Counting elliptic curves with a cyclic $m$-isogeny over $\mathbb{Q}$

Abstract

Using methods from analytic number theory, for $m>5$ and for $m=4$, we obtain asymptotics with power-saving error terms for counts of elliptic curves with a cyclic $m$-isogeny up to quadratic twist over the rational numbers. For $m>5$, we then apply a Tauberian theorem to achieve asymptotics with power saving error for counts of elliptic curves with a cyclic $m$-isogeny up to isomorphism over the rational numbers.

PhD thesis. The official manuscript for this thesis can be found here: https://digitalcommons.dartmouth.edu/dissertations/214/ 285 pages, 14 graphics