Computing the Level of a Fiber for 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.