Abbey Bourdon

activity

20132026

most citedTorsion points and Galois representations on CM elliptic curves

27 citations · 44 across the 10 of their papers we have counts for

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15 papers · 1 filter

math.NT2026

On the Finiteness of Isolated $j$ -invariants for $X_{1} (N)$

Abbey Bourdon

Characterizing isolated points on the modular curve $X_{1} (N)$ is a key obstruction to classifying all points of a fixed degree. These points do not lie in infinite parameterized fam…

math.NT2025

Rational isolated $j$ -invariants from $X_{1} (ℓ^{n})$ and $X_{0} (ℓ^{n})$

Abbey Bourdon, Özlem Ejder

Let $ℓ$ and $n$ be positive integers with $ℓ$ prime. The modular curves $X_{1} (ℓ^{n})$ and $X_{0} (ℓ^{n})$ are algebraic curves over $Q$ whose non-cuspidal points param…

math.NT2024

Uniform polynomial bounds on torsion from rational geometric isogeny classes

Abbey Bourdon, Tyler Genao

In 1996, Merel showed there exists a function $B : Z^{+} \to Z^{+}$ such that for any elliptic curve $E / F$ defined over a number field of degree $d$ , one h…

math.NT2024

Minimal torsion curves in geometric isogeny classes

Abbey Bourdon, Nina Ryalls, Lori D. Watson

In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $Q$ -isogeny class $E$ of elliptic curves…

math.NT2023★ 1 cited

Towards a classification of isolated $j$ -invariants

Abbey Bourdon, Sachi Hashimoto, Timo Keller +5

We develop an algorithm to test whether a non-CM elliptic curve $E / Q$ gives rise to an isolated point of any degree on any modular curve of the form $X_{1} (N)$ . This builds…

math.NT2021

Torsion for CM elliptic curves defined over number fields of degree 2p

Abbey Bourdon, Holly Paige Chaos

For a prime number p, we characterize the groups that may arise as torsion subgroups of an elliptic curve with complex multiplication defined over a number field of degree 2p. In p…

math.NT2021

Sporadic points of odd degree on $X_{1} (N)$ coming from $Q$ -curves

Abbey Bourdon, Filip Najman

We say a closed point $x$ on a curve $C$ is sporadic if there are only finitely many points on $C$ of degree at most deg $(x)$ . In the case where $C$ is the modular curve $X_{1} (N)$ ,…

math.NT2020★ 3 cited

Odd degree isolated points on $X_{1} (N)$ with rational $j$ -invariant

Abbey Bourdon, David R. Gill, Jeremy Rouse +1

Let $C$ be a curve defined over a number field $k$ . We say a closed point $x \in C$ of degree $d$ is isolated if it does not belong to an infinite family of degree $d$ points parame…

math.NT2019★ 10 cited

Torsion points and isogenies on CM elliptic curves

Abbey Bourdon, Pete L. Clark

Let $O$ be an order in the imaginary quadratic field $K$ . For positive integers $M ∣ N$ , we determine the least degree of an $O$ -CM point on the modular curv…

math.NT2018

On the level of modular curves that give rise to isolated $j$ -invariants

Abbey Bourdon, Ozlem Ejder, Yuan Liu +2

We say a closed point $x$ on a curve $C$ is sporadic if $C$ has only finitely many closed points of degree at most $deg (x)$ and that $x$ is isolated if it is not in…

math.NT2016★ 27 cited

Torsion points and Galois representations on CM elliptic curves

Abbey Bourdon, Pete L. Clark

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result comput…

math.NT2016

Torsion subgroups of CM elliptic curves over odd degree number fields

Abbey Bourdon, Paul Pollack

Let $G_{CM} (d)$ denote the collection of groups (up to isomorphism) that appear as the torsion subgroup of a CM elliptic curve over a degree $d$ number field. We comp…

math.NT2015★ 3 cited

Anatomy of torsion in the CM case

Abbey Bourdon, Pete L. Clark, Paul Pollack

Let $T_{CM} (d)$ denote the maximum size of a torsion subgroup of a CM elliptic curve over a degree $d$ number field. We initiate a systematic study of the asymptotic behav…

math.NT2014

Torsion Points on CM Elliptic Curves Over Real Number Fields

Abbey Bourdon, Pete L. Clark, James Stankewicz

We study torsion subgroups of elliptic curves with complex multiplication (CM) defined over number fields which admit a real embedding. We give a complete classification of the gro…

math.NT2013

A Uniform Version of a Finiteness Conjecture for CM Elliptic Curves

Abbey Bourdon

Let A be an abelian variety defined over a number field F. For a prime number $ℓ$ , we consider the field extension of F generated by the $ℓ$ -powered torsion points of A. Acco…

On the Finiteness of Isolated j-invariants for X1​(N)

Rational isolated j-invariants from X1​(ℓn) and X0​(ℓn)

Uniform polynomial bounds on torsion from rational geometric isogeny classes

Minimal torsion curves in geometric isogeny classes

Towards a classification of isolated j-invariants

Torsion for CM elliptic curves defined over number fields of degree 2p

Sporadic points of odd degree on X1​(N) coming from Q-curves

Odd degree isolated points on X1​(N) with rational j-invariant

Torsion points and isogenies on CM elliptic curves

On the level of modular curves that give rise to isolated j-invariants

Torsion points and Galois representations on CM elliptic curves

Torsion subgroups of CM elliptic curves over odd degree number fields

Anatomy of torsion in the CM case

Torsion Points on CM Elliptic Curves Over Real Number Fields

A Uniform Version of a Finiteness Conjecture for CM Elliptic Curves

On the Finiteness of Isolated $j$ -invariants for $X_{1} (N)$

Rational isolated $j$ -invariants from $X_{1} (ℓ^{n})$ and $X_{0} (ℓ^{n})$

Towards a classification of isolated $j$ -invariants

Sporadic points of odd degree on $X_{1} (N)$ coming from $Q$ -curves

Odd degree isolated points on $X_{1} (N)$ with rational $j$ -invariant

On the level of modular curves that give rise to isolated $j$ -invariants