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

Grant Molnar

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.

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

Abstract

Using methods from analytic number theory, for and for , we obtain asymptotics with power-saving error terms for counts of elliptic curves with a cyclic -isogeny up to quadratic twist over the rational numbers. For , we then apply a Tauberian theorem to achieve asymptotics with power saving error for counts of elliptic curves with a cyclic -isogeny up to isomorphism over the rational numbers.
Paper Structure (41 sections, 111 theorems, 631 equations)

This paper contains 41 sections, 111 theorems, 631 equations.

Table of Contents

  1. Introduction
  2. Motivation and setup
  3. Results
  4. Our approach
  5. Contents
  6. Historical background
  7. A primer on elliptic curves
  8. Counting elliptic curves: a brief history
  9. Previous results for m <= 5
  10. Technical preliminaries
  11. Height, minimality, and defect
  12. Parameterizing elliptic curves equipped with a cyclic m-isogeny
  13. Lattices and the principle of Lipschitz
  14. Analytic ingredients
  15. Our approach revisited
  16. ...and 26 more sections

Key Result

Theorem 1.2.1

Let $m \in \left\{6, 7, 8, 9\right\}$. Then there are effectively computable constants $\widetilde{c}_{m}$, $\widetilde{c}^\prime_{m}$, $c_{m}$, and $c^\prime_{m}$ such that for any $\epsilon > 0$, we have and for $X \geq 1$. The implicit constant depends on $m$ and $\epsilon$.

Theorems & Definitions (253)

  • Theorem 1.2.1
  • Theorem 1.2.4
  • Theorem 1.2.8: \ref{['Theorem: asymptotic for NQ(X) for m of nonzero genus']} and \ref{['Theorem: asymptotic for NQ(X) for m of nonzero genus given RH']}
  • Theorem 1.2.10
  • Remark 1.3.2
  • Remark 1.3.4
  • Theorem 2.1.1: Faltings's Theorem
  • proof
  • Theorem 2.1.2: Silverman
  • Remark 2.1.8
  • ...and 243 more