Splitting of primes in number fields generated by points on some modular curves

TL;DR

This paper investigates how primes split in number fields generated by points on specific modular curves, expanding understanding of prime behavior in quadratic and cubic fields linked to these curves.

Contribution

It provides new results on prime splitting in quadratic fields from hyperelliptic modular curves and in cubic fields from points on $X_1(2,14)$, extending prior work.

Findings

Prime splitting patterns in quadratic fields from hyperelliptic $X_0(n)$ curves.

Prime splitting in cubic fields generated by points on $X_1(2,14)$.

Generalization of Momose and Krumm's results on prime splitting.

Abstract

We study the splitting of primes in number fields generated by points on modular curves. Momose was the first to notice that quadratic points on $X_1(n)$ generate quadratic fields over which certain primes split in a particular way and his results were later expanded upon by Krumm. We prove results about the splitting behaviour of primes in quadratic fields generated by points on the modular curves $X_0(n)$ which are hyperelliptic (except for $n=37$) and in cubic fields generated by points on $X_1(2, 14)$.