Abbey Bourdon

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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

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.NT2025

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

Abbey Bourdon, Ãzlem 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.NT2025

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.NT2025

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…

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

Minimal torsion curves in geometric isogeny classes

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

Uniform polynomial bounds on torsion from rational geometric isogeny classes

Towards a classification of isolated j-invariants

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