Investigation of Siegel Modular Forms from Weil Representations in SL₂(ℝ) and GL₂(ℝ) Cases

arXiv Math · · 2 min read · Natural Sciences

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Key Takeaways

  • Investigation of explicit modular forms of weights $1/2$ and $3/2$, including classical, minus, and fermionic theta series.
  • These forms arise from the classical Weil representation associated to $\operatorname{SL}_2(\mathbb{R})$ via specific $2$-cocycles.
  • The forms were reorganized using (tensor) induction.
  • The study was extended to the similitude group $\operatorname{GL}_2(\mathbb{R})$.

Overview

This research focuses on the explicit investigation of Siegel modular forms. The primary objects of study are modular forms of weights $1/2$ and $3/2$, encompassing classical, minus, and fermionic theta series. These modular forms are derived from the classical Weil representation, specifically as it is associated with the group $\operatorname{SL}_2(\mathbb{R})$. The investigation incorporates $2$-cocycles documented by researchers including Rao, Kudla, Perrin, Lion–Vergne, and Satake–Takase.

The methodology involved reorganizing these forms using a process referred to as (tensor) induction. Subsequently, the scope of the study was extended to include the similitude group $\operatorname{GL}_2(\mathbb{R})$, applying the same framework of investigation.

Research Context

The foundational concept for this study is the Weil representation. The classical Weil representation, in particular, serves as the origin for the modular forms under examination. This representation is specifically associated with the group $\operatorname{SL}_2(\mathbb{R})$. The context further involves the application of $2$-cocycles, which are mathematical constructs contributing to the definition and properties of these representations. The named contributors to the understanding of these $2$-cocycles are Rao, Kudla, Perrin, Lion–Vergne, and Satake–Takase.

The study differentiates between various types of theta series: classical, minus, and fermionic. These distinctions are relevant to the classification of the investigated modular forms. The weights of the modular forms, specifically $1/2$ and $3/2$, indicate their specific mathematical properties within the theory of modular forms.

Approach

The research approach commenced with an investigation into modular forms originating from the classical Weil representation associated with $\operatorname{SL}_2(\mathbb{R})$. This initial phase explicitly considered the role of specified $2$-cocycles.

A key procedural step involved the reorganization of the identified modular forms. This reorganization was achieved through the application of (tensor) induction. Following this, the methodology was extended. The extended study then applied the same investigative framework to the similitude group $\operatorname{GL}_2(\mathbb{R})$, building upon the initial findings from the $\operatorname{SL}_2(\mathbb{R})$ case.

Findings

  • Explicit modular forms of weights $1/2$ and $3/2$ were investigated.
  • These forms included classical, minus, and fermionic theta series.
  • The investigated forms were found to arise from the classical Weil representation associated with $\operatorname{SL}_2(\mathbb{R})$.
  • The derivation of these forms incorporated $2$-cocycles attributed to Rao, Kudla, Perrin, Lion–Vergne, and Satake–Takase.
  • The study reorganized these forms using (tensor) induction.
  • The investigation was extended to the similitude group $\operatorname{GL}_2(\mathbb{R})$.

Research Information

Institution
arXiv
Original Study
View Publication
Source
arXiv Math

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