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

TL;DR

This paper develops methods to compute the number of rational points with a specific $j$-invariant on modular curves, with applications to elliptic curves' isogenies and CM points, advancing understanding of rational points in number theory.

Contribution

It introduces a general approach for counting $K$-rational points with a given $j$-invariant on modular curves and applies it to classify possible isogeny counts and CM points.

Findings

Determined possible numbers of cyclic $n$-isogenies for elliptic curves over number fields.

Calculated possible point counts on Cartan modular curves for prime power levels.

Devised an algorithm to count rational CM points on modular curves.

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.