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On the Classification of Modular Congruence Families

Nicolas Allen Smoot

Abstract

Congruence families, i.e., $\ell$-adic convergence for well-defined arithmetic subsequences, is a commonplace phenomenon for the coefficients of modular forms. Such families superficially resemble one another, but they often vary substantially in difficulty. Moreover, the critical difficulties associated with a given family will generally manifest themselves at the end stages of an attempted proof. We give a conjectured classification system of congruence families for the coefficients of modular eta quotients by studying the topology of the associated modular curve.

On the Classification of Modular Congruence Families

Abstract

Congruence families, i.e., -adic convergence for well-defined arithmetic subsequences, is a commonplace phenomenon for the coefficients of modular forms. Such families superficially resemble one another, but they often vary substantially in difficulty. Moreover, the critical difficulties associated with a given family will generally manifest themselves at the end stages of an attempted proof. We give a conjectured classification system of congruence families for the coefficients of modular eta quotients by studying the topology of the associated modular curve.
Paper Structure (17 sections, 6 theorems, 40 equations)

This paper contains 17 sections, 6 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Background
  3. General Method
  4. Genus
  5. Genus 0
  6. Genus 1
  7. Genus $\ge 2$
  8. Cusp Count
  9. Prime $N$
  10. Composite $N$: Cusp Count 4
  11. Composite $N$: Cusp Count 6
  12. Composite $N$: Odd Cusp Count
  13. Classification
  14. Summary
  15. Sporadic Families
  16. ...and 2 more sections

Key Result

Theorem 1.1

with

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2: Rø dseth
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 3.1: Weierstrass
  • Lemma 4.1