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Siegel modular forms associated to Weil representations: $\operatorname{SL}_2(\mathbb{R}) \& \operatorname{GL}_2(\mathbb{R})$ cases

Chun-Hui Wang

TL;DR

The paper develops an explicit framework for half-integral weight modular forms arising from the real Weil representation, connecting very concrete theta-series to cocycle data and automorphic factors on SL$_2(\mathbb{R})$ and GL$_2(\mathbb{R})$. By organizing the Weil representation through (tensor) induction and carefully analyzing $2$-cocycles, the authors produce explicit multiplier systems for theta-series of weights $\tfrac{1}{2}$ and $\tfrac{3}{2}$, extend these to congruence subgroups via matrix-valued theta-functions, and extend the construction to GL$_2$ using the Weil–Deligne framework. They establish irreducibility results for induced representations on various discrete subgroups, derive detailed automorphic factors, and provide explicit realizations in Schrödinger, lattice, and Fock models, thereby linking automorphic forms of half-integral weight to the representation-theoretic language of induction and transfer. The methods offer a blueprint for pursuing higher-rank Siegel-type objects and place the concrete half-integral weight theta-forms within the broader Langlands program, offering both computational tools and structural insight for automorphic forms tied to Weil representations.

Abstract

We investigate explicit modular forms of weights $1/2$ and $3/2$-classical, minus, and fermionic theta series-arising from the classical Weil representation associated to $\operatorname{SL}_2(\mathbb{R})$ via the $2$-cocycles of Rao, Kudla, Perrin, Lion--Vergne and Satake--Takase. We reorganize these forms using (tensor) induction, and subsequently extend our study to the similitude group $\operatorname{GL}_2(\mathbb{R})$.

Siegel modular forms associated to Weil representations: $\operatorname{SL}_2(\mathbb{R}) \& \operatorname{GL}_2(\mathbb{R})$ cases

TL;DR

The paper develops an explicit framework for half-integral weight modular forms arising from the real Weil representation, connecting very concrete theta-series to cocycle data and automorphic factors on SL and GL. By organizing the Weil representation through (tensor) induction and carefully analyzing -cocycles, the authors produce explicit multiplier systems for theta-series of weights and , extend these to congruence subgroups via matrix-valued theta-functions, and extend the construction to GL using the Weil–Deligne framework. They establish irreducibility results for induced representations on various discrete subgroups, derive detailed automorphic factors, and provide explicit realizations in Schrödinger, lattice, and Fock models, thereby linking automorphic forms of half-integral weight to the representation-theoretic language of induction and transfer. The methods offer a blueprint for pursuing higher-rank Siegel-type objects and place the concrete half-integral weight theta-forms within the broader Langlands program, offering both computational tools and structural insight for automorphic forms tied to Weil representations.

Abstract

We investigate explicit modular forms of weights and -classical, minus, and fermionic theta series-arising from the classical Weil representation associated to via the -cocycles of Rao, Kudla, Perrin, Lion--Vergne and Satake--Takase. We reorganize these forms using (tensor) induction, and subsequently extend our study to the similitude group .
Paper Structure (85 sections, 145 theorems, 651 equations)

This paper contains 85 sections, 145 theorems, 651 equations.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. Weil representations
  4. Modular forms of half-integral weight
  5. The results
  6. The relevant covering groups: $\operatorname{SL}_2(\mathbb R)$ case
  7. The cocycles $\overline{c}_{X^{\ast}}$ and $\widetilde{c}_{X^{\ast}}$
  8. $K=\operatorname{SO}_2(\mathbb R)$
  9. The cocycle $\widehat{c}_z$
  10. Link $\widehat{c}_{z_0}$ to $\overline{c}_{X^{\ast}}$ and $\widetilde{c}_{X^{\ast}}$
  11. The discrete group $\operatorname{SL}_2(\mathbb Z)$
  12. The discrete group $\Gamma_{\theta}$
  13. Automorphic factor I
  14. The relevant covering groups: $\operatorname{GL}_2(\mathbb R)$ case
  15. $\mu_8$-covering
  16. ...and 70 more sections

Key Result

Theorem 1.1

$\theta(\tfrac{az+b}{cz+d})=\nu(r)\sqrt{cz+d} \theta(z)$, for $r=\in \Gamma_0(4), \nu( r)= \left(\frac{c}{d}\right)\epsilon^{-1}_d$.

Theorems & Definitions (320)

  • Theorem 1.1
  • Theorem 1.2: LiVe
  • Definition 1.3
  • Theorem 1.4: LiVe
  • Definition 1.5
  • Definition 1.6
  • proof
  • proof
  • proof
  • Lemma 2.1
  • ...and 310 more