Research

My research area is Number Theory and I am particulary interested in Diophantine equations, elliptic curves, Galois representations and modular forms.

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.

1. On the sum of fifth powers in arithmetic progression.
   Submmited [arXiv].

2. On some asymptotic results for the equations  x⁴+dy²=zp and x²+dy⁶=zp, with Franco Golfieri and Ariel Pacetti.
   Submmited [arXiv].

3. Asymptotic Fermat for signature (4,2,p) over number fields.
   Journal of Number Theory, Vol. 250 (2023),  124-138 [DOI | arXiv].

4. 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), 1129-1165 [DOI | arXiv].

5. Q-curves, Hecke characters and some Diophantine equations II, with Ariel Pacetti.
   Publicacions Matemàtiques, Vol. 67, No. 2 (2023),  569-599 [DOI | arXiv].

6. Q-curves, Hecke characters and some Diophantine equations, with Ariel Pacetti.
   Mathematics of Computation, Vol. 91, No. 338 (2022),  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)