Alain Kraus · wovepaper

activity

20162025

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

7 citations · 16 across the 8 of their papers we have counts for

collaborators

10 papers

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★ 3 cited

On the unit equation over cyclic number fields of prime degree

Nuno Freitas, Alain Kraus, Samir Siksek

Let $ℓ \neq = 3$ be a prime. We show that there are only finitely many cyclic number fields $F$ of degree $ℓ$ for which the unit equation $$λ+ μ= 1, \qquad λ,~μ\in \mathcal{O}_F…

math.NT2020★ 7 cited

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

Nuno Freitas, Alain Kraus, Samir Siksek

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 o…

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★ 5 cited

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…

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…

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…

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…

math.NT2016★ 1 cited

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

Nuno Freitas, Alain Kraus

Let $p \geq 3$ be a prime. Let $E / Q$ and $E^{'} / Q$ be elliptic curves with isomorphic $p$ -torsion modules $E [p]$ and $E^{'} [p]$ . Assume further that either (i) every $G…

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

Nuno Freitas, Alain Kraus

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…