Computing quadratic points on modular curves 𝑋₀(𝑁)

Adžaga, Nikola; Keller, Timo; Michaud-Jacobs, Philippe; Najman, Filip; Ozman, Ekin; Vukorepa, Borna

doi:10.1090/mcom/3902

Computing quadratic points on modular curves $X_0(N)$
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by Nikola Adžaga, Timo Keller, Philippe Michaud-Jacobs, Filip Najman, Ekin Ozman and Borna Vukorepa;

Math. Comp. 93 (2024), 1371-1397

DOI: https://doi.org/10.1090/mcom/3902

Published electronically: October 3, 2023

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Abstract:

In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves $X_0(N)$ of genus up to $8$, and genus up to $10$ with $N$ prime, for which they were previously unknown. The values of $N$ we consider are contained in the set \begin{equation*} \mathcal {L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}. \end{equation*} We obtain that all the non-cuspidal quadratic points on $X_0(N)$ for $N\in \mathcal {L}$ are complex multiplication (CM) points, except for one pair of Galois conjugate points on $X_0(103)$ defined over $\mathbb {Q}(\sqrt {2885})$. We also compute the $j$-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.

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Mathematics of Computation

Computing quadratic points on modular curves $X_0(N)$
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Abstract: