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arXiv · 2512.24817

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

Ivan Novak
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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.

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Ivan Novak. 2025-12-31. Number of $K$-rational points with given $j$-invariant on modular curves. https://arxiv.org/abs/2512.24817

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