Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Degrees of isogenies over prime degree number fields of non-CM elliptic curves with rational $j$-invariant

Ivan Novak

Abstract

We determine all possible degrees of cyclic isogenies of non-CM elliptic curves with rational $j$-invariant over number fields of degree $p$, where $p$ is an odd prime. The question had been answered for $p=2$, so this paper completes the classification in case when the degree of the number field is prime.

Degrees of isogenies over prime degree number fields of non-CM elliptic curves with rational $j$-invariant

Abstract

We determine all possible degrees of cyclic isogenies of non-CM elliptic curves with rational -invariant over number fields of degree , where is an odd prime. The question had been answered for , so this paper completes the classification in case when the degree of the number field is prime.

Paper Structure

This paper contains 14 sections, 35 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Images of $\ell$-adic Galois representations
  3. Growth of cyclic isogeny degree for prime powers
  4. Subgroups with uppertriangular tendencies
  5. Upper bounds on the size of $UTT$ subgroups
  6. Isogeny degrees over cubic extensions
  7. Prime powers
  8. Powers of two
  9. Powers of odd primes
  10. Odd isogeny degrees
  11. Even isogeny degrees
  12. Isogeny degrees over extensions of prime degree $p>3$
  13. Case $p>5$
  14. Case $p=5$

Key Result

Theorem 1.2

There exists a non-CM elliptic curve with rational $j$-invariant and an $n$-isogeny defined over a cubic extension of $\mathbb{Q}$ if and only if

Theorems & Definitions (69)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 59 more