Computing the Level of a Fiber for Points on Modular Curves

2025/08/24 by Maxwell, Hailey
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.17463

Abstract

The modular curves in the family X₁(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 X₁(ℓⁿ) for a prime ℓ 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 X₁(ℓ^k+1) is as large as possible given the degree of its image on X₁(ℓ^k), then its lifts on X₁(ℓⁿ) 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 ℓ-adic Galois representations.

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