Alexander J. Barrios

activity

20192026

most citedMinimal models of rational elliptic curves with non-trivial torsion

6 citations · 13 across the 15 of their papers we have counts for

collaborators

15 papers

math.NT2026

Isogeny graphs of elliptic curves in characteristic zero

Alexander J. Barrios, Enrique González-Jiménez, Ivan Novak

For an elliptic curve $E$ defined over a field $K$ of characteristic $0$ with $End_{K} E ≅ Z$ , we classify which isogeny graphs $G (E / K)$ can…

math.NT2025

On the Birch and Swinnerton-Dyer formula modulo squares for certain quadratic twists of elliptic curves

Alexander J. Barrios, Chung Pang Mok

Let $E / Q$ be an elliptic curve with conductor $N = N_{+} N_{-}$ , where $N_{+}$ and $N_{-}$ are coprime and $N_{-}$ is squarefree. Let $D$ be a positive fundamental discriminant satisfy…

math.NT2025

Local data of elliptic curves under quadratic twist

Alexander J. Barrios, Manami Roy, Nandita Sahajpal +3

Let $K$ be the field of fractions of a complete discrete valuation ring with a perfect residue field. In this article, we investigate how the Tamagawa number of $E / K$ changes under…

math.NT2024

Tamagawa Numbers of Elliptic Curves with an $ℓ$ -isogeny

Alexander Barrios, John Cullinan

Let $ℓ$ be an odd prime, and suppose $E$ is an elliptic curve defined over the rational numbers $Q$ . If $E$ has an $ℓ$ -torsion point, then there has been significant…

math.NT2024

Prime isogenous discriminant ideal twins

Alexander J. Barrios, Maila Brucal-Hallare, Alyson Deines +2

Let $E_{1}$ and $E_{2}$ be elliptic curves defined over a number field $K$ . We say that $E_{1}$ and $E_{2}$ are discriminant ideal twins if they are not $K$ -isomorphic and have the…

math.CO2023

Symmetric tensor powers of graphs

Weymar Astaiza, Alexander J. Barrios, Henry Chimal-Dzul +5

The symmetric tensor power of graphs is introduced and its fundamental properties are explored. A wide range of intriguing phenomena occur when one considers symmetric tensor power…

math.NT2023

Reduced minimal models and torsion

Alexander J. Barrios

Let $E / Q$ be an elliptic curve. The reduced minimal model of $E$ is a global minimal model $y^{2} + a_{1} x y + a_{3} y = x^{3} + a_{2} x^{2} + a_{4} x + a_{6}$ which satisfies the additio…

math.NT2023

On $ab c$ triples of the form $(1, c - 1, c)$

Elise Alvarez-Salazar, Alexander J. Barrios, Calvin Henaku +1

By an $ab c$ triple, we mean a triple $(a, b, c)$ of relatively prime positive integers $a, b,$ and $c$ such that $a + b = c$ and $rad (ab c) < c$ , where $rad (n)$ …

math.NT2022★ 2 cited

Explicit classification of isogeny graphs of rational elliptic curves

Alexander J. Barrios

Let $n > 1$ be an integer such that $X_{0} (n)$ has genus $0$ , and let $K$ be a field of characteristic $0$ or relatively prime to $6 n$ . In this article, we explicitly…

math.NT2021

Representations attached to elliptic curves with a non-trivial odd torsion point

Alexander J. Barrios, Manami Roy

We give a classification of the cuspidal automorphic representations attached to rational elliptic curves with a non-trivial torsion point of odd order. Such elliptic curves are pa…

math.NT2021

Lower bounds for the modified Szpiro ratio

Alexander J. Barrios

Let $E / Q$ be an elliptic curve. The modified Szpiro ratio of $E$ is the quantity $σ_{m} (E) = lo g max {c_{4}^{3}, c_{6}^{2}} / lo g N_{E}$ …

math.NT2021★ 4 cited

Local data of rational elliptic curves with non-trivial torsion

Alexander J. Barrios, Manami Roy

By Mazur's Torsion Theorem, there are fourteen possibilities for the non-trivial torsion subgroup $T$ of a rational elliptic curve. For each $T$ , such that $E$ may have additive re…

math.NT2020★ 1 cited

Good elliptic curves with a specified torsion subgroup

Alexander J. Barrios

An elliptic curve $E$ over $Q$ is said to be good if $N_{E}^{6} < max {c_{4}^{3}, c_{6}^{2}}$ where $N_{E}$ is the conductor of $E$ an…

math.NT2020★ 6 cited

Minimal models of rational elliptic curves with non-trivial torsion

Alexander J. Barrios

In this paper, we explicitly classify the minimal discriminants of all elliptic curves $E / Q$ with a non-trivial torsion subgroup. This is done by considering various param…

math.NT2019

A Constructive Proof of Masser's Theorem

Alexander J. Barrios

The Modified Szpiro Conjecture, equivalent to the $ab c$ Conjecture, states that for each $ε > 0$ , there are finitely many rational elliptic curves satisfying $N_{E}^{6+ε}<\max\!\left…

Isogeny graphs of elliptic curves in characteristic zero

On the Birch and Swinnerton-Dyer formula modulo squares for certain quadratic twists of elliptic curves

Local data of elliptic curves under quadratic twist

Tamagawa Numbers of Elliptic Curves with an ℓ-isogeny

Prime isogenous discriminant ideal twins

Symmetric tensor powers of graphs

Reduced minimal models and torsion

On abc triples of the form (1,c−1,c)

Explicit classification of isogeny graphs of rational elliptic curves

Representations attached to elliptic curves with a non-trivial odd torsion point

Lower bounds for the modified Szpiro ratio

Local data of rational elliptic curves with non-trivial torsion

Good elliptic curves with a specified torsion subgroup

Minimal models of rational elliptic curves with non-trivial torsion

A Constructive Proof of Masser's Theorem

Tamagawa Numbers of Elliptic Curves with an $ℓ$ -isogeny

On $ab c$ triples of the form $(1, c - 1, c)$