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

TL;DR

This paper studies explicit modular forms of weights 1/2 and 3/2 derived from Weil representations related to SL(2,R) and GL(2,R), using tensor induction and cocycle techniques.

Contribution

It introduces a reorganization of classical theta series through tensor induction and extends the analysis from SL(2,R) to GL(2,R).

Findings

Explicit modular forms of weights 1/2 and 3/2 are constructed.

Reorganization of forms using tensor induction provides new insights.

Extension of the study to the similitude group GL(2,R).

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 $\operatorname{GL}_2(\mathbb{R})$.