# Abbey Bourdon

## Wake Forest University

H-index: 8

North America-United States

## About Abbey Bourdon

Abbey Bourdon, With an exceptional h-index of 8 and a recent h-index of 7 (since 2020), a distinguished researcher at Wake Forest University, specializes in the field of Arithmetic Geometry.

His recent articles reflect a diverse array of research interests and contributions to the field:

Odd degree isolated points on with rational j-invariant

Towards a classification of isolated ????-invariants

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

Sporadic points of odd degree on coming from -curves

Torsion points and Galois representations on CM elliptic curves

Torsion points and isogenies on CM elliptic curves

### Abbey Bourdon Information

University | Wake Forest University |
---|---|

Position | ___ |

Citations(all) | 203 |

Citations(since 2020) | 178 |

Cited By | 106 |

hIndex(all) | 8 |

hIndex(since 2020) | 7 |

i10Index(all) | 6 |

i10Index(since 2020) | 6 |

University Profile Page | Wake Forest University |

#### Abbey Bourdon Skills & Research Interests

Arithmetic Geometry

## Top articles of Abbey Bourdon

### Odd degree isolated points on with rational j-invariant

Authors

Abbey Bourdon,David R Gill,Jeremy Rouse,Lori D Watson

Journal

Research in Number Theory

Published Date

2024/3

Let C be a curve defined over a number field k. We say a closed point 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 subvariety of the curve’s Jacobian. Building on work of Bourdon et al. (Adv Math 357(33):106824, 2019), we characterize elliptic curves with rational j-invariant which give rise to an isolated point of odd degree on for some positive integer N.

### Towards a classification of isolated ????-invariants

Authors

Abbey Bourdon,Sachi Hashimoto,Timo Keller,Zev Klagsbrun,David Lowry-Duda,Travis Morrison,Filip Najman,Himanshu Shukla

Journal

Mathematics of Computation

Published Date

2024

We develop an algorithm to test whether a non-complex multiplication elliptic curve gives rise to an isolated point of any degree on any modular curve of the form . This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to . Running this algorithm on all elliptic curves presently in the -functions and Modular Forms Database and the Stein–Watkins Database gives strong evidence for the conjecture that gives rise to an isolated point on if and only if , , or . References

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

Authors

Abbey Bourdon,Holly Paige Chaos

Journal

Proceedings of the American Mathematical Society

Published Date

2023/3

For a prime number , we characterize the groups that may arise as torsion subgroups of an elliptic curve with complex multiplication defined over a number field of degree . In particular, our work shows that a classification in the strongest sense is tied to determining whether there exist infinitely many Sophie Germain primes. References

### Sporadic points of odd degree on coming from -curves

Authors

Abbey Bourdon,Filip Najman

Journal

arXiv preprint arXiv:2107.10909

Published Date

2021/7/22

We say a closed point on a curve is sporadic if there are only finitely many points on of degree at most deg. In the case where is the modular curve , most known examples of sporadic points come from elliptic curves with complex multiplication (CM). We seek to understand all sporadic points on corresponding to -curves, which are elliptic curves isogenous to their Galois conjugates. This class contains not only all CM elliptic curves, but also any elliptic curve -isogenous to one with a rational -invariant, among others. In this paper, we show that all non-CM -curves giving rise to a sporadic point of odd degree lie in the -isogeny class of the elliptic curve with -invariant . In addition, we show that a stronger version of this finiteness result would imply Serre's Uniformity Conjecture.

### Torsion points and Galois representations on CM elliptic curves

Authors

Abbey Bourdon,Pete L Clark Clark

Journal

Pacific Journal of Mathematics

Published Date

2020/3/17

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 computes the degree in which such an elliptic curve has a rational point of order N, refining results of Silverberg (Compositio Math. 68: 3 (1988), 241–249; Contemp. Math. 133 (1992)). Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a K-CM elliptic curve E defined over K (j (E)).

### Torsion points and isogenies on CM elliptic curves

Authors

Abbey Bourdon,Pete L Clark

Journal

Journal of the London Mathematical Society

Published Date

2020/10

Let be an order in the imaginary quadratic field . For positive integers , we determine the least degree of an ‐CM point on the modular curve and also on the modular curve : that is, we treat both the case in which the complex multiplication is rationally defined and the case in which we do not assume that the complex multiplication is rationally defined. To prove these results, we establish several new theorems on rational cyclic isogenies of complex multiplication elliptic curves. In particular, we extend a result of Kwon (J. Korean Math. Soc. 36 (1999) 945–958) that determines the set of positive integers for which there is an ‐CM elliptic curve admitting a cyclic, ‐rational ‐isogeny.

## Abbey Bourdon FAQs

### What is Abbey Bourdon's h-index at Wake Forest University?

The h-index of Abbey Bourdon has been 7 since 2020 and 8 in total.

### What are Abbey Bourdon's top articles?

The articles with the titles of

Odd degree isolated points on with rational j-invariant

Towards a classification of isolated ????-invariants

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

Sporadic points of odd degree on coming from -curves

Torsion points and Galois representations on CM elliptic curves

Torsion points and isogenies on CM elliptic curves

are the top articles of Abbey Bourdon at Wake Forest University.

### What are Abbey Bourdon's research interests?

The research interests of Abbey Bourdon are: Arithmetic Geometry

### What is Abbey Bourdon's total number of citations?

Abbey Bourdon has 203 citations in total.