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Yves Aubry

Publications and source records attributed to Yves Aubry.

At least 19 recordsLinked to original sources
arXiv2026

Minimal degree of an isogeny between a supersingular elliptic curve and its conjugate

Let $E$ be a supersingular elliptic curve defined over $\bar{\mathbb{F}}_p$ and $E^{(p)}$ be its conjugate. We give a bound on the minimal degree of an isogeny from $E$ to $E^{(p)}$ depending on $p$, and show that this bound is both asymptotically optimal as well as sharp in many cases. This bound is obtained by developing a new technique to compute the degree of certain isogenies from a supersingular elliptic curve to its conjugate, and we present extensive computations of the successive minima of the lattice containing these isogenies. Following this, we give several conjectures supported by the data we have obtained, including some on the set of primes $p$ for which the bound we give in this article is attained.

math.NT↗
arXiv2026

Rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields

We improve a bound due to the second author on number of rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields and look at families of surfaces that achieve or nearly achieve this bound, for which we compute their exact number of rational points. These computations may have independent interest.

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arXiv2026

Inseparable endomorphisms and rank-2 sublattices of the Gross lattice

We answer a question posed by Love asking about a correspondence between isogenies from a supersingular elliptic curve to its Frobenius base-change and rank-2 sublattices of its Gross lattice. We recast the question as one about the inseparable endomorphisms of the curve, and show that the correspondence holds when the trace of the endomorphism is zero, and may not hold otherwise.

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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

Maximal curves with respect to quadratic extensions over finite fields

We propose a detailed study of a canonical bound which relates the numbers of rational points of a curve over a finite field with that over its quadratic extension. Alternative proofs which make a connection with the variance enable to obtain optimal refinements. We focus on the curves reaching the bound, which we call Hallouin-Perret-maximal curves. We provide different characterizations and stress natural links with the curves which attain the Ihara bound. As consequences, we establish the list of such curves with low genus and we outline a maximality result which involves the Suzuki curves. At last we determine which polynomials correspond to the Jacobian of a Hallouin-Perret-maximal curve of genus 2.

math.AG↗
arXiv2024

Differential uniformity of polynomials of degree 10

We prove that polynomials of degree 10 over finite fields of even characteristic with some conditions on theirs coefficients have a differential uniformity greater than or equal to 6 over $\mathbb{F}_{2^n}$ for all $n$ sufficiently large.

math.NT↗
arXiv2024

Trinomials with high differential uniformity

Comparisons of arithmetic and geometric monodromy groups coupled with the Chebotarev density theorem enable to obtain families of trinomials defined over finite fields of even characteristic with high differential uniformity when the base field is large enough.

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.

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arXiv2023

Closed points on curves over finite fields

We are interested in the quantity $ρ$(q, g) defined as the smallest positive integer such that r $\ge$ $ρ$(q, g) implies that any absolutely irreducible smooth projective algebraic curve defined over F q of genus g has a closed point of degree r. We provide general upper bounds for this number and its exact value for g = 1, 2 and 3. We also improve the known upper bounds on the number of closed points of degree 2 on a curve.

math.AG↗
arXiv2022

Polynomials with maximal differential uniformity and the exceptional APN conjecture

We contribute to the exceptional APN conjecture by showing that no polynomial of degree m = 2 r (2 {\ell} + 1) where gcd(r, {\ell}) 2, r 2, {\ell} 1 with a nonzero second leading coefficient can be APN over infinitely many extensions of the base field. More precisely, we prove that for n sufficiently large, all polynomials of F 2 n [x] of such a degree with a nonzero second leading coefficient have a differential uniformity equal to m -- 2.

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arXiv2021

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↗
arXiv2020

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↗
arXiv2018

Maximal differential uniformity polynomials

We provide an explicit infinite family of integers $m$ such that all the polynomials of ${\mathbb F}_{2^n}[x]$ of degree $m$ have maximal differential uniformity for $n$ large enough. We also prove a conjecture of the third author in these cases.

math.NT↗
arXiv2017

Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory

We consider the question of determining the maximum number of $\mathbb{F}_q$-rational points that can lie on a hypersurface of a given degree in a weighted projective space over the finite field $\mathbb{F}_q$, or in other words, the maximum number of zeros that a weighted homogeneous polynomial of a given degree can have in the corresponding weighted projective space over $\mathbb{F}_q$. In the case of classical projective spaces, this question has been answered by J.-P. Serre. In the case of weighted projective spaces, we give some conjectures and partial results. Applications to coding theory are included and an appendix providing a brief compendium of results about weighted projective spaces is also included.

math.AG↗
arXiv2017

Differential uniformity and second order derivatives for generic polynomials

For any polynomial $f$ of ${\mathbb F}\_{2^n}[x]$ we introduce the following characteristic of the distribution of its second order derivative,which extends the differential uniformity notion:$$δ^2(f):=\max\_{\substack{α\in {\mathbb F}\_{2^n}^{\ast} ,α' \in {\mathbb F}\_{2^n}^{\ast} ,β\in {\mathbb F}\_{2^n} α\not=α'}} \sharp\{x\in{\mathbb F}\_{2^n} \mid D\_{α,α'}^2f(x)=β\}$$where $D\_{α,α'}^2f(x):=D\_{α'}(D\_αf(x))=f(x)+f(x+α)+f(x+α')+f(x+α+α')$ is the second order derivative.Our purpose is to prove a density theorem relative to this quantity,which is an analogue of a density theorem proved by Voloch for the differential uniformity.

math.AG↗
arXiv2015

Optimal and maximal singular curves

Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field $\mathbb F\_q$.This bound enables us to provide explicit conditions on $q, g$ and $π$ for the non-existence of absolutely irreducible projective algebraic curves defined over $\mathbb F\_q$ of geometric genus $g$, arithmetic genus $π$ and with $N\_q(g)+π-g$ rational points.Moreover, for $q$ a square, we study the set of pairs $(g,π)$ for which there exists a maximal absolutely irreducible projective algebraic curve defined over $\mathbb F\_q$ of geometric genus $g$ and arithmetic genus $π$, i.e. with $q+1+2g\sqrt{q}+π-g$ rational points.

math.AG↗
arXiv2015

Cyclotomy of Weil Sums of Binomials

The Weil sum $W_{K,d}(a)=\sum_{x \in K} ψ(x^d + a x)$ where $K$ is a finite field, $ψ$ is an additive character of $K$, $d$ is coprime to $|K^\times|$, and $a \in K^\times$ arises often in number-theoretic calculations, and in applications to finite geometry, cryptography, digital sequence design, and coding theory. Researchers are especially interested in the case where $W_{K,d}(a)$ assumes three distinct values as $a$ runs through $K^\times$. A Galois-theoretic approach, combined with $p$-divisibility results on Gauss sums, is used here to prove a variety of new results that constrain which fields $K$ and exponents $d$ support three-valued Weil sums, and restrict the values that such Weil sums may assume.

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