Counting elliptic curves with a rational N-isogeny for small N

Brandon Boggess; Soumya Sankar

doi:10.1016/j.jnt.2024.03.004

Counting elliptic curves with a rational N-isogeny for small N

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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	https://doi.org/10.1016/j.jnt.2024.03.004
Publication status	Published - Sept 2024

Keywords

Elliptic curves
Isogenies
Moduli stacks
Rational points

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10.1016/j.jnt.2024.03.004Licence: CC BY

1-s2.0-S0022314X24000751-mainFinal published version, 618 KBLicence: CC BY

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N2 - 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.

AB - 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.

KW - Elliptic curves

KW - Isogenies

KW - Moduli stacks

KW - Rational points

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JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

Counting elliptic curves with a rational N-isogeny for small N

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Keywords

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