Item Details

title

Computing the Level of a Fiber for Points on Modular Curves

author

Maxwell, Hailey

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 thesis 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^{n+1}) is as large as possible given the degree of its image on X_1(ell^n), then its lifts on X_1(ell^k) have degree as large as possible for all k > n. 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.

contributor

Bourdon, Abbey (advisor)

Howards, Hugh (committee member)

date

2025-06-24T08:36:28Z (accessioned)

2025-06-24T08:36:28Z (available)

2025 (issued)

degree

Mathematics (discipline)

identifier

http://hdl.handle.net/10339/111004 (uri)

language

en (iso)

publisher

Wake Forest University

type

Thesis