On quadratic points of classical modular curves

TL;DR

This survey explores quadratic points on classical modular curves, highlighting the role of Fumiyuki Momose's work, correcting literature inaccuracies, and providing detailed arithmetical results on hyperelliptic and bielliptic curves.

Contribution

It consolidates and clarifies existing results on quadratic points of modular curves, correcting inaccuracies and making difficult-to-find arithmetical results accessible.

Findings

Identification of modular curves with infinite quadratic points over Q

Clarification of automorphism groups of modular curves

Compilation of arithmetical results on hyperelliptic and bielliptic curves

Abstract

Classical modular curves are of deep interest in arithmetic geometry. In this survey we show how the work of Fumiyuki Momose is involved in order to list the classical modular curves which satisfy that the set of quadratic points over $\mathbb{Q}$ is infinite. In particular we recall results of Momose on hyperelliptic modular curves and on automorphisms groups of modular curves. Moreover, we fix some inaccuracies of the existing literature in few statements concerning automorphism groups of modular curves and we make available different results that are difficult to find a precise reference, for example: arithmetical results on hyperelliptic and bielliptic curves (like the arithmetical statement of a Harris and Silverman theorem (or the case $d=2$ of a Abramovich and Harris theorem)) and on the conductor of elliptic curves over $\mathbb{Q}$ parametrized by X(N).