Linear codes in the Hamming metric have played a central role in error correction since the 1950s and have been extensively studied. In contrast, the theory of codes in the sum-rank metric is still in its early stages, with only a few known constructions.
A cornerstone of coding theory in the Hamming metric is the family of Reed–Solomon (RS) codes, which are constructed by evaluating univariate polynomials at distinct elements of a finite field $F_{q}$ . RS codes have optimal parameters, however, their length is by definition limited by the size of $ F_{q}$. Two classical approaches to overcome this limitation, while maintaining control on the parameters, are considering multivariate polynomials, giving rise to Reed–Muller (RM) codes, and evaluating rational function at points on algebraic curves, leading to Algebraic Geometry (AG) codes.
The sum-rank analogue of RS codes is the family of linearized Reed–Solomon (LRS) codes (see U. Martínez-Peñas 2018), which also achieve optimal parameters but face a similar length restriction as RS codes. In this talk, inspired by the similarities between RS and LRS codes,we will introduce analogues of RM and AG codes in the sum-rank metric, known as linearized Reed–Muller (LRM) codes (see E. Berardini and X. Caruso 2025) and linearized Algebraic Geometry (LAG) codes (see E. Berardini and X. Caruso 2024).
We will begin by reviewing key background on sum-rank metric codes and univariate Ore polynomials. Afterwards, we will introduce the theory of multivariate Ore polynomials and their evaluation, leading to the construction of linearized Reed–Muller codes and an analysis of their parameters. Then, we will develop the theory of Riemann–Roch spaces over Ore polynomial rings with coefficients in the function field of a curve, leveraging the classical framework of divisors and Riemann–Roch spaces on curves. Using this foundation, we will construct linearized AG codes, providing lower bounds on their dimension and minimum distance. We will conclude the talk by sketching some related works in progress.[-]

Let $V$ be an $(n+1)$-dimensional vector space over an arbitrary field $\mathbb{K}$ and denote by $\mathrm{PG}(V)$ the corresponding projective space. Define $\Gamma$ as the point-hyperplane geometry of $\mathrm{PG}(V)$, whose points are the pairs $(p, H)$, where $p$ is a point, $H$ is a hyperplane of $\mathrm{PG}(V)$ and $p \in H$ and whose lines are the sets $\ell_{p, *}:=\{(p, U): p \in U\}$ or $\ell_{*, H}=\{(x, H): x \in H\}$. The geometry $\Gamma$ is also known as the long root geometry for the special linear group $\mathrm{SL}(n+1, \mathbb{K})$ and admits an embedding (the Segre embedding of $\Gamma$ ) in the projective space $\mathrm{PG}\left(M_0\right)$, where $M_0$ is the vector space of the traceless square matrices of order $n+1$ with entries in the field $\mathbb{K}$. Since $M_0$ is isomorphic to a hyperplane of the vector space $V \otimes V^*$, we explicitly have

$$
\varepsilon: \Gamma \rightarrow \mathrm{PG}\left(M_0\right), \quad \varepsilon((\langle x\rangle,\langle\xi\rangle))=\langle x \otimes \xi\rangle,
$$

with $x \in V \backslash\{0\}, \xi \in V^* \backslash\{0\}$. The image $\Lambda_1:=\varepsilon(\Gamma)$ of $\varepsilon$ is represented by the pure tensors $x \otimes \xi$ with $x \in V$ and $\xi \in V^*$ such that $\xi(x)=0$.

If the underlying field $\mathbb{K}$ admits non-trivial automorphisms, for $1 \neq \sigma \in \operatorname{Aut}(\mathrm{K})$, then it is possible to define a 'twisted version' $\varepsilon_\sigma$ of $\varepsilon$ as follows

$$
\varepsilon_\sigma: \Gamma \rightarrow \mathrm{PG}\left(V \otimes V^*\right), \varepsilon_\sigma((\langle x\rangle,\langle\xi\rangle))=\left\langle x^\sigma \otimes \xi\right\rangle,
$$

where $x^\sigma:=\left(x_i{ }^\sigma\right)_{i=1}^{n+1}$.
Consequently, the points of $\Lambda_\sigma:=\varepsilon_\sigma(\Gamma)$ are represented by pure tensors of the form $x^\sigma \otimes \xi$, under the condition $\xi(x)=0$.

In the first part of the talk I will address the problem of the universality of the Segre embedding $\varepsilon$ for $\Gamma$ proving that the answer to this question depends on the underlying field $\mathbb{K}$ and generalizing a previous result for $n=2$ (see recent work of I. Cardinali, L. Giuzzi, A. Pasini).

In the second part of the talk, I shall focus on the case where $\mathbb{K}=\mathbb{F}_q$ is a finite field of order $q$. Thus, regarding $\Lambda_1$ and $\Lambda_\sigma$ as projective systems of $\mathrm{PG}\left(M_0\right)$ respectively $\mathrm{PG}\left(V \otimes V^*\right)$, I will consider the linear codes $\mathcal{C}\left(\Lambda_1\right)$ and $\mathcal{C}\left(\Lambda_\sigma\right)$ arising from them. I shall determine the parameters of $\mathcal{C}(\Lambda)$ and $\mathcal{C}\left(\Lambda_\sigma\right)$ as well as their weight list. I will also give a (geometrical) characterization of some of the words of these codes having minimum or maximal weight (see recent work of I. Cardinali, L. Giuzzi).[-]

Algebraic curves in positive characteristic and their function fields have been a source of great interest ever since the seminal work of Hasse and Weil in the 1930s and 1940s. Many important and fruitful ideas have arisen out of this area, where number theory and algebraic geometry meet, including the famous application to error-correcting codes given by Goppa's AG codes.

Let $\mathcal{X}$ be a projective, geometrically irreducible, non-singular algebraic curve defined over an algebraically closed field $\mathbb{K}$ of positive characteristic $p$. Let $\mathbb{K}(\mathcal{X})$ be the field of rational functions on $\mathcal{X}$ (i.e. the function field of $\mathcal{X}$ over $\mathbb{K}$ ). The $\mathbb{K}$-automorphism group $\operatorname{Aut}(\mathcal{X})$ of $\mathcal{X}$ is defined as the automorphism group of $\mathbb{K}(\mathcal{X})$ fixing $\mathbb{K}$ element-wise. The group $\operatorname{Aut}(\mathcal{X})$ has a faithful action on the set of points of $\mathcal{X}$.

By a classical result by Schnid (1938), Aut( $\mathcal{X}$ ) is finite whenever the genus $g$ of $\mathcal{X}$ is at least two. Furthermore it is known that every finite group occurs in this way, since, for any ground field $\mathbb{K}$ and any finite group $G$, there exists an algebraic curve $\mathcal{X}$ defined over $\mathbb{K}$ such that $\operatorname{Aut}(\mathcal{X}) \cong G$ (see for example the work of Valentini-Madden, 1982).

This result raised a general problem for groups and curves, namely, that of determining the finite groups that can be realized as the $\mathbb{K}$-automorphism group of some curve with a given invariant. The most important such invariant is the genus $g$ of the curve. In positive characteristic, another important invariant is the so-called $p$-rank of the curve, which is the integer $0 \leq \gamma \leq g$ such that the Jacobian of $\mathcal{X}$ has $p^7 p$-torsion points.

Several results on the interaction between the automorphism group, the genus and the $p$-rank of a curve can be found in the literature. A remarkable example is the work of Nakajima (1987) who showed that the value of the $p$-rank deeply influences the order of a $p$-Sylow subgroup of $A u t(\mathcal{X})$. Extremal examples with respect to Nakajima's bound are known from the work of Korchmáros-Giulietti (2017) and Stichtenoth (1973). The following open problem arose naturally:

Open Problem 1: How large can a d-group of aulomorphisms $G$ of an algebraic curve $\mathcal{X}$ of genus $g \geq 2$ be when $d \neq p$ is a prime number? Is there a method to construct extremal cxamples as for the case $d=p$ ?

In his work Nakajima also analyzed the case of curves for which the $p$-rank is the largest possible (the so-called ordinary curves), namely $\gamma=g$, proving that they can have at most $84\left(g^2-g\right)$ automorphisms. Since no extremal examples for this bound were found by Naka.jima, also the following open problem arose naturally:

Open Problem 2: Is Nakajima's bound $\mid$ Aut $(\mathcal{X}) \mid \leq 84\left(g^2-g\right)$, sharp for an ordinary curve $\mathcal{X}$ of genus $g \geq 2$ ?

Hurwitz (1893) showed that if $\mathcal{X}$ is defined over $\mathbb{C}$ then $|A u t(\mathcal{X})| \leq 84(g-1)$, which is
known as the Hurwitz bound. This bound is sharp, i.e., there exist algebraic curves over $\mathbb{C}$ of arbitrarily high genus $g$ whose automorphism group has order exactly $84(g-1)$. Well-known examples are the Klein quartic and the Fricke-Macbeath curve.

Roquette (1970) showed that Hurwitz bound also holds in positive characteristic $p$, if $p$ does not divide $|\operatorname{Aut}(\mathcal{X})|$. A general bound in positive characteristic is $|\operatorname{Aul}(\mathcal{X})| \leq 16 g^4$ with one exception: the so-called Hermitian curve. This result is due to Stichtenoth (1973). The quartic bound $|A u t(\mathcal{X})| \leq 16 g^4$ was improved by Henn (1978). Henn's result shows that if $|A u t(\mathcal{X})|>8 g^3$ then $\mathcal{X}$ is $\mathbb{K}$-isomorphic to one of 4 explicit exceptional curves, all having $p$-rank equal to zero. A third natural open problem arose as a consequence of this result:

Open Problem 3: Is it possible to find a (optimal) function $f(g)$ such that the existence of an automorphism group $G$ of $\mathcal{X}$ with $|G|>f(g)$ implies that $\mathcal{X}$ has p-rank zero?

Henn's result clearly implies that $f(g) \leq 8 g^3$, but it is pleausible to believe that a quadratic bound with respect to $g$ could also be found.

In this talk, we will describe our main contributions to the three problems mentioned above and more generally in understanding the relation between automorphism groups of algebraic curves in positive characteristic and the other invariants mentioned above. If time allows, applications of these results in determining isomorphism classes of algebraic curves over finite fields will also be discussed.[-]

Supersingular elliptic curve isogeny graphs have isomorphism classes of supersingular elliptic curves over a finite field as their vertices and isogenies of some fixed degree between them as their edges. Due to their apparent "random" nature, supersingular isogeny graphs - which are optimal expander graphs - have been used as a setting for certain cryptographic schemes that are resistant to attacks by quantum computers. Hidden structures in these graphs may have implications to the security of these systems. In this talk, we analyze a number of graph theoretic structural properties of supersingular isogeny graphs over a finite field $\mathbb{F}_{p^2}$ and their subgraphs induced by the vertices defined over $\mathbb{F}_p$. This is joint work with Sarah Arpin (Virginia Tech) and our jointly supervised undergraduate student Taha Hedayat (University of Calgary).[-]

The classical Brauer-Siegel theorem can be seen as one of the first instances of description of asymptotical arithmetic: it states that, for a family of number fields $K_i$, under mild conditions (e.g. bounded degree), the product of the regulator by the class number behaves asymptotically like the square root of the discriminant.
This can be reformulated as saying that the Brauer-Siegel ratio log($hR$)/ log$\sqrt{D}$ has limit 1.
Even if some of the fundamental problems like the existence or non-existence of Siegel zeroes remains unsolved, several generalisations and analog have been developed: Tsfasman-Vladuts, Kunyavskii-Tsfasman, Lebacque-Zykin, Hindry-Pacheco and lately Griffon. These analogues deal with number fields for which the limit is different from 1 or with elliptic curves and abelian varieties either for a fixed variety and varying field or over a fixed field with a family of varieties.[-]

Consider an ordinary isogeny class of elliptic curves over a finite, prime field. Inspired by a random matrix heuristic (which is so strong it's false), Gekeler defines a local factor for each rational prime. Using the analytic class number formula, he shows that the associated infinite product computes the size of the isogeny class.
I'll explain a transparent proof of this formula; it turns out that this product actually computes an adelic orbital integral which visibly counts the desired cardinality. Moreover, the new perspective allows a natural generalization to higher-dimensional abelian varieties. This is joint work with Julia Gordon and S. Ali Altug.[-]

For a long time people have been interested in finding and constructing curves over finite fields with many points. For genus 1 and genus 2 curves, we know how to construct curves over any finite field of defect less than 1 or 3 (respectively), i.e. with a number of points at distance at most 1 or 3 to the upper bound given by the Hasse-Weil-Serre bound. The case of genus 3 is still open after more than 40 years of research. In this talk I will take a different approach based on the random matrix theory of Katz-Sarnak, that describe the distribution of the number of points, to prove the existence, for all $\epsilon>0$, of curves of genus $g$ over $\mathbb{F}_{q}$ with more than $1+q+(2 g-\epsilon) \sqrt{q}$ points for $q$ big enough. I will also discuss some explicit constructions as well as some details about the asymmetric of the distribution of the trace of the Frobenius for curves of genus 3 .This is a joint work with J. Bergström, E. Howe and C. Ritzenthaler.[-]

Algebraic curves over a finite field $\mathbb{F}_{q}$ and their function fields have been a source of great fascination for number theorists and geometers alike, ever since the seminal work of Hasse and Weil in the 1930s and 1940s. Many important and fruitful ideas have arisen out of this area, where number theory and algebraic geometry meet. For a long time, the study of algebraic curves and their function fields was the province of pure mathematicians. But then, in a series of three papers in the period 1977-1982, Goppa found important applications of algebraic curves over finite fields to coding theory. The key point of Goppa's construction is that the code parameters are essentially expressed in terms of arithmetic and geometric features of the curve, such as the number $N_{q}$ of $\mathbb{F}_{q}$-rational points and the genus $g$. Goppa codes with good parameters are constructed from curves with large $N_{q}$ with respect to their genus $g$. Given a smooth projective, algebraic curve of genus $g$ over $\mathbb{F}_{q}$, an upper bound for $N_{q}$ is a corollary to the celebrated Hasse-Weil Theorem,$$N_{q} \leq q+1+2 g \sqrt{q} .$$Curves attaining this bound are called $\mathbb{F}_{q}$-maximal. The Hermitian curve $\mathcal{H}$, that is, the plane projective curve with equation$$X^{\sqrt{q}+1}+Y^{\sqrt{q}+1}+Z^{\sqrt{q}+1}=0,$$is a key example of an $\mathbb{F}_{q}$-maximal curve, as it is the unique curve, up to isomorphism, attaining the maximum possible genus $\sqrt{q}(\sqrt{q}-1) / 2$ of an $\mathbb{F}_{q^{-}}$ maximal curve. Other important examples of maximal curves are the Suzuki and the Ree curves. It is a result commonly attributed to Serre that any curve which is $\mathbb{F}_{q}$-covered by an $\mathbb{F}_{q}$-maximal curve is still $\mathbb{F}_{q}$-maximal. In particular, quotient curves of $\mathbb{F}_{q}$-maximal curves are $\mathbb{F}_{q}$-maximal. Many examples of $\mathbb{F}_{q}$-maximal curves have been constructed as quotient curves $\mathcal{X} / G$ of the Hermitian/Ree/Suzuki curve $\mathcal{X}$ under the action of subgroups $G$ of the full automorphism group of $\mathcal{X}$. It is a challenging problem to construct maximal curves that cannot be obtained in this way for some $G$. In this talk, we will describe our main contributions to both the theory of maximal curves over finite fields and to applications of algebraic curves with many points in coding theory. In particular, the following three topics will be discussed:
1. Construction of maximal curves
2. Weierstrass semigroups and points on maximal curves;
3. Algebraic curves with many rational points and coding theory.[-]

We are interested in the behaviour of Frobenius roots when the base field is fixed and the genus of the curve or the dimension of the abelian variety tends to infinity. I shall explain how to put the question and what are the answers. This happens to be a question in algebraic number theory and harmonic analysis. For curves (and for number fields) these are my old results with Serge Vladuts, for abelian varieties those of J.-P. Serre (séminaire Bourbaki, 2018) and my work in progress with Nicolas Nadirashvili.[-]