Alain Kraus

activity

20162025

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math.NT2025

Symplectic criteria for elliptic curves, revisited

Alain Kraus, Nuno Freitas, Ignasi Sánchez-Rodríguez

Let $ℓ$ and $p \geq 3$ be different primes. Let $E / Q_{ℓ}$ and $E^{'} / Q_{ℓ}$ be elliptic curves with isomorphic $p$ -torsion. Assume that $E$ has potentially mult…

math.NT2020

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

Nuno Freitas, Alain Kraus, Samir Siksek

Let $F$ be a number field and $O_{F}$ its ring of integers. We use Chevalley's ambiguous class number formula to give a criterion for the non-existence of solutions to the…

math.NT2020

On Asymptotic Fermat over $Z_{p}$ extensions of $Q$

Nuno Freitas, Alain Kraus, Samir Siksek

Let $p \geq 5$ be a prime and let $Q_{n, p}$ denote the $n$ -th layer of the cyclotomic $Z_{p}$ -extension of $Q$ . We show that $Q_{n, p}$ has no excep…

math.NT2019

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

Nuno Freitas, Alain Kraus, Samir Siksek

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 bui…

math.NT2018

On the degree of the $p$ -torsion field of elliptic curves over $Q_{ℓ}$ for $ℓ \neq = p$

Nuno Freitas, Alain Kraus

Let $ℓ$ and $p \geq 3$ be distinct prime numbers. Let $E / Q_{ℓ}$ be an elliptic curve with $p$ -torsion module $E_{p}$ . Let $Q_{ℓ} (E_{p})$ be the $p$ -torsion f…

math.NT2017

Quartic points on the Fermat quintic

Alain Kraus

In this paper, we study the algebraic points of degree $4$ over $Q$ on the Fermat curve $F_{5} / Q$ of equation $x^{5} + y^{5} + z^{5} = 0$ . A geometrical description of these po…

Symplectic criteria for elliptic curves, revisited

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

On Asymptotic Fermat over Zp​ extensions of Q

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

On the degree of the p-torsion field of elliptic curves over Qℓ​ for ℓ=p

Quartic points on the Fermat quintic

On Asymptotic Fermat over $Z_{p}$ extensions of $Q$

On the degree of the $p$ -torsion field of elliptic curves over $Q_{ℓ}$ for $ℓ \neq = p$