Gonality of the modular curve X1(N)
Maarten Derickx, Mark van Hoeij
TL;DR
This paper computes the gonality of modular curves X1(N) over Q for N up to 40, provides bounds up to 250, and identifies N with infinitely many points of degree ≤ 8, contributing to understanding rational points on these curves.
Contribution
It offers the first comprehensive computation of gonality for X1(N) over Q for N ≤ 40 and establishes bounds for larger N, along with a conjecture on modular units generation.
Findings
Gonality values for N ≤ 40 are explicitly computed.
Upper bounds for gonality are provided for N ≤ 250.
Identifies N with infinitely many points of degree ≤ 8.
Abstract
In this paper we compute the gonality over Q of the modular curve X1(N) for all N <= 40 and give upper bounds for each N <= 250. This allows us to determine all N for which X1(N) has infinitely points of degree <= 8. We conjecture that the modular units of Q(X1(N)) are freely generated by f_2,...,f_{[N/2]+1} where f_k is obtained from the equation for X1(k).
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Commutative Algebra and Its Applications · Meromorphic and Entire Functions