arXiv subjects

Lucas Villagra Torcomian

Publications and source records attributed to Lucas Villagra Torcomian.

10 recordsLinked to original sources

arXiv2026

On the conductor of a family of Frey hyperelliptic curves

In his breakthrough article, Darmon presented a program to study Generalized Fermat Equations (GFE) via abelian varieties of $\text{GL}_2$-type over totally real fields. So far, only Jacobians of some Frey hyperelliptic curves have been used with that purpose. In the present article, we show how most of the known Frey hyperelliptic curves are particular instances of a more general biparametric family of hyperelliptic curves $C(z,s)$. Then, we apply the cluster picture methodology to compute the conductor of $C(z,s)$ at all odd places. As a Diophantine application, we specialize $C(z,s)$ in some particular values $z_0$ and $s_0$, and we find the conductor exponent at odd places of the natural Frey hyperelliptic curves attached to $Ax^p+By^p=Cz^r$ and $Ax^r+By^r=Cz^p$, generalizing the results due to Azon, Curcó-Iranzo, Khawaja, Maistret and Mocanu, and opening the door for future research in GFE with coefficients. Moreover, we show how a new Frey hyperelliptic curve for $Ax^2+By^r=Cz^p$ can be constructed in this way, giving new results on the conductor exponents for this equation. Finally, following the recent approach by Golfieri and Pacetti, we consider the Frey representations attached to a general signature $(q,r,p)$ via hypergeometric motives and, using $C(z,s)$, we compute the wild part of the conductor exponent at primes above $q$ and $r$ of the residual representation modulo a prime above $p$.

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arXiv2025

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

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 an explicit description of the ramification behavior at primes dividing $2qr$ and analyze the contribution of trivial solutions. We identify a general obstruction to the modular method that accounts for its failure in many instances. As an application, assuming a standard large image conjecture, we prove that the previous equation admits no nontrivial primitive solutions $(a,b,c)$ with $3 \nmid c$, when $q=5,$ $r=3$ and $p$ is a prime sufficiently large.

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arXiv2025

A note on conductors of Frey representations at $2$

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 program is the computation of their conductors, with the case at the prime $2$ presenting particular challenges. In this article we study the conductor exponent at $2$ for Frey representations of signatures $(p,p,r)$, $(r,r,p)$, $(2,r,p)$, and $(3,5,p)$, all of which have hyperelliptic realizations. In particular we are able to determine the conductor at $2$ for even degree Frey representations of signature $(p,p,r)$ and $(3,5,p)$ and all rational parameters.

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arXiv2025

On endomorphism algebras of $\text{GL}_2$-type abelian varieties and Diophantine applications

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 the Galois representations attached to $f$ and to $g$ for a large prime $p$ implies an isomorphism between the endomorphism algebras of the abelian varieties $A_f$ and $A_g$ attached to $f$ and $g$ by the Eichler-Shimura construction. This implies important relations between their building blocks. A non-trivial application of our result is that for all prime numbers $d$ congruent to $3$ modulo $8$ satisfying that the class number of $\mathbb{Q}(\sqrt{-d})$ is prime to $3$, the equation $x^4+dy^2 =z^p$ has no non-trivial primitive solutions when $p$ is large enough. We prove a similar result for the equation $x^2+dy^6=z^p$.

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arXiv2024

On the sum of fifth powers in arithmetic progression

In this paper we study equation $$(x-dr)^5+\cdots+x^5+\cdots+(x+dr)^5=y^p$$ under the condition $\gcd(x,r)=1$. We present a recipe for proving the non-existence of non-trivial integer solutions of the above equation, and as an application we obtain explicit results for the cases $d=2,3$ (the case $d=1$ was already solved). We also prove an asymptotic result for $d\equiv 1, 7\pmod9$. Our main tools include the modular method, employing Frey curves and their associated modular forms, as well as the symplectic argument.

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arXiv2022

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

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 conjectures of the Langlands programme when $K$ has at least one complex embedding. On the other hand, we give unconditional results in the case of totally real extensions having odd narrow class number and a unique prime above $2$. When modularity of elliptic curves over $K$ is known, for example when $K$ is real quadratic or the $r$-layer of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, effective asymptotic results hold.

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arXiv2022

${\mathbb Q}$-curves, Hecke characters and some Diophantine equations II

In the article [PV] a general procedure to study solutions of the equations $x^4-dy^2=z^p$ was presented for negative values of $d$. The purpose of the resent article is to extend our previous results to positive values of $d$. On doing so, we give a description of the extension $\mathbb{Q}(\sqrt{d},\sqrtε)/\mathbb{Q}(\sqrt{d})$ (where $ε$ is a fundamental unit) needed to prove the existence of a Hecke character over $\mathbb{Q}(\sqrt{d})$ with prescribed local conditions. We also extend some "large image" results due to Ellenberg regarding images of Galois representations coming from $\mathbb{Q}$-curves from imaginary to real quadratic fields.

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arXiv2022

$\mathbb{Q}$-curves, Hecke characters and some Diophantine equations

In this article we study the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$ for positive square-free values of $d$. A Frey curve over $\mathbb{Q}(\sqrt{-d})$ is attached to each primitive solution, which happens to be a $\mathbb{Q}$-curve. Our main result is the construction of a Hecke character $χ$ satisfying that the Frey elliptic curve representation twisted by $χ$ extends to $\text{Gal}_\mathbb{Q}$, therefore (by Serre's conjectures) corresponds to a newform in $S_2(n,\varepsilon)$ for explicit values of $n$ and $\varepsilon$. Following some well known results and elimination techniques (together with some improvements) it provides a systematic procedure to study solutions of the above equations and allows us to prove non-existence of non-trivial primitive solutions for large values of $p$ of both equations for new values of $d$.

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arXiv2021

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

In this article we finish the study of solutions of the equation $x^2-2^m=y^n$ for $m\in\mathbb{Z}$ and $n\geq3$. This is achieved using the modularity method in unsolved cases, namely, we prove that the only integer solutions of $x^2-2=y^n$ for $n\geq2$ are $(\pm1,-1)$ when $n$ is odd.

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arXiv2021

On the equation $x^2+dy^6=z^p$ for square-free $1\le d\le 20$

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 equation $x^2+dy^6=z^p$ for square-free values $1 \le d \le 20$ following the approach of [PT]. The main innovation is to make use of the symplectic argument over ramified extensions to discard solutions, together with a multi-Frey approach to deduce large image of Galois representations.

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