Alex Barrios

My primary research focuses on the explicit arithmetic of elliptic curves and related questions in algebraic number theory and arithmetic geometry. A recurring theme in my work is the study of how arithmetic invariants vary in families, including local reduction data, minimal discriminants, and other quantities that enter into the Birch and Swinnerton-Dyer (BSD) conjecture. In its strong form, BSD asserts that the leading term of the L-function of an elliptic curve is governed by a collection of arithmetic invariants, so understanding how these quantities vary in families is an important part of understanding BSD. I am also interested in questions motivated by the abc conjecture, and its equivalent formulation in the realm of elliptic curves, the modified Szpiro conjecture, where explicit constructions and divisibility results provide concrete evidence and insight. Below is a list of my research publications and submitted articles in chronological order, followed by my expository publications.

1. On the Birch and Swinnerton-Dyer formula modulo squares for certain quadratic twists of elliptic curves, submitted (with Chung Pang Mok)

2. Tamagawa numbers of elliptic curves with an l-isogeny, submitted (with John Cullinan).

3. Reduced minimal models and torsion, submitted.

4. Prime isogenous discriminant ideal twins, to appear in Journal of Number Theory (with Alyson Deines, Maila Brucal-Hallare, Piper Harris, Manami Roy).

5. Symmetric tensor powers of graphs, Sci. Ser. A Math. Sci. (N.S.) 36 (2026), 12–34. (with Weymar Astaiza, Henry Chimal-Dzul, Stephan Ramon Garcia, Jaaziel de la Luz, Victor H. Moll, Yunied Puig, Diego Villamizar) DOI

6. Local data of elliptic curves under quadratic twist, Res. Number Theory 11 (2025), no. 3, Paper No. 75, 39 pp. (with Manami Roy, Nandita Sahajpal, Darwin Tallana, Bella Tobin, Hanneke Wiersema). DOI

7. On abc triples of the form (1,c-1,c),  Integers 23 (2023), Paper No. A65, 22 pp. (with Elise Alvarez-Salazar, Calvin Henaku, Summer Soller). DOI

8. Lower bounds for the modified Szpiro ratio, Acta Arithmetica 208 (2023), 51-68. DOI

9. Explicit classification of isogeny graphs of rational elliptic curves, Int. J. Number Theory 19 (2023), no. 4, 913–936. DOI

10. Good elliptic curves with a specified torsion subgroup, Journal of Number Theory 242 (2023), 21–43. DOI

11. Representations attached to elliptic curves with a non-trivial odd torsion point, Bull. London Math. Soc. 2022;54:1846–1861 (with Manami Roy). DOI

12. Local data of rational elliptic curves with non-trivial torsion, Pacific J. Math. 318 (2022) 1-42 (with Manami Roy). DOI

13. Minimal models of rational elliptic curves with non-trivial torsion, Res. Number Theory 8 (2022), no. 1, Paper No. 4., 39 pp. DOI

14. A constructive proof of Masser’s Theorem, Contemp. Math., Vol 759, pp. 51-61, 2020. DOI

15. Minimal models of rational elliptic curves with non-trivial torsion, Ph.D. Thesis - Purdue University, 2018, 372 pp., ProQuest LLC, Ann Arbor, MI.

Expository publications

1. From Proofs to Discovery, A Junior's Path to Math Graduate School (Part II), to appear in MAA FOCUS Jun./Jul. 2026 (with Natasa Dragovic).

2. From Proofs to Discovery, A Junior's Path to Math Graduate School (Part I), to appear in MAA FOCUS Apr./May 2026  (with Natasa Dragovic).

3. Book chapter in Math Alliance: Investing in Tomorrow Today. Count Me In: Community and Belonging in Mathematics by Della Dumbaugh and Deanna Haunsperger, MAA Press, 2022. (with Ranthony A.C. Edmonds, Roberto Soto).