Alex Barrios

My research interests are in Algebraic Number Theory, Class Field Theory, and Diophantine Geometry.

One of my primary focuses is on bridging the gap between our theoretical understanding of elliptic curves and the explicit construction of examples, and the calculation of data pertaining to elliptic curves. In this direction, my recent work includes the explicit classification of rational elliptic curves with non-trivial isogeny, as well as the classification of minimal discriminants and local data (joint work with Manami Roy) of rational elliptic curves with a non-trivial torsion subgroup. The latter also explicitly classifies all rational elliptic curves with a non-trivial torsion point which have global Tamagawa number equal to one. The global Tamagawa number is one of the quantities that occur in the famed Birch and Swinnerton-Dyer conjecture. In addition, Roy and I have explicitly classified the cuspidal automorphic representation attached to rational elliptic curves with a non-trivial odd torsion point.

Part of my work on elliptic curves is motivated by the Modified Szpiro Conjecture, which is equivalent to Masser and Oesterlé‘s ABC Conjecture. To this end, I have shown that there are infinitely many good elliptic curves with a specified torsion subgroup. This generalizes Masser’s Theorem on the existence of infinitely many good elliptic curves with full 2-torsion. I have also established lower bounds for the modified Szpiro ratio. In fact, these lower bounds are sharp.

Undergraduate Research

I am also passionate about introducing undergraduates to research topics in algebraic geometry and number theory. Through the Pomona Research in Mathematics Experience (PRiME) and the Mathematical Sciences Research Institute Undergraduate Program (MSRI-UP), I have directed the following undergraduate research projects:

PRiME 2022

Good ABC Triples and Good Elliptic Curves

Students: Elise Alvarez-Salazar (University of California, Santa Barbara), Barry Henaku (Washington University in St. Louis), and Summer Soller (University of Utah)

• Recipients of the MAA MathFest 2022 Poster Presentation Award

PRiME 2021

Minimal Discriminants of Rational Elliptics with Prescribed Isogeny Graphs

Students: Alyssa Brasse (Hunter College), Nevin Etter (Washington and Lee University), Gustavo Flores (Carleton College), Drew Miller (University of California, Santa Barbara), and Summer Soller (University of Utah)

• Recipients of the MAA MathFest 2021 Number Theory Undergraduate Research Presentation

MSRI-UP 2020

Automorphism and Monodromy Groups of Classical Modular Curves (co-advised with Edray Goins)

Students: Samuel Heard (University of Oklahoma), Fabian Ramirez (Sonoma State University), Vanessa Sun (Hunter College)

Explicit Constructions of Finite Groups as Monodromy Groups (co-advised with Edray Goins)

Students: Ra-Zakee Muhammad (Pomona College), Javier Santiago (University of Puerto Rico, Río Piedras), Eyob Tsegaye (Stanford University)

Dessin d’Enfants from Cartographic Groups (co-advised with Edray Goins and Sofía Martinez)

Students: Nicholas Arosemena (Morehouse College), Yaren Euceda (University of Minnesota, Twin Cities), Ashly Powell (University of the Virgin Islands)

Carleton College Towsley Fund for Winter Research 2019

Minimal Discriminants of Rational Elliptic Curves Separated by a 4-isogeny

Student: Abigail Loe (Carleton College)

PRiME 2019

Minimal Discriminants of Rational Elliptic Curves with a non-trivial Rational Isogeny

Students: Alvaro Cornejo (University of California, Santa Barbara), Owen Ekblad (University of Michigan, Dearborn), Marietta Geist (Carleton College), Kayla Harrison (Eckerd College), and Abigail Loe (Carleton College)

• Recipients of the MAA MathFest 2019 Number Theory Undergraduate Research Presentation