Abstract
We count the number of rational elliptic curves of bounded naive height that have a rational N-isogeny, for N∈{2,3,4,5,6,8,9,12,16,18}. For some N, this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack.
| Original language | English |
|---|---|
| Pages (from-to) | 471-505 |
| Number of pages | 35 |
| Journal | Journal of Number Theory |
| Volume | 262 |
| DOIs | |
| Publication status | Published - Sept 2024 |
Bibliographical note
Publisher Copyright:© 2024 The Author(s)
Funding
We would like to thank Jordan Ellenberg for his valuable help and support. We would also like to thank David Zureick-Brown, John Voight and Jeremy Rouse for many helpful conversations and comments. We are also grateful to Andrew Snowden, Peter Bruin and Filip Najman for comments on the early draft of this paper. We thank Aaron Landesman for pointing out an error in the previous proof of Lemma 5.8 , and Brandon Alberts for his help in fixing it. The second author would also like to thank Wanlin Li and Libby Taylor. We also thank the anonymous referee for their comments on the paper, which helped improve its quality. During the preparation of this article, the authors were partially supported by NSF DMS-1700884. The second author was partially supported by NSF DMS-1928930 while they participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2020 semester.
| Funders | Funder number |
|---|---|
| National Science Foundation | DMS-1700884, DMS-1928930 |
Keywords
- Elliptic curves
- Isogenies
- Moduli stacks
- Rational points