arXiv subjects

Alain Kraus

Publications and source records attributed to Alain Kraus.

At least 19 recordsLinked to original sources

arXiv2025

Symplectic criteria for elliptic curves, revisited

Let $\ell$ and $p \geq 3$ be different primes. Let $E/\mathbb{Q}_\ell$ and $E'/\mathbb{Q}_\ell$ be elliptic curves with isomorphic $p$-torsion. Assume that $E$ has potentially multiplicative reduction. We classify when all $G_{\mathbb{Q}_\ell}$-isomorphisms $\phi : E[p] \to E'[p]$ have the same symplectic type and prove two new criteria to determine the type in that case. In particular, when both curves have multiplicative reduction, our results cover the case of unramified $p$-torsion which is not covered by the original criterion due to Kraus and Oesterl\'e. We also give a variant of a symplectic criterion for the case when both $E$ and~$E'$ have good reduction and provide an algorithm to apply it. As an application, we determine the symplectic type of all the mod $p \geq 5$ congruences between rational elliptic curves with conductor $\leq 500 000$ that satisfy the hypothesis of either of our criteria at some prime~$\ell$.

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arXiv2025

Courbes de Fermat et principe de Hasse

Let $p\geq 3$ be a prime number. A Fermat curve over $\mathbb{Q}$ of exponent $p$ is defined by an equation of the shape $ax^p+by^p+cz^p=0$, where $a,b,c$ are non-zero rational numbers. We prove in this article that there exist infinitely many Fermat curves defined over $\mathbb{Q}$, of exponent $p$, pairwise non $\mathbb{Q}$-isomorphic, contradicting the Hasse principle.

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arXiv2024

Points totalement r\'eels de la courbe $x^5+y^5+z^5=0$

Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$ and $\mathbb{Q}^{tr}$ be the subfield of $\overline{\mathbb{Q}}$ obtained by taking the union of all totally real number fields. For any prime $p\geq 3$, let $F_p/\mathbb{Q}$ be the Fermat curve of equation $x^p+y^p+z^p=0$. In 1996, Pop has shown that the field $\mathbb{Q}^{tr}$ is large. In particular, the set $F_p(\mathbb{Q}^{tr})$ of the points of $F_p$ rational over $\mathbb{Q}^{tr}$ is infinite. How to explicit non-trivial points $(xyz\neq 0$) in $F_p(\mathbb{Q}^{tr})$ ? If one has $p\geq 5$, it seems that the only points already known in $F_p(\mathbb{Q}^{tr})$ are those of $F_p(\mathbb{Q})$ and they are trivial. In this paper, we investigate this question in case $p=5$. There are no totally real fields whose degree over $\mathbb{Q}$ is at most $5$ over which $F_5$ has non-trivial points. We propose here to explicit infinitely many points of $F_5$ rational over totally real fields of degree $6$ over $\mathbb{Q}$.

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arXiv2022

On the symplectic type of isomorphims of the p-torsion of elliptic curves

Let $p \geq 3$ be a prime. Let $E/\mathbb{Q}$ and $E'/\mathbb{Q}$ be elliptic curves with isomorphic $p$-torsion modules $E[p]$ and $E'[p]$. Assume further that either (i) every $G_\mathbb{Q}$-modules isomorphism $ϕ: E[p] \to E'[p]$ admits a multiple $λ\cdot ϕ$ with $λ\in \mathbb{F}_p^\times$ preserving the Weil pairing; or (ii) no $G_\mathbb{Q}$-isomorphism $ϕ: E[p] \to E'[p]$ preserves the Weil pairing. This paper considers the problem of deciding if we are in case (i) or (ii). Our approach is to consider the problem locally at a prime $\ell \neq p$. Firstly, we determine the primes $\ell$ for which the local curves $E/\mathbb{Q}_\ell$ and $E'/\mathbb{Q}_\ell$ contain enough information to decide between (i) or (ii). Secondly, we establish a collection of criteria, in terms of the standard invariants associated to minimal Weierstrass models of $E/\mathbb{Q}_\ell$ and $E'/\mathbb{Q}_\ell$, to decide between (i) and (ii). We show that our results give a complete solution to the problem by local methods away from $p$. We apply our methods to show the non-existence of rational points on certain hyperelliptic curves of the form $y^2 = x^p - \ell$ and $y^2 = x^p - 2\ell$ where $\ell$ is a prime; we also give incremental results on the Fermat equation $x^2 + y^3 = z^p$. As a different application, we discuss variants of a question raised by Mazur concerning the existence of symplectic isomorphisms between the $p$-torsion of two non-isogenous elliptic curves defined over $\mathbb{Q}$.

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arXiv2021

Local criteria for the unit equation and the asymptotic Fermat's Last Theorem

Let F be a totally real number field of odd degree. We prove several purely local criteria for the asymptotic Fermat's Last Theorem to hold over F, and also for the non-existence of solutions to the unit equation over F. For example, if 2 totally ramifies and 3 splits completely in F, then the asymptotic Fermat's Last Theorem holds over F.

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arXiv2020

On the unit equation over cyclic number fields of prime degree

Let $\ell \ne 3$ be a prime. We show that there are only finitely many cyclic number fields $F$ of degree $\ell$ for which the unit equation $$λ+ μ= 1, \qquad λ,~μ\in \mathcal{O}_F^\times$$ has solutions. Our result is effective. For example, we deduce that the only cyclic quintic number field for which the unit equation has solutions is $\mathbb{Q}(ζ_{11})^+$.

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arXiv2020

Idéaux premiers totalement décomposés et sommes de Newton

Let $K$ be a number field and $f\in K[X]$ an irreducible monic polynomial with coefficients in $O_K$, the ring of integers of $K$. We aim to enounce an effective criterion, in terms of the Galois group of $f$ over $K$ and a linear recurrence sequence associated to $f$, allowing sometimes to characterize the prime ideals of $O_K$ modulo which $f$ completely splits. If $α$ is a root of $f$, this criterion therefore gives a characterization of the prime ideals of $O_K$ which split completely in $K(α)$. It does apply if the degree of $f$ is at least $4$ and the Galois group of $f$ is the symmetric group or the alternating group.

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arXiv2020

On asymptotic Fermat over the Z_2 extension of Q

In a recent work the authors prove the effective asymptotic Fermat's Last Theorem for the infinite family of fields $\mathbb{Q}(ζ_{2^{r+2}})^+$ where $r \ge 0$. A crucial step in their proof is the following conjecture of Kraus. Let $K$ be a number field having odd narrow class number and a unique prime $λ$ above $2$. Then there are no elliptic curves defined over $K$ with conductor $λ$ and a $K$-rational point of order $2$. In this note we give a new elementary proof of Kraus' conjecture that makes use only of basic facts about elliptic curves, Tate curves and Tate modules.

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arXiv2020

Chevalley's class number formula, unit equations and the asymptotic Fermat's Last Theorem

Let $F$ be a number field and $\mathcal{O}_F$ its ring of integers. We use Chevalley's ambiguous class number formula to give a criterion for the non-existence of solutions to the unit equation $λ+ μ= 1$, $λ, μ\in \mathcal{O}_F^\times$. This is then used to strengthen a criterion for the asymptotic Fermat's Last Theorem due to Freitas and Siksek.

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arXiv2020

On Asymptotic Fermat over $\mathbb{Z}_p$ extensions of $\mathbb{Q}$

Let $p \ge 5$ be a prime and let $\mathbb{Q}_{n,p}$ denote the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. We show that $\mathbb{Q}_{n,p}$ has no exceptional units. We use this to prove the effective asymptotic Fermat's Last Theorem over $\mathbb{Q}_{n,p}$ for all $n \ge 1$ and all primes $p \ge 5$ that are non-Wieferich, i.e. $2^{p-1} \not \equiv 1 \pmod{p^2}$. The effectivity in our result builds on recent work of Thorne proving modularity of elliptic curves over $\mathbb{Q}_{n,p}$.

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arXiv2019

Class field theory, Diophantine analysis and the asymptotic Fermat's Last Theorem

Recent results of Freitas, Kraus, Sengun and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number field. Those works in turn build on many deep theorems in arithmetic geometry. In this paper we combine the aforementioned results with techniques from class field theory, the theory of p-groups and p-extensions, Diophantine approximation and linear forms in logarithms, to establish the asymptotic Fermat's Last Theorem for many infinite families of number fields, and for thousands of number fields of small degree. For example, we prove the effective asymptotic Fermat's Last Theorem for the infinite family of fields $\mathbb{Q}(ζ_{2^r})^+$.

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arXiv2018

On the degree of the $p$-torsion field of elliptic curves over $\mathbb{Q}_\ell$ for $\ell \neq p$

Let $\ell$ and $p \geq 3$ be distinct prime numbers. Let $E/\mathbb{Q}_{\ell}$ be an elliptic curve with $p$-torsion module $E_p$. Let $\mathbb{Q}_{\ell}(E_p)$ be the $p$-torsion field of $E$. We provide a complete description of the degree of the extension $\mathbb{Q}_{\ell}(E_p)/\mathbb{Q}_{\ell}$. As a consequence, we obtain a recipe to determine the discriminant ideal of the extension $\mathbb{Q}_{\ell}(E_p)/\mathbb{Q}_\ell$ in terms of standard information on $E$.

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arXiv2017

Le théorème de Fermat sur certains corps de nombres totalement réels

Let $K$ be a totally real number field. For all prime number $p\geq 5$, let us denote by $F_p$ the Fermat curve of equation $x^p+y^p+z^p=0$. Under the assumption that $2$ is totally ramified in $K$, we establish some results about the set $F_p(K)$ of points of $F_p$ rational over $K$. We obtain a criterion so that the asymptotic Fermat's Last Theorem is true over $K$, criterion related to the set of Hilbert modular cusp newforms over $K$, of parallel weight $2$ and of level the prime ideal above $2$. It is often simply testable numerically, particularly if the narrow class number of $K$ is $1$. Furthermore, using the modular method, we prove Fermat's Last Theorem effectively, over some number fields whose degrees over $\mathbb{Q}$ are $3,4,5,6$ and $8$.

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arXiv2017

Quartic points on the Fermat quintic

In this paper, we study the algebraic points of degree $4$ over $\mathbb{Q}$ on the Fermat curve $F_5/\mathbb{Q}$ of equation $x^5+y^5+z^5=0$. A geometrical description of these points has been given in 1997 by Klassen and Tzermias. Using their result, as well as Bruin's work about diophantine equations of signature $(5,5,2)$, we give here an algebraic description of these points. In particular, we prove there is only one Galois extension of $\mathbb{Q}$ of degree $4$ that arises as the field of definition of a non-trivial point of $F_5$.

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arXiv2016

An application of the symplectic argument to some Fermat-type Equations

Let $p$ be a prime number. In the early 2000s, it was proved that the Fermat equations with coefficients \[3x^p + 8y^p + 21z^p =0\quad \text{ and } \quad 3x^p + 4y^p + 5z^p=0 \] do not admit non-trivial solutions for a set of exponents $p$ with Dirichlet density ${1/4}$ and ${1/8}$, respectively. In this note, using a recent criterion to decide if two elliptic curves over $\mathbb{Q}$ with certain types of additive reduction at 2 have symplectically isomorphic $p$-torsion modules, we improve these densities to ${3/8}$.

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arXiv2016

Contre-exemples au principe de Hasse pour les courbes de Fermat

Let $p$ be an odd prime number. In this paper, we are concerned with the behaviour of Fermat curves defined over ${\bf Q}$ given by equations $ax^p+by^p+cz^p=0$, with respect to the local-global Hasse principle. It is conjectured that there exist infinitely many Fermat curves of exponent $p$ which are counterexamples to the Hasse principle. It is a consequence of the abc-conjecture if $p\geq 5$. Using a cyclotomic approach due to H. Cohen and Chebotarev's density theorem, we obtain a partial result towards this conjecture, by proving it for $p\leq 19$.

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arXiv2014

Équation de Fermat et nombres premiers inertes

Let $K$ be a number field and $p$ a prime number $\geq 5$. Let us denote by $μ_p$ the group of the $p$th roots of unity. We define $p$ to be $K$-regular if $p$ does not divide the class number of the field $K(μ_p)$. Under the assumption that $p$ is $K$-regular and inert in $K$, we establish the second case of Fermat's Last Theorem over $K$ for the exponent $p$. We use in the proof classical arguments, as well as Faltings' theorem stating that a curve of genus at least two over $K$ has a finite number of $K$-rational points. Moreover, if $K$ is an imaginary quadratic field, other than ${\bf Q}(\sqrt{-3})$, we deduce a statement which allows often in practice to prove Fermat's Last Theorem over $K$ for the $K$-regular exponents.

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arXiv2014

Sur le théorème de Fermat sur ${\bf Q}(\sqrt{5})$

Let $p$ be an odd prime number. Using modular arguments, we give an easy testable condition which allows often to prove Fermat's Last Theorem over the quadratic field ${\bf Q}(\sqrt{5})$ for the exponent $p$. It is related to the Wendt's resultant of the polynomials $X^n-1$ and $(X+1)^n-1$. We deduce Fermat's Last Theorem over this field in case one has $5\leq p<10^7$, and we obtain analogous results on Sophie Germain type criteria.

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