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Analogue of the theta group $Γ_θ,$ II

Kazuhide Matsuda

Abstract

In this series of papers, we introduce higher level versions of the theta group $Γ_θ.$ In this paper, we treat the theta group of level $5$, $Γ_{θ,5},$ and construct modular forms on $Γ_{θ,5}$. Moreover we compute their multiplier systems. For this purpose, we derive transformation formula of theta function with characteristics.

Analogue of the theta group $Γ_θ,$ II

Abstract

In this series of papers, we introduce higher level versions of the theta group In this paper, we treat the theta group of level , and construct modular forms on . Moreover we compute their multiplier systems. For this purpose, we derive transformation formula of theta function with characteristics.
Paper Structure (28 sections, 22 theorems, 110 equations)

This paper contains 28 sections, 22 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. The multiplier system of $\eta(\tau)$
  3. Farkas and Kra's theory of theta functions
  4. Definitions
  5. Basic properties
  6. Jacobi's triple product identity
  7. Transformation formula
  8. Modular forms on $\Gamma_{\theta,5}$ (1)
  9. The case where $M \equiv 1001 \bmod{5}.$
  10. The case where $M \equiv -100-1 \bmod{5}.$
  11. The case where $M \equiv 0-110 \bmod{5}.$
  12. The case where $M \equiv 01-10 \bmod{5}.$
  13. Summary
  14. The case where $k\equiv 0 \bmod{10}$
  15. The case where $k\equiv 5 \bmod{10}$
  16. ...and 13 more sections

Key Result

Theorem 3.1

For each $M = \in \Gamma(1),$ we have

Theorems & Definitions (42)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • ...and 32 more