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Analogue of the theta group $Γ_θ$

Kazuhide Matsuda

Abstract

In this paper, we introduce higher level versions of the theta group $Γ_θ.$ In particular, we treat level 3 and 4 versions of the theta group, $Γ_{θ,3}$ and $Γ_{θ,4}$ and prove that $\displaystyle F(τ)=η\left(\frac{τ-1}{3} \right) η\left(\frac{τ+1}{3} \right)$ and $\displaystyle G(τ)=η\left(\frac{τ-1}{4} \right) η\left(\frac{τ+1}{4} \right)$ are modular forms on $Γ_{θ,3}$ and $Γ_{θ,4}$ respectively. Moreover we compute their multiplier systems, $ν_{F}$ and $ν_{G}$.

Analogue of the theta group $Γ_θ$

Abstract

In this paper, we introduce higher level versions of the theta group In particular, we treat level 3 and 4 versions of the theta group, and and prove that and are modular forms on and respectively. Moreover we compute their multiplier systems, and .
Paper Structure (26 sections, 28 theorems, 146 equations)

This paper contains 26 sections, 28 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. The multiplier system of $\eta(\tau)$
  3. Modular form on $\Gamma_{\theta,3}$
  4. The case where $M = abcd \equiv \pm 1001 \bmod{3}$ and $c$ is odd
  5. The case where $M = abcd \equiv \pm 0-110 \bmod{3}$ and $c$ is odd
  6. The case where $M = abcd \equiv \pm 1001 \bmod{3}$ and $c$ is even
  7. The case where $M = abcd \equiv \pm 0-110 \bmod{3}$ and $c$ is even
  8. Summary
  9. The case where $k\equiv 0 \bmod{12}$
  10. The case where $k\equiv 6 \bmod{12}$
  11. The case where $k\equiv \pm 3 \bmod{12}$
  12. The case where $k\equiv \pm 4 \bmod{12}$
  13. The case where $k\equiv \pm 2 \bmod{12}$
  14. The case where $k\equiv \pm 1, \pm 5 \bmod{12}$
  15. Modular form on $\Gamma_{\theta,4}$
  16. ...and 11 more sections

Key Result

Proposition 3.1

For every $\tau\in\mathscr{H},$ set The multiplier system $\nu_{F}$ is a character.

Theorems & Definitions (58)

  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 48 more