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An Analogue of the Dedekind Eta Function for Hecke Groups $H(\sqrt{D})$

Debmalya Basak, Dorian Goldfeld, Winston Heap, Nicolas Robles, Alexandru Zaharescu

Abstract

Let $D\equiv 1\bmod{4}$ be a fundamental discriminant of a real quadratic field. We construct an analogue of the classical Dedekind eta function for the Hecke group $H(\sqrt{D})$. This gives rise to a new family of holomorphic modular functions for $H(\sqrt{D})$ which vanish at the cusp at $\infty$. We establish results on the asymptotic growth and sign patterns of the Fourier coefficients associated to these modular forms.

An Analogue of the Dedekind Eta Function for Hecke Groups $H(\sqrt{D})$

Abstract

Let be a fundamental discriminant of a real quadratic field. We construct an analogue of the classical Dedekind eta function for the Hecke group . This gives rise to a new family of holomorphic modular functions for which vanish at the cusp at . We establish results on the asymptotic growth and sign patterns of the Fourier coefficients associated to these modular forms.

Paper Structure

This paper contains 12 sections, 7 theorems, 103 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Modular Relations between Products Twisted by Characters
  3. The function $F_{{{_D}}}(z)$.
  4. The dual function $F_{_D}^\#(z)$
  5. Proof of Theorem \ref{['Thm : Modular Relation']}
  6. Modular Forms on $H(\sqrt{D})$: Proofs of Theorems \ref{['Theorem: Delta5isModular']} and \ref{['Theorem: D>5']}
  7. Preliminaries
  8. Proof of Theorem \ref{['Theorem: Delta5isModular']}
  9. Proof of Theorem \ref{['Theorem: D>5']}: The Case $D>5$
  10. Fourier Coefficients and Partitions : Proof of Theorem \ref{['Thm: Bounds on Fourier Coefficients']}
  11. Graphical and Numerical Data
  12. Acknowledgements

Key Result

Theorem 1.2

Let $z\in\mathfrak h$. The function $\Delta_5(z)$ is a holomorphic modular form for the Hecke group $H(\sqrt{5})$, that is, for all $\left(\right)\in H(\sqrt{5})$ and all $z\in\mathfrak h.$ Furthermore $\Delta_5(z)$ vanishes at the cusp $\infty$, that is, $\lim\limits_{y\to\infty} \Delta_5(iy) = 0.$

Figures (5)

  • Figure 1: Fundamental domain of the Hecke group $H(\lambda)$ for $\lambda > 2$.
  • Figure 2: Modularity of $\eta_5(z)$ under the transformation $z \mapsto -1/z$, with the product over $n$ truncated at $n=300$. For $z = x + iy$ with $x \in (-6,6)$ and $y \in (0.1,1.1)$, the contour plots of $\operatorname{Re}(\eta_5(z))$ and $\operatorname{Re}(\eta_5(-1/z))$ coincide, as do the contour plots of $\operatorname{Im}(\eta_5(z))$ and $\operatorname{Im}(\eta_5(-1/z))$.
  • Figure 3: Top: Plot of $a_5(N)$, $1 \leq N \leq 100$. Bottom: Plot of $a_{13}(N), 1 \leq N \leq 100$.
  • Figure 4: Growth of $\log \, \lvert a_5(N) \rvert$ (in red) compared to $\sqrt{N}$ (in blue), for $N =1$ to $800$.
  • Figure 5: Comparison between the growths of $\log \, \lvert a_{_D}(N) \rvert$ for $D=5$ (in red), $D=13$ (in green) and $D=17$ (in orange), for $N =1$ to $100$.

Theorems & Definitions (21)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 11 more