at noon
in M1040

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 Adrien-Marie 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.

It is an open question of Baker whether the Dirichlet L-values at 1 with fixed modulus are linearly independent over the rational numbers. The best-known result is due to Baker, Birch and Wirsing, which affirms this when the modulus of the associated Dirichlet character is co-prime 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.

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.

at noon

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 well-known 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 $p-1$.

In this talk, we present equivariant relations inferred by Hamza (2022) and give explicit examples in an arithmetical context.

at noon
in M1040

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.

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.

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.

at noon

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 non-splitting of the Hilbert exact sequence.

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.

Given coprime integers $a,b$, the classical identity of Bezout provides integers $u,v$ such that $au-bv = 1$. 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.

at noon
in M1040

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 well-known class group action in the setting of ordinary elliptic curves.

at noon
in M1040

In his famous '86 paper, Andrews made several conjectures on the function $\sigma(q)$ of Ramanujan, including that it has coefficients (which count certain partition-theoretic objects) whose sup grows in absolute value, and that it has infinitely many Fourier coefficients that vanish. These conjectures were famously proved by Andrews-Dyson-Hickerson 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 Nahm-type sum and whose coefficients count certain differences of partition-theoretic objects. We give explanations of four conjectures made by Andrews on $v_1$, which require a blend of novel and well-known techniques, and reveal that $v_1$ should be intimately linked to the arithmetic of the imaginary quadratic field $\mathbb{Q}[\sqrt{-3}]$.

at noon
in M1040

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 p-torsion 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.

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 Matomaki-Radziwill. 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 well-known results by Tao and Tao-Teravainen. 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 $n \leq N$).