
in M1040

(Institut Élie Cartan de Lorraine, France)
PIMS Distinguished Speaker Series

The Legendre symbol is one of the most basic, mysterious and extensively studied objects in number theory. It is a multiplicative function that encodes information about whether an integer is a square modulo an odd prime $p$. The Legendre symbol was introduced by AdrienMarie Legendre in 1798, and has since found countless applications in various areas of mathematics as well as in other fields including cryptography. In this talk, we shall explore what we call "Legendre paths", which encode information about the values of the Legendre symbol. The Legendre path modulo $p$ is defined as the polygonal path in the plane formed by joining the partial sums of the Legendre symbol modulo $p$. In particular, we will attempt to answer the following questions as we vary over the primes $p$: how are these paths distributed? how do their maximums behave? and what proportion of the path is above the real axis? Among our results, we prove that these paths converge in law, in the space of continuous functions, to a certain random Fourier series constructed using Rademakher random multiplicative functions. Part of this work is joint with Ayesha Hussain.



Jan 16 
Neelam Kandhil 
On linear independence of Dirichlet Lvalues 

at 9:30am
via zoom

(Institute of Mathematical Sciences, Chennai, India)

It is an open question of Baker whether the Dirichlet Lvalues at 1 with fixed modulus are linearly
independent over the rational numbers. The bestknown result is due to Baker, Birch and Wirsing, which affirms
this when the modulus of the associated Dirichlet character is coprime to its Euler's phi value. In this talk,
we will discuss an extension of this result to any arbitrary family of moduli. The interplay between the
resulting ambient number fields brings new technical issues and complications hitherto absent in the context of
a fixed modulus. We will also investigate the linear independence of such values at integers greater than 1.



Jan 23 
Antonella Perucca 
Recent advances in Kummer theory 

at 9:30am
via zoom

(University of Luxembourg)

Kummer theory is a classical theory about radical extensions of fields in the case where suitable roots of unity are present in the base field. Motivated by problems close to Artin's primitive root conjecture, we have investigated the degree of families of general Kummer extensions of number fields, providing parametric closed formulas. We present a series of papers that are in part joint work with Christophe Debry, Fritz Hörmann, Pietro Sgobba, and Sebastiano Tronto.



Jan 30 
Oussama Hamza 
Filtrations, arithmetic and explicit examples in an equivariant context 

at noon
via zoom

(University of Western Ontario)

Pro$p$ groups arise naturally in number theory as quotients of absolute Galois groups over number fields. These groups are quite mysterious. During the 60's, Koch gave a presentation of some of these quotients. Furthermore, around the same period, Jennings, Golod, Shafarevich and Lazard introduced two integer sequences $(a_n)$ and $(c_n)$, closely related to a special filtration of a finitely generated pro$p$ group $G$, called the Zassenhaus filtration. These sequences give the cardinality of $G$, and characterize its topology. For instance, we have the wellknown Gocha's alternative (Golod and Shafarevich): There exists an integer $n$ such that $a_n=0$ (or $c_n$ has a polynomial growth) if and only if $G$ is a Lie group over $p$adic fields.
In 2016, Minac, Rogelstad and Tan inferred an explicit relation between $a_n$ and $c_n$. Recently (2022), considering geometrical ideas of Filip and Stix, Hamza got more precise relations in an equivariant context: when the automorphism group of $G$ admits a subgroup of order a prime $q$ dividing $p1$.
In this talk, we present equivariant relations inferred by Hamza (2022) and give explicit examples in an arithmetical context.



Feb 6 
Cristhian Garay 
Generalized valuations and idempotization of schemes 

at noon
in M1040

(CIMAT Guanajuato, Mexico)

Classical valuation theory has proved to be a valuable tool in number theory, algebraic geometry and singularity theory. For example, one can enrich spectra of rings with new points coming from valuations defined on them and taking values in totally ordered abelian groups.
Totally ordered groups are examples of idempotent semirings, and generalized valuations appear when we replace totally ordered abelian groups with more general idempotent semirings. An important example of idempotent semiring is the tropical semifield.
As an application of this set of ideas, we show how to associate an idempotent version of the structure sheaf of a scheme, which behaves particularly well with respect to idempotization of closed subschemes.
This is a joint work with Félix Baril Boudreau.



Feb 6 
Cristhian Garay 
An invitation to the algebraic geometry over idempotent semirings 

3:10–4:45pm
in B716

(CIMAT Guanajuato, Mexico)

Minicourse, Lecture 1
Idempotent semirings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization.
They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry.
However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semirings.
In this mini course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semirings and modules over them.



Feb 9 
Cristhian Garay 
An invitation to the algebraic geometry over idempotent semirings 

3:10–4:45pm
in B716

(CIMAT Guanajuato, Mexico)

Minicourse, Lecture 2
Idempotent semirings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization.
They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry.
However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semirings.
In this mini course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semirings and modules over them.



Feb 13 
Kelly Emmrich 
The principal Chebotarev density theorem 

at noon
via zoom

(Colorado State University)

Let $K/k$ be a finite Galois extension. We define a principal version of the Chebotarev density theorem which represents the density of prime ideals of $k$ that factor into a product of principal prime ideals in $K$. We find explicit equations to express the principal density in terms of the invariants of $K/k$ and give an effective bound which can be used to verify the nonsplitting of the Hilbert exact sequence.



Feb 27 
Florent Jouve 
Fluctuations in the distribution of Frobenius automorphisms in number field extensions 

at 9:30am
via zoom

(Université de Bordeaux, France)

Given a Galois extension of number fields $L/K$, the Chebotarev Density Theorem asserts that, away from ramified primes, Frobenius automorphisms equidistribute in the set of conjugacy classes of ${\rm Gal}(L/K)$. In this talk we report on joint work with D. Fiorilli in which we study the variations of the error term in Chebotarev's Theorem as $L/K$ runs over certain families of extensions. We shall explain some consequences of this analysis: regarding first “Linnik type problems” on the least prime ideal in a given Frobenius set, and second, the existence of unconditional “Chebyshev biases” in the context of number fields. Time permitting we will mention joint work with R. de La Bretèche and D. Fiorilli in which we go one step further and study moments of the distribution of Frobenius automorphisms.



Mar 6 
John Voight 
A norm refinement of Bezout's Lemma, and quaternion orders 

at noon
via zoom

(Dartmouth College, USA)

Given coprime integers $a,b$, the classical identity of Bezout provides
integers $u,v$ such that $aubv = 1$. We consider refinements to this
identity, where we ask that $u,v$ are norms from a quadratic extension.
We then find ourselves counting optimal embeddings of a quadratic
order in a quaternion order, for which we give explicit formulas in
many cases. This is joint work with Donald Cartwright and Xavier
Roulleau.



Mar 13 
Renate Scheidler 
Orienteering on Supersingular Isogeny Volcanoes Using One Endomorphism 

at noon
in M1040

(University of Calgary)

Elliptic curve isogeny path finding has many applications in number theory and cryptography. For supersingular curves, this problem is known to be easy when one small endomorphism or the entire endomorphism ring are known. Unfortunately, computing the endomorphism ring, or even just finding one small endomorphism, is hard. How difficult is path finding in the presence of one (not necessarily small) endomorphism? We use the volcano structure of the oriented supersingular isogeny graph to answer this question. We give a classical algorithm for path finding that is subexponential in the degree of the endomorphism and linear in a certain class number, and a quantum algorithm for finding a smooth isogeny (and hence also a path) that is subexponential in the discriminant of the endomorphism. A crucial tool for navigating supersingular oriented isogeny volcanoes is a certain class group action on oriented elliptic curves which generalizes the wellknown class group action in the setting of ordinary elliptic curves.



Mar 20 
Joshua Males 
Forgotten conjectures of Andrews for Nahmtype sums 

at noon
in M1040

(University of Manitoba)

In his famous '86 paper, Andrews made several conjectures on
the function $\sigma(q)$ of Ramanujan, including that it has
coefficients (which count certain partitiontheoretic objects) whose
sup grows in absolute value, and that it has infinitely many Fourier
coefficients that vanish. These conjectures were famously proved by
AndrewsDysonHickerson in their '88 Invent. paper, and the function
$\sigma$ has been related to the arithmetic of $\mathbb{Z}[\sqrt{6}]$
by Cohen (and extensions by Zwegers), and is an important first
example of quantum modular forms introduced by Zagier.
A closer inspection of Andrews' '86 paper reveals several more
functions that have been a little left in the shadow of their sibling
$\sigma$, but which also exhibit extraordinary behaviour. In an
ongoing project with Folsom, Rolen, and Storzer, we study the function
$v_1(q)$ which is given by a Nahmtype sum and whose coefficients
count certain differences of partitiontheoretic objects. We give
explanations of four conjectures made by Andrews on $v_1$, which
require a blend of novel and wellknown techniques, and reveal that
$v_1$ should be intimately linked to the arithmetic of the imaginary
quadratic field $\mathbb{Q}[\sqrt{3}]$.



Mar 27 
Douglas Ulmer 
$p$torsion of Jacobians for unramified $\mathbb{Z}/p\mathbb{Z}$covers of curves 

at noon
in M1040

(University of Arizona)

It is a classical problem to understand the set of Jacobians of curves among all abelian varieties, i.e., the image of the map $M_g\to A_g$ which sends a curve $X$ to its Jacobian $J_X$. In characteristic $p$, $A_g$ has interesting filtrations, and we can ask how the image of $M_g$ interacts with them. Concretely, which groups schemes arise as the ptorsion subgroup $J_X[p]$ of a Jacobian? We consider this problem in the context of unramified $Z/pZ$ covers $Y\to X$ of curves, asking how $J_Y[p]$ is related to $J_X[p]$. Translating this into a problem about de Rham cohmology yields some results using classical ideas of Chevalley and Weil. This is joint work with Bryden Cais.



Apr 3 
Harald Andrés Helfgott 
Expansion, divisibility and parity 

at 10:30am
via zoom

(University of Göttingen, Germany, and Institut de Mathématiques de Jussieu, France)

We will discuss a graph that encodes the divisibility properties of integers by primes. We prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier, by combining our result with MatomakiRadziwill. For instance: for $\lambda$ the Liouville function (that is, the completely multiplicative function with $\lambda(p) = 1$ for every prime),
$$ \frac{1}{\log x} \sum_{n\leq x} \frac{\lambda(n) \, \lambda(n+1)}{n} = O \left( \frac{1}{\sqrt{\log \log x}} \right) , $$ which is stronger than wellknown results by Tao and TaoTeravainen. We also manage to prove, for example, that $\lambda(n+1)$ averages to $0$ at almost all scales when $n$ restricted to have a specific number of prime divisors $\Omega(n)=k$, for any "popular" value of $k$ (that is, $k = \log \log N + O \bigl( \sqrt{\log \log N} \, \bigr)$ for $n \leq N$).

