Quadratic Points on Modular Curves

TL;DR

This paper classifies quadratic points on specific modular curves X_0(N) with genus 3 to 5, non-hyperelliptic, and finite Mordell-Weil group, providing explicit j-invariants and CM or quadratic erences.

Contribution

It explicitly determines quadratic points on X_0(N) for certain N with specified properties, including their j-invariants and CM or -curves.

Findings

Identified quadratic points on X_0(N) for N in {34, 38, 42, 44, 45, 51, 52, 54, 55, 56, 63, 64, 72, 75, 81}.

Provided explicit j-invariants for the associated elliptic curves.

Determined whether these elliptic curves have complex multiplication or are quadratic -curves.

Abstract

In this paper we determine the quadratic points on the modular curves X_0(N), where the curve is non-hyperelliptic, the genus is 3, 4 or 5, and the Mordell--Weil group of J_0(N) is finite. The values of N are 34, 38, 42, 44, 45, 51, 52, 54, 55, 56, 63, 64, 72, 75, 81. As well as determining the non-cuspidal quadratic points, we give the j-invariants of the elliptic curves parametrized by those points, and determine if they have complex multiplication or are quadratic \Q-curves.