Computing the Level of a Fiber for Points on Modular Curves

TL;DR

This paper introduces a new way to measure the level of fibers on modular curves and proves results about the degrees of points on these curves, extending understanding of their arithmetic properties.

Contribution

It defines the level of a fiber on modular curves and establishes conditions under which degrees of points are maximized across lifts, inspired by Galois representation techniques.

Findings

Degree of points on $X_1( ext{ell}^{k+1})$ can be maximized given the degree on $X_1( ext{ell}^k)$

Lifts of points maintain maximal degree under certain conditions

Results extend to all $n > k$ for the degrees of points on modular curves

Abstract

The modular curves in the family $X_1(N)$ for natural numbers $N$ parametrize elliptic curves over the complex numbers with a distinguished point of order $N$. The purpose of this paper is to better understand how to calculate the degrees of points on $X_1(\ell^n)$ for a prime $\ell$ and arbitrary positive integer $n$. In analogy with the definition of the level of a Galois representation, we construct a new definition: the level of a fiber of a closed point on a modular curve. Using this definition, we prove that, under certain conditions, if the degree of a point on $X_1(\ell^{k+1})$ is as large as possible given the degree of its image on $X_1(\ell^k),$ then its lifts on $X_1(\ell^n)$ have degree as large as possible for all $n > k$. We prove this result using techniques inspired by work of Lang and Trotter which gives a similar result for the image of $\ell$-adic Galois representations.