Quadratic points on modular curves $X_0(N)$ for $N\leq 100$

TL;DR

This paper determines all quadratic points on modular curves $X_0(N)$ for specific $N eq 66,70,78,82,84,86,87,88,90,96,99$, extending previous classifications by refining the 'going down method' with moduli descriptions.

Contribution

The paper advances the classification of quadratic points on $X_0(N)$ for certain $N$ by improving the 'going down method' using moduli descriptions.

Findings

Complete list of quadratic points for specified $N$.

Enhanced 'going down method' for analyzing quadratic points.

New techniques for modular curve analysis.

Abstract

We determine the quadratic points on the modular curves $X_0(N)$ for $N\leq 100$ for which this has not been previously done, namely the cases $$N\in\{66,70,78,82,84,86,87,88,90,96,99\}.$$ We accomplish this by improving on the ``going down method," which uses the fact that we have a moduli description of all the (infinitely many) quadratic points on $X_0(n)$ for some divisor $n$ of $N$.