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Marc Perret

Publications and source records attributed to Marc Perret.

14 recordsLinked to original sources
arXiv2025

Corrigendum to the paper "Maximum number of rational points on hypersurfaces in weighted projective spaces over finite fields", Journal of Algebra and Its Applications, Vol. 24, No. 13n14, 2541015(2025)

The statement of item (ii) of Proposition 3.2 of the article referenced in the title is not correct. We provide a corrected version and show that, under the assumption that $\gcd(a_i, a_j, q-1)=1$ for any pair $i\neq j$ in $\{0, \cdots, n\}$ (with the notations of the paper), our initial statement becomes valid, as does the remainder of the paper.

math.AG↗
arXiv2025

Refinements on higher order Weil-Oesterl\'e bounds via a Serre type argument

Weil's theorem gives the most standard bound on the number of points of a curve over a finite field. This bound was improved by Ihara and Oesterl\'e for larger genus. Recently, Hallouin and Perret gave a new point of view on these bounds, that can be obtained by solving a sequence of semi-definite programs, and the two first steps of this hierarchy recover Weil's and Ihara's bounds. On the other hand, by taking into account arithmetic constraints, Serre obtained a refinement on Weil's bound. In this article, we combine these two approaches and propose a strengthening of Ihara's bound, based on an argument similar to Serre's refinement. We show that this generically improves upon Ihara's bound, even in the range where it was the best bound so far. Finally we discuss possible extensions to higher order Weil-Oesterl\'e bounds.

math.NT↗
arXiv2024

Maximum number of rational points on hypersurfaces in weighted projective spaces over finite fields

An upper bound for the maximum number of rational points on an hypersurface in a projective space over a finite field has been conjectured by Tsfasman and proved by Serre in 1989. The analogue question for hypersurfaces on weighted projective spaces has been considered by Castryck, Ghorpade, Lachaud, O'Sullivan, Ram and the first author in 2017. A conjecture has been proposed there and proved in the particular case of the dimension 2. We prove here the conjecture in any dimension provided the second weight is also equal to one.

math.AG↗
arXiv2020

Number of points of curves over finite fields in some relative situations from an euclidean point of vue

We study the number of rational points of smooth projective curves over finite fields in some relative situations in the spirit of a previous paper from an euclidean point of vue. We prove some kinds of relative Weil bounds, derived from Schwarz inequality for some "relative parts" of the diagonal and of the graph of the Frobenius on some euclidean sub-spaces of the numerical space of the squared curve endowed with the opposite of the intersection product.

math.AG↗
arXiv2020

Toward good families of codes from towers of surfaces

We introduce in this article a new method to estimate the minimum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be ``liftable'' under finite morphisms, paving the way toward the construction of good codes from towers of surfaces. In the same direction, we establish a criterion for a surface with a fixed finite set of closed points $\mathcal P$ to have an infinite tower of $\ell$--\'etale covers in which $\mathcal P$ splits totally. We conclude by stating several open problems. In particular, we relate the existence of asymptotically good codes from general type surfaces with a very ample canonical class to the behaviour of their number of rational points with respect to their $K^2$ and coherent Euler characteristic.

math.AG↗
arXiv2019

Bounds on the minimum distance of algebraic geometry codes defined over some families of surfaces

We prove lower bounds for the minimum distance of algebraic geometry codes over surfaces whose canonical divisor is either nef or anti-strictly nef and over surfaces without irreducible curves of small genus. We sharpen these lower bounds for surfaces whose arithmetic Picard number equals one, surfaces without curves with small self-intersection and fibered surfaces. Finally we specify our bounds to the case of surfaces of degree $d\geq 3$ embedded in $\mathbb{P)^3$.

math.AG↗
arXiv2019

Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus

We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that the abelian surface does not contain low genus curves. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization.

cs.IT↗
arXiv2015

A Graph Aided Strategy to Produce Good Recursive Towers over Finite Fields

We propose a systematic method to produce potentially good recursive towers over finite fields. The graph point of view, so as some magma and sage computations are used in this process. We also establish some theoretical functional criterion ensuring the existence of many rational points on a recursive tower. Both points are illustrated on an example, from the production process, to the theoretical study, using this functional criterion, of the parameters of the obtained potentially good tower.

math.NT↗
arXiv2014

From Hodge Index Theorem to the number of points of curves over finite fields

We push further the classical proof of Weil upper bound for the number of rational points of an absolutely irreducible smooth projective curve $X$ over a finite field in term of euclidean relationships between the Neron Severi classes in $X\times X$ of the graphs of iterations of the Frobenius morphism. This allows us to recover Ihara's bound, which can be seen as a {\em second order} Weil upper bound, to establish a new {\em third order} Weil upper bound, and using {\tt magma} to produce numerical tables for {\em higher order} Weil upper bounds. We also give some interpretation for the defect of exact recursive towers, and give several new bounds for points of curves in relative situation $X \rightarrow Y$.

math.NT↗
arXiv2012

Recursive towers of curves over finite fields using graph theory

We give a new way to study recursive towers of curves over a finite field, defined from a bottom curve $\Cun$ and a correspondence $\Cdeux$ on $\Cun$.In particular, we study their asymptotic behavior. A close examination of singularities leads to a necessary condition for a tower to be asymptotically good. Then, spectral theory on a directed graph and considerations on the class of $\Cdeux$ in $\NS (\Cun \times \Cun)$ lead to the fact that, under some mild assumptions, a recursive tower which does not reach Drinfeld-Vladut bound cannot be optimal in Tsfasmann-Vladut sense. Results are applied to the Bezerra-Garcia-Stichtenoth tower along the paper for illustration.

math.NT↗
arXiv2000

Sur les invariants d'Iwasawa des \Z_p^*/+-1-extensions

In this paper we give bounds for the Iwasawa invariants of the Igusa tower of curves investigated by Mazur and Wiles. We give an upper bound for the mu invariants and a lower bound for the sum of the lambda invariants, in terms of the genus and number of supersingular points of the base curve.

math.NT↗