Lucas Villagra Torcomian

Research

I am beginning to believe that nothing can ever be proved. These are honest hypotheses which take the facts into account: but I sense so definitely that they come from me, and that they are simply a way of unifying my own knowledge [...].Slow, lazy, sulky, the facts adapt themselves to the rigour of the order I wish to give them; but it remains outside of them. I have the feeling of doing a work of pure imagination.

J.P. Sartre, Nausea.

Publications and preprints

All the codes used in the papers are available at https://github.com/lucasvillagra. The versions posted in arxiv may not be in final form, please see the published versions.

On the generalized Fermat equation of signature (5,p,3), with Ariel Pacetti.
Submitted [arXiv].
A note on the conductors of Frey representations at 2, with Imin Chen.
Submitted [arXiv].
On the conductor of a family of Frey hyperelliptic curves, with Pedro-José Cazorla García.
Research in Number Theory, Vol. 12, No. 13 (2026). [DOI |arXiv].
On the sum of fifth powers in arithmetic progression.
The Ramanujan Journal, Vol. 68, No. 29 (2025). [DOI | arXiv].
On endomorphism algebras of GL_2-type abelian varieties and Diophantine applications, with Franco Golfieri and Ariel Pacetti.
Revista Matemática Iberoamericana, Vol. 41, No. 5 (2025), pp. 1897–1924. [DOI | arXiv].
Asymptotic Fermat for signature (4,2,p) over number fields.
Journal of Number Theory, Vol. 250 (2023), pp. 124-138 [DOI | arXiv].
On the equation x²+dy⁶=zp for square-free 1≤d≤20, with Franco Golfieri and Ariel Pacetti.
International Journal of Number Theory, Vol. 19, No. 05 (2023), pp. 1129-1165 [DOI | arXiv].
Q-curves, Hecke characters and some Diophantine equations II, with Ariel Pacetti.
Publicacions Matemàtiques, Vol. 67, No. 2 (2023), pp. 569-599 [DOI | arXiv].
Q-curves, Hecke characters and some Diophantine equations, with Ariel Pacetti.
Mathematics of Computation, Vol. 91, No. 338 (2022), pp. 2817-2865 [DOI | arXiv].

Other surveys (in Spanish)

Q-curvas y caracteres de Hecke en el método modular, PhD Thesis (pdf)
Correspondencia de Langlands en dimensión 1, trabajo final de licenciatura (pdf)
Teorema de Lax-Milgram y aplicaciones al método de elementos finitos, with Agustina Czenky, concurso de monografía de la UMA (pdf)

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