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).[-]

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.[-]

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.[-]

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.[-]

In the field of coding theory, Goppa's construction of error-correcting codes on algebraic curves has been widely studied and applied. As noticed by M. Tsfasman and S. Vlădut¸, this construction can be generalized to any algebraic variety. This talk aims to shed light on the case of surfaces and expand the understanding of Goppa's construction beyond curves. After discussing the motivations for considering codes from higher–dimensional varieties, we will compare and contrast codes from curves and codes from surfaces, notably regarding the computation of their parameters, their local properties, and asymptotic constructions.[-]