# Quadratic points on modular curves with infinite Mordell--Weil group

## TL;DR

This paper extends the classification of quadratic points on modular curves $X_0(N)$ for genus 2 to 5 with positive Mordell--Weil rank, using a relative symmetric Chabauty method and Mordell--Weil sieve.

## Contribution

It provides a complete determination of quadratic points on $X_0(N)$ for specific N with positive rank, including cases with infinite quadratic points due to degree 2 maps.

## Key findings

- Identifies quadratic points on $X_0(N)$ for N=37, 43, 53, 61, 57, 65, 67, 73.
- Describes the structure of quadratic points arising from the map to $X_0(N)^+$.
- Determines the $j$-invariants and properties of associated elliptic curves.

## Abstract

Bruin--Najman and Ozman--Siksek have recently determined the quadratic points on all modular curves $X_0(N)$ of genus 2, 3, 4, and 5 whose Mordell--Weil group has rank 0. In this paper we do the same for the $X_0(N)$ of genus 2, 3, 4, and 5 and positive Mordell--Weil rank. The values of $N$ are 37, 43, 53, 61, 57, 65, 67 and 73. The main tool used is a relative symmetric Chabauty method, in combination with the Mordell--Weil sieve. Often the quadratic points are not finite, as the degree 2 map $X_0(N)\to X_0(N)^+$ can be a source of infinitely many such points. In such cases, we describe this map and the rational points on $X_0(N)^+$, and we specify the exceptional quadratic points on $X_0(N)$ not coming from $X_0(N)^+$. In particular we determine the $j$-invariants of the corresponding elliptic curves and whether they are $\mathbb{Q}$-curves or have complex multiplication.