Cartier duality of spherical group rings at higher height

background-Cartier Duality

in Pontrajagin-Cartier duality for Spherical group rings, we saw that for a compact abelian Lie group $G$, we have

$\{\text{Functions on the stack }B{G}_R\}\simeq Hom(\hat{G},{\mathbb{G}}_{m,R})$ Replacing the height-1 object ${\mathbb{G}}_{m,R}$ with a height-2 object: $\mathbb{E}$, a spectral elliptic curve, we can define

$El{l}_{\mathbb{E}}(BG):=Hom(\hat{G},\mathbb{E})$ Where LHS can be read as “$\mathbb{E}$-elliptic functions on $BG$“.

Example Taking $G=\mathbb{T}$ recovers underlying elliptic curve

$El{l}_E(B\mathbb{T})=Hom(\mathbb{Z},E)=E$ This corresponds to how in the height 1 case, taking $G=\mathbb{T}$ recovered ${\mathbb{G}}_m$.

Example Taking $G=C_p$ recovers $p$-torsion points of $E$.

Functoriality: this is functorial over $BG\in Or{b}^{ab}$ (abelian part of the orbit category), so Kan extends to a functor

$Orb\stackrel{El{l}_E:=Hom(\widehat{-},E)}{\to }Fun(CAlg(Sp),S)$ (in fact we see it lands in very nice spectral schemes)