Lucas Villagra Torcomian

activity

20202025

collaborators

9 papers

math.NT2025

On the generalized Fermat equation of signature $(5, p, 3)$

Ariel Pacetti, Lucas Villagra Torcomian

In this article we study solutions to the generalized Fermat equation $x^{q} + y^{p} + z^{r} = 0$ using hypergeometric motives within the framework of the modular method. In doing so, we give…

math.NT2025

A note on conductors of Frey representations at $2$

Imin Chen, Lucas Villagra Torcomian

In 2000, Darmon introduced the notion of Frey representations within the framework of the modular method for studying the generalized Fermat equation. A central step in this progra…

math.NT2025

On the conductor of a family of Frey hyperelliptic curves

Pedro-José Cazorla García, Lucas Villagra Torcomian

In his breakthrough article, Darmon presented a program to study Generalized Fermat Equations (GFE) via abelian varieties of $GL_{2}$ -type over totally real fields. So far, on…

math.NT2022

On endomorphism algebras of $GL_{2}$ -type abelian varieties and Diophantine applications

Franco Golfieri Madriaga, Ariel Pacetti, Lucas Villagra Torcomian

Let $f$ and $g$ be two different newforms without complex multiplication having the same coefficient field. The main result of the present article proves that a congruence between…

math.NT2022

Asymptotic Fermat for signature $(4, 2, p)$ over number fields

Lucas Villagra Torcomian

Let $K$ be a number field. Using the modular method, we prove asymptotic results on solutions of the Diophantine equation $x^{4} - y^{2} = z^{p}$ over $K$ , assuming some deep but standard co…

math.NT2021

On the equation $x^{2} + d y^{6} = z^{p}$ for square-free $1 \leq d \leq 20$

Franco Golfieri, Ariel Pacetti, Lucas Villagra Torcomian

The purpose of the present article is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the…

math.NT2021

$Q$ -curves, Hecke characters and some Diophantine equations II

Ariel Pacetti, Lucas Villagra Torcomian

In the article [PV] a general procedure to study solutions of the equations $x^{4} - d y^{2} = z^{p}$ was presented for negative values of $d$ . The purpose of the resent article is to extend…

math.NT2020

On the equation $x^{2} - 2^{m} = y^{n}$

Lucas Villagra Torcomian

In this article we finish the study of solutions of the equation $x^{2} - 2^{m} = y^{n}$ for $m \in Z$ and $n \geq 3$ . This is achieved using the modularity method in unsolved cases, na…

math.NT2020

$Q$ -curves, Hecke characters and some Diophantine equations

Ariel Pacetti, Lucas Villagra Torcomian

In this article we study the equations $x^{4} + d y^{2} = z^{p}$ and $x^{2} + d y^{6} = z^{p}$ for positive square-free values of $d$ . A Frey curve over $Q (- d)$ is attached to each primi…

On the generalized Fermat equation of signature (5,p,3)

A note on conductors of Frey representations at 2

On the conductor of a family of Frey hyperelliptic curves

On endomorphism algebras of GL2​-type abelian varieties and Diophantine applications

Asymptotic Fermat for signature (4,2,p) over number fields

On the equation x2+dy6=zp for square-free 1≤d≤20

Q-curves, Hecke characters and some Diophantine equations II

On the equation x2−2m=yn

Q-curves, Hecke characters and some Diophantine equations

On the generalized Fermat equation of signature $(5, p, 3)$

A note on conductors of Frey representations at $2$

On endomorphism algebras of $GL_{2}$ -type abelian varieties and Diophantine applications

Asymptotic Fermat for signature $(4, 2, p)$ over number fields

On the equation $x^{2} + d y^{6} = z^{p}$ for square-free $1 \leq d \leq 20$

$Q$ -curves, Hecke characters and some Diophantine equations II

On the equation $x^{2} - 2^{m} = y^{n}$

$Q$ -curves, Hecke characters and some Diophantine equations