Ariel Martín Pacetti

Principal Researcher in Number Theory · Universidade de Aveiro (CIDMA Chair)

About Me

My name is Ariel Martín Pacetti. I am originally from Argentina (born in Buenos Aires). I finished my undergraduate studies in Mathematics at the University of Buenos Aires (UBA) in 1999 under the supervision of Professor Fernando Cukierman.

In 2003, I completed my PhD in Mathematics at the University of Texas at Austin under the supervision of Professor Fernando Rodriguez Villegas.

Between 2003 and 2015, I held various academic positions at the University of Buenos Aires (including a visiting position during the 2009–2010 academic year at the Centre de Recerca Matemàtica in Barcelona). In 2016, I held a Leverhulme Trust Visiting Professorship at the University of Warwick, working with Professor John Cremona. From 2017 to 2021, I was a Professor at the Universidad Nacional de Córdoba.

Since 2021, I am a Principal Researcher at the Universidade de Aveiro, holding a CIDMA Chair in Number Theory since July 2025.

I am an editor of the Revista de la Unión Matemática Argentina. You can submit your articles by clicking on this link.

If you have comments or suggestions, please write to: apacetti [at] ua [dot] pt.

📄 View Curriculum Vitae (PDF)

Research & Publications

My main area of research is Number Theory, specializing in explicit arithmetic geometry, $Q$-curves, Diophantine equations, Galois representations, $L$-functions, and computational modular forms.

Publications

  1. Serre's conjectures for Q(\sqrt{13})
    with Luis Dieulefait
    Submitted. [pdf]
  2. On rank 2 hypergeometric motives
    with Franco Golfieri Madriaga and Fernando Rodriguez Villegas
    Submitted. [pdf]
  3. On the generalized Fermat equation of signature (5,p,3)
    with Lucas Villagra Torcomian
    Submitted. [pdf]
  4. On transformation properties of hypergeometric motives and Diophantine equations
    Submitted. [pdf] | Code: [link]
  5. Hypergeometric Motives and the Generalized Fermat Equation
    with Franco Golfieri (with an appendix by Ariel Pacetti and Fernando Rodriguez Villegas)
    Submitted. [pdf]
  6. K-varieties and Galois representations
    with Enric Florit
    Submitted. [pdf]
  7. On the 2-Selmer group of Jacobians of hyperelliptic curves
    with Daniel Barrera Salazar and Gonzalo Tornaría
    Submitted. [pdf]
  8. On computing finite index subgroups of PSL(2,Z)
    with Nicolás Mayorga Uruburu and Leandro Vendramin
    Journal of Algebra and Its Applications 24 (2025), n.9. [pdf]
  9. On endomorphism algebras of GL2-type abelian varieties and Diophantine applications
    with Franco Golfieri and Lucas Villagra Torcomian
    Revista Matemática Iberoamericana 41 (2025), n.5, 1897-1924. [pdf]
  10. Quinary forms and Paramodular forms
    with Neil Dummigan, Gustavo Rama and Gonzalo Tornaría
    Mathematics of Computation 93 (2024), n.348, 1805-1858. [pdf]
  11. On the equation $x^2 + dy^6 = z^p$ for square-free $1 \le d \le 20$
    with Franco Golfieri and Lucas Villagra Torcomian
    International Journal of Number Theory 19 (2023), no.5, 1129-1165. [pdf]
    Computations: [mazur.mg], [ellenberg.gp]. Outputs: [this file]
  12. A simplified proof of Serre's conjecture
    with Luis Dieulefait
    Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 117 (2023), n.4. [pdf]
  13. $Q$-curves, Hecke characters and some diophantine equations II
    with Lucas Villagra Torcomian
    Publicacions Matemàtiques 67 (2023), no.2, 569-599. [pdf]
    Scripts: [RemoveCM.gp], [Mazur42p.mg], [Case_d10.mg], [Case_d11.mg], [Case_d129.mg], [Case_d19.mg], [Case_d6.mg]. Outputs: [Outputs.txt], [SageOutputs.txt]. Tables verified with: [table3.gp], [table4.gp], [table5.gp]
  14. On Galois representations of superelliptic curves
    with Angel Villanueva
    Glasgow Mathematical Journal Vol. 65 Issue 2 (2023), 356-382. [pdf]
  15. An exotic example of a tensor product of a CM elliptic curve and a weight $1$ form
    Revista de la Unión Matemática Argentina 65 (2023) n.2, 225-228. [pdf]
  16. $Q$-curves, Hecke characters and some diophantine equations
    with Lucas Villagra Torcomian
    Math. Comp. 91 (2022), no. 338, 2817-2865. [pdf]
    Scripts: [Ellenberg.gp], [Mazur42p.mg], [Mazur26p.mg], [Eq1d5.mg], [Eq1d6.mg], [Eq1d7.mg], [Eq2d5.mg], [Eq2d6.mg], [Eq2d7.mg]. Outputs: [OutputsEq1.txt], [OutputsEq2.txt]
  17. On 2-Selmer groups and quadratic twists of elliptic curves
    with Daniel Barrera Salazar and Gonzalo Tornaría
    Math. Res. Lett. 28 (2021), no. 6, 1633-1659. [pdf]
  18. On rational Bianchi newforms and abelian surfaces with quaternionic multiplication
    with John Cremona, Lassina Dembélé, Ciaran Schembri and John Voight
    Arithmetic geometry, number theory, and computation, 343-363, Simons Symp., Springer, Cham, 2021. [pdf]
  19. On the number of Galois orbits of newforms
    with Luis Dieulefait and Paganiotis Tsaknias
    Journal of the European Mathematical Society, Volume 23 Number 8 (2021) 2833-2860. [pdf]
  20. On the paramodularity of typical abelian surfaces
    with Armand Brumer, Cris Poor, Gonzalo Tornaría, John Voight, and David S. Yuen
    Algebra and Number Theory, Volume 13 Number 5 (2019) 1145-1195. [pdf]
  21. Anticyclotomic p-adic L-functions for additive reduction primes
    with Daniel Kohen
    Comptes Rendus Mathematique 356, Issue 10 (2018) 973-983. [pdf] | [Published]
  22. On Elliptic Curves of prime power conductor over imaginary quadratic fields with class number one
    with John Cremona
    Proceedings of the London Mathematical Society 118, Issue 5 (2019) 1245-1276. [pdf]
  23. Non-paritious Hilbert modular forms
    with David Loeffler and Lassina Dembélé
    Mathematische Zeitschrift 292, Issue 1-2 (2019) 361-385. [pdf] | [Published]
  24. On Heegner points for primes of additive reduction ramifying in the base field
    with Daniel Kohen (and an appendix by Marc Masdeu)
    Trans. of the AMS 370 (2018) 911-926. [pdf]
  25. Theta Lifts of Bianchi Modular Forms and Applications to Paramodularity
    with Tobias Berger, Lassina Dembele and Mehmet Haluk Sengun
    Journal of the London Mathematical Society 92 (2015) 353-370. [pdf] | [Published]
  26. Heegner Points on Cartan non-split curves
    with Daniel Kohen
    Canadian Journal of Mathematics 68 (2016) 422-444. [pdf]
  27. Congruences between modular forms modulo prime powers
    with Maximiliano Camporino
    Revista Matemática Iberoamericana Volume 34 Issue 4 (2018), 1609-1643. [pdf]
    Files: [CongruencesRoutines.gp], [E17a1example.txt], Output: [E17a1example.output.txt]
  28. Connectedness of Hecke Algebras and the Rayuela conjecture: a path to functoriality and modularity
    with Luis Dieulefait
    Arithmetic and geometry, 193-216, London Math. Soc. Lecture Note Ser., 420, Cambridge University Press (2015). [pdf]
  29. Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields
    with José Ignacio Burgos Gil
    Math. Comp. 86 (2017), 1527-1552. [pdf]
  30. Computing ideal classes representatives in quaternion algebras
    with Nicolás Sirolli
    Math. Comp. (2014). [pdf]
  31. On the change of root numbers under twisting and applications
    Proceedings of the AMS 141 (2013) n.8. [pdf]
  32. Modularity of the Consani-Scholten quintic
    with Luis Dieulefait and Matthias Schuett (and an appendix with Jose Burgos Gil)
    Documenta Mathematica 17 (2012). [pdf]
  33. Shimura correspondence for level $p^2$ and the central values of L-series II
    with Gonzalo Tornaría
    International Journal of Number Theory Vol. 10, N. 07 (2014) 1595-1635. [pdf]
  34. Proving Modularity for a given elliptic curve over an imaginary quadratic field
    with Luis Dieulefait and Lucio Guerberoff
    Math. Comp. 79 N. 270 (2010). [pdf]
    Code: [here]
  35. Shimura Correspondence and Central values of twisted L-series
    Proceedings of the "Segundas Jornadas de Teoría de Números". [pdf] (ISBN: 978-84-612-8007-0)
  36. Computing central values of twisted L-series, the case of composite levels
    with Gonzalo Tornaría
    Exp. Math. 17 N. 4 (2008). [pdf]
  37. On the embedding problem for $2^+S_4$ representations
    Math. Comp. 76 N. 260 (2007). [pdf]
  38. Shimura Correspondance for level p^2 and the central values of L-series
    with Gonzalo Tornaría
    Journal of Number Theory 124 N. 2 (2007). [pdf]
  39. Newton-Hensel Interpolation Lifting
    with Martín Avendaño and Teresa Krick
    Found. of Comp. Math. 6 (1) 2006. [pdf]
  40. Examples of Shimura Correspondance for level p^2 and real quadratic twists
    with Gonzalo Tornaría
    Ranks of elliptic curves and random matrix theory, pp. 289-314. [pdf]
  41. A formula for the central value of certain Hecke L-series (my Ph.D. thesis)
    Journal of Number Theory 113 (2005). [dvi]
  42. Computing Modular Forms of level p^2
    with Fernando Rodriguez Villegas
    Math. Comp. 74 (2005). [dvi]

Unpublished Works

  • Preimages for the Shimura map on Hilbert modular forms (survey article) with Nicolás Sirolli. [pdf]

Ph.D. Thesis

  • My Ph.D. thesis at the University of Texas at Austin. [pdf]

Scientific Events & Conferences

Conferences, research schools, and scientific meetings organized or co-organized:

Jornadas de Teoría de Números

Portuguese Meetings in Number Theory

  • Portuguese Number Theory Meetings & Summer Schools (including the First Meeting in Aveiro — Nov 2, 2022; Second Meeting in Porto — Sep 4–8, 2023; Third Meeting in Aveiro — Sep 9, 2024; and Fourth Meeting / Aveiro Summer School — Sep 1–4, 2025).

Students & Supervision

Student supervision in Number Theory, Algebra, and Arithmetic Geometry:

Graduate Students

  • Patricia Quattrini (Ph.D. co-advisor) — El efecto de la torsión en la distribución de Sha de los twists cuadráticos de una curva elíptica, July 2008.
  • Lucio Guerberoff (joint student with Michael Harris) — Modularity lifting theorems for unitary groups, July 2011.
  • Nicolás Sirolli — Calculo de formas de Hilber de pesos entero y medio entero, July 2013.
  • Maximiliano Camporino — Congruences between modular forms modulo prime powers, December 2015.
  • Daniel Kohen — Constructing Heegner points, August 2017.
  • Lucas Villagra Torcomian — Q-curvas y caracteres de Hecke en el método modular, February 2023.
  • Nicolás Mayorga Uruburu — Diagramas de Árbol en la clasificación de curvas modulares, April 2025.
  • Franco Golfieri Madriaga — Estudo de algumas equações diofantinas, September 2025.

Undergraduate Students

  • Lucio Guerberoff (Licenciatura thesis) — Teoría de Cuerpos de Clases, February 2007.
  • Marcelo Valdettaro (Licenciatura thesis) — Elliptic curves over $\mathbb{Q}(t)$ with high rank, November 2008.
  • Maximiliano Camporino (Licenciatura thesis) — Conjeturas de Serre, March 2011.
  • Daniel Kohen (Licenciatura thesis) — Puntos de Heegner, March 2013.
  • Nicolás Vescovo (Licenciatura thesis) — Sumas de cuatro cuadrados y Formas modulares, March 2013.
  • Lucas Villagra Torcomian (Licenciatura thesis) — Correspondencia de Langlands en dimensión 1, March 15, 2019 (Universidad Nacional de Córdoba).
  • Franco Anibal Golfieri Madriaga (Licenciatura thesis) — El último Teorema de Fermat, March 22, 2021 (Universidad Nacional de Córdoba).

Course Notes & Manuscripts

Here is a list some course notes I prepared:

  • Álgebra I (with Matías Graña), Universidad de Buenos Aires pdf
  • Topología, Universidad Nacional de Córdoba pdf
  • Formas de Hilbert (AGRA II school, 2015) pdf
  • Representaciones de Galois (AGRA III school, 2018) with Luis Dieulefait and Fernando Rodriguez-Villegas pdf

Please submit comments/mistakes to my email address.

Teaching

Courses lectured together with their problem sheets:

Advanced Courses at Universidad Nacional de Córdoba

Advanced Courses at Universidad de Buenos Aires

Please submit comments/mistakes to my email address.