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Number of $K$-rational points with given $j$-invariant on modular curves

Ivan Novak

Abstract

In this article, we study how to compute the number of $K$-rational points with a given $j$-invariant on an arbitrary modular curve. As an application, for each positive integer $n$, we determine the list of possible numbers of cyclic $n$-isogenies an elliptic curve over some number field can admit. Similarly, for an odd prime power $p^k$, we calculate the possible values for the number of points above some $j$-invariant on Cartan modular curves $X_{\mathrm s}(p^k)$, $X_{\mathrm{ns}}(p^k)$ and their normalizers. Combining known results about images of Galois representations of CM elliptic curves with our work, we also devise a simple algorithm to determine the number of rational CM points on any modular curve.

Number of $K$-rational points with given $j$-invariant on modular curves

Abstract

In this article, we study how to compute the number of -rational points with a given -invariant on an arbitrary modular curve. As an application, for each positive integer , we determine the list of possible numbers of cyclic -isogenies an elliptic curve over some number field can admit. Similarly, for an odd prime power , we calculate the possible values for the number of points above some -invariant on Cartan modular curves , and their normalizers. Combining known results about images of Galois representations of CM elliptic curves with our work, we also devise a simple algorithm to determine the number of rational CM points on any modular curve.
Paper Structure (5 sections, 23 theorems, 79 equations)

This paper contains 5 sections, 23 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Number of $K$-rational points on a modular curve with a given $j$-invariant
  3. Number of $p^k$-isogenies
  4. Possible number of points above a $j$-invariant on some classes of modular curves
  5. Number of rational CM points on a modular curve

Key Result

Proposition 1.1

Let $p$ be an odd prime and $k$ a positive integer. Let $E$ be an elliptic curve over a number field $K$. The number of $p^k$-isogenies of $E$ which are defined over $K$ belongs to the set Furthermore, each of the numbers from the set is achieved for some number field $K$ and some elliptic curve $E/K$.

Theorems & Definitions (50)

  • Proposition 1.1
  • Proposition 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • ...and 40 more