Abbey Bourdon
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View article: Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$
Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$ Open
Let $\ell$ and $n$ be positive integers with $\ell$ prime. The modular curves $X_1(\ell^n)$ and $X_0(\ell^n)$ are algebraic curves over $\mathbb{Q}$ whose non-cuspidal points parameterize elliptic curves with a distinguished point of order…
View article: Uniform polynomial bounds on torsion from rational geometric isogeny classes
Uniform polynomial bounds on torsion from rational geometric isogeny classes Open
In 1996, Merel showed there exists a function $B\colon \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for any elliptic curve $E/F$ defined over a number field of degree $d$, one has the torsion group bound $\# E(F)[\textrm{tors}]\leq B(d)…
View article: Minimal torsion curves in geometric isogeny classes
Minimal torsion curves in geometric isogeny classes Open
In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $\overline{\mathbb{Q}}$-isogeny class $\mathcal{E}$ of elliptic curves and $N \in \mathbb{Z}^+$, we wish to determine the least d…
View article: Towards a classification of isolated $j$-invariants
Towards a classification of isolated $j$-invariants Open
We develop an algorithm to test whether a non-CM elliptic curve $E/\mathbb{Q}$ gives rise to an isolated point of any degree on any modular curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a method for computing …
View article: Torsion for CM elliptic curves defined over number fields of degree 2p
Torsion for CM elliptic curves defined over number fields of degree 2p Open
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 particular, our work shows that a classification in the stro…
View article: Sporadic points of odd degree on $X_1(N)$ coming from $\mathbb{Q}$-curves
Sporadic points of odd degree on $X_1(N)$ coming from $\mathbb{Q}$-curves Open
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 $X_1(N)$, most known examples of sporadic points come from elliptic c…
View article: On Isolated Points of Odd Degree
On Isolated Points of Odd Degree Open
Let C be a curve defined over a number field $k$, and suppose $C(k)$ is nonempty. We say a closed point $x$ on $C$ of degree $d$ is isolated if it does not belong to an infinite family of degree d points parametrized by the projective line…
View article: Odd degree isolated points on $X_1(N)$ with rational $j$-invariant
Odd degree isolated points on $X_1(N)$ with rational $j$-invariant Open
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 parametrized by the projective line or a positive rank abelian su…
View article: Torsion Subgroups of CM Elliptic Curves over Odd Degree Number Fields:
Torsion Subgroups of CM Elliptic Curves over Odd Degree Number Fields: Open
Let $\mathscr{G}_{\rm CM}(d)$ 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 completely determine $\mathscr{G}_{\rm CM}(d)$ for odd integers…
View article: Torsion points on CM elliptic curves over real number fields
Torsion points on CM elliptic curves over real number fields Open
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 groups which arise up to isomorphism as the torsion subgroup o…
View article: Anatomy of torsion in the CM case
Anatomy of torsion in the CM case Open
Let $T_{\mathrm{CM}}(d)$ 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 behavior of $T_{\mathrm{CM}}(d)$ as an "arithmetic function". Wh…
View article: A uniform version of a finiteness conjecture for CM elliptic curves
A uniform version of a finiteness conjecture for CM elliptic curves Open
Let A be an abelian variety defined over a number field F. For a prime number $\ell$, we consider the field extension of F generated by the $\ell$-powered torsion points of A. According to a conjecture made by Rasmussen and Tamagawa, if we…