Abbey Bourdon
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Abbey Bourdon has a total of 18 co-authors. Here are the top co-authors by number of shared publications:
- Bianca Viray
Department of Mathematics, University of Washington, Seattle, WA, United States
1 shared publication - David Lowry-Duda
Institute of Computational and Experimental Research in Mathematics
1 shared publication - David R. Gill
Department of Mathematics, Wake Forest University, Winston-Salem, USA
1 shared publication - Filip Najman
Faculty of Science, Department of Mathematics, University of Zagreb, Zagreb, Croatia
1 shared publication - Frances Odumodu1 shared publication
- Himanshu Shukla
Mathematisches Institut, Uiversität Bayreuth, Universitätstrasse 30, 95444 Bayreuth, Germany
1 shared publication
Latest Publications
- Open AccessRational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$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 $\ell^n$ or a distinguished cyclic subgroup of order $\ell^n$, respectively. We wish to understand isolated points on these curves, which are roughly those not belonging to an infinite parameterized family of points having the same degree. Our first main result is that there are precisely 15 $j$-invariants in $\mathbb{Q}$ which arise as the image of an isolated point $x\in X_1(\ell^n)$ under the natural map $j:X_1(\ell^n) \rightarrow X_1(1)$. This completes a prior partial classification of Ejder. We also identify the 19 rational $j$-invariants which correspond to isolated points on $X_0(\ell^n)$.Open AccessRational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$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 $\ell^n$ or a distinguished cyclic subgroup of order $\ell^n$, respectively. We wish to understand isolated points on these curves, which are roughly those not belonging to an infinite parameterized family of points having the same degree. Our first main result is that there are precisely 15 $j$-invariants in $\mathbb{Q}$ which arise as the image of an isolated point $x\in X_1(\ell^n)$ under the natural map $j:X_1(\ell^n) \rightarrow X_1(1)$. This completes a prior partial classification of Ejder. We also identify the 19 rational $j$-invariants which correspond to isolated points on $X_0(\ell^n)$.
- 25 April 2024
- Published by American Mathematical Society (AMS) in Mathematics of Computation
- Vol. 94 (351) , 447-473
Towards a classification of isolated 𝑗-invariants
We develop an algorithm to test whether a non-complex multiplication elliptic curve E / Q E/\mathbf {Q} gives rise to an isolated point of any degree on any modular curve of the form X 1 ( N ) X_1(N) . This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to E E . Running this algorithm on all elliptic curves presently in the L L -functions and Modular Forms Database and the Stein–Watkins Database gives strong evidence for the conjecture that E E gives rise to an isolated point on X 1 ( N ) X_1(N) if and only if j ( E ) = − 140625 / 8 , − 9317 j(E)=-140625/8, -9317 , 351 / 4 351/4 , or − 162677523113838677 -162677523113838677 .
- 25 April 2024
- Published by American Mathematical Society (AMS) in Mathematics of Computation
- Vol. 94 (351) , 447-473
Towards a classification of isolated 𝑗-invariants
We develop an algorithm to test whether a non-complex multiplication elliptic curve E / Q E/\mathbf {Q} gives rise to an isolated point of any degree on any modular curve of the form X 1 ( N ) X_1(N) . This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to E E . Running this algorithm on all elliptic curves presently in the L L -functions and Modular Forms Database and the Stein–Watkins Database gives strong evidence for the conjecture that E E gives rise to an isolated point on X 1 ( N ) X_1(N) if and only if j ( E ) = − 140625 / 8 , − 9317 j(E)=-140625/8, -9317 , 351 / 4 351/4 , or − 162677523113838677 -162677523113838677 .
- 17 December 2023
- Published by Springer Nature in Research in Number Theory
- Vol. 10 (1) , 1-32
Odd degree isolated points on $X_1(N)$ with rational j-invariant- 17 December 2023
- Published by Springer Nature in Research in Number Theory
- Vol. 10 (1) , 1-32
Odd degree isolated points on $X_1(N)$ with rational j-invariant- 15 June 2020
- Published by Wiley in Journal of the London Mathematical Society
- Vol. 102 (2) , 580-622
Torsion points and isogenies on CM elliptic curves
- 15 June 2020
- Published by Wiley in Journal of the London Mathematical Society
- Vol. 102 (2) , 580-622
Torsion points and isogenies on CM elliptic curves
- Open Access
On the level of modular curves that give rise to isolated $j$-invariants
We say a closed point $x$ on a curve $C$ is sporadic if $C$ has only finitely many closed points of degree at most $\operatorname{deg}(x)$ and that $x$ is isolated if it is not in a family of effective degree $d$ divisors parametrized by $\mathbb{P}^1$ or a positive rank abelian variety (see Section 4 for more precise definitions and a proof that sporadic points are isolated). Motivated by well-known classification problems concerning rational torsion of elliptic curves, we study sporadic and isolated points on the modular curves $X_1(N)$. In particular, we show that any non-cuspidal non-CM sporadic, respectively isolated, point $x \in X_1(N)$ maps down to a sporadic, respectively isolated, point on a modular curve $X_1(d)$, where $d$ is bounded by a constant depending only on $j(x)$. Conditionally, we show that $d$ is bounded by a constant depending only on the degree of $\mathbb{Q}(j(x))$, so in particular there are only finitely many $j$-invariants of bounded degree that give rise to sporadic or isolated points.Open AccessOn the level of modular curves that give rise to isolated $j$-invariants
We say a closed point $x$ on a curve $C$ is sporadic if $C$ has only finitely many closed points of degree at most $\operatorname{deg}(x)$ and that $x$ is isolated if it is not in a family of effective degree $d$ divisors parametrized by $\mathbb{P}^1$ or a positive rank abelian variety (see Section 4 for more precise definitions and a proof that sporadic points are isolated). Motivated by well-known classification problems concerning rational torsion of elliptic curves, we study sporadic and isolated points on the modular curves $X_1(N)$. In particular, we show that any non-cuspidal non-CM sporadic, respectively isolated, point $x \in X_1(N)$ maps down to a sporadic, respectively isolated, point on a modular curve $X_1(d)$, where $d$ is bounded by a constant depending only on $j(x)$. Conditionally, we show that $d$ is bounded by a constant depending only on the degree of $\mathbb{Q}(j(x))$, so in particular there are only finitely many $j$-invariants of bounded degree that give rise to sporadic or isolated points. - 1 September 2016
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 285 (3-4) , 795-820
Anatomy of torsion in the CM case
- 1 September 2016
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 285 (3-4) , 795-820
Anatomy of torsion in the CM case