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

We investigate explicit modular forms of weights $1\mathrm{/}2$ and $3\mathrm{/}2$-classical, minus, and fermionic theta series-arising from the classical Weil representation associated to ${\mathrm{SL}}_2(\mathbb{R})$ 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 ${\mathrm{GL}}_2(\mathbb{R})$.