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

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 \[ \mathcal{L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}.\] We obtain that all the non-cuspidal quadratic points on $X_0(N)$ for $N\in \mathcal{L}$ are 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.

arXiv commentMinor corrections, to appear in Mathematics of Computation. (25 pages, 16 tables)

References in corpus (6)

• The Mordell-Weil sieve: Proving non-existence of rational points on curves
• Hyperelliptic modular curves $X_0(n)$ and isogenies of elliptic curves over quadratic fields
• Fermat's Last Theorem over some small real quadratic fields
• Torsion points on elliptic curves over number fields of small degree
• Fermat's Last Theorem and modular curves over real quadratic fields
• Quadratic Points on Non-Split Cartan Modular Curves