On the genera of X₀(N)

2000/06/13 by János Csirik, János A. Csirik, Joseph L. Wetherell +4 · 1 citation
Mathematics · #14G35 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.AG #math.NT #msc:14G35

paper · pdf · doi:10.48550/arxiv.math/0006096

9 pages; minor changes in v2

openalex publication_date 2000/06/13 · arxiv created 2000/09/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let g₀(N) be the genus of the modular curve X₀(N). We record several properties of the sequence g₀(N). Even though the average size of g₀(N) is 1.25N/pi², a random positive integer has probability zero of being a value of g₀(N). Also, if N is a random positive integer then g₀(N) is odd with probability one. The smallest non-negative integer not occuring as g₀(N) for any N is 150; the smallest such odd integer is 49267.

Cited by

• Dimension sequences of modular forms

Related

• The expected genus of a random chord diagram
• Genus formulas for families of modular curves
• The non-ordinary locus of the TTV family of curves
• Genus 0 Modular curves of prime power level with a point defined over number fields other than \Q
• Bielliptic modular curves X₀^*(N) with square-free levels