Algebraic structures for $\bigoplus \sum _{n\geq 1}L^{2}(Z/n)$ compatible with the finite Fourier transform

Abstract:

Let ${Z / n}$ denote the integers $\bmod n$ and let ${\mathcal {F}_n}$ denote the finite Fourier transform on ${L^2}({Z / n})$. We let $\oplus \Sigma {{\mathcal {F}_n}} = F$ operate on $\oplus \Sigma {L^2}({Z / n})$ and show that $\oplus \Sigma {L^2}({Z / n})$ can be given a graded algebra structure (with no zero divisors) such that $\mathcal {F}(fg) = \mathcal {F}(f)\mathcal {F}(g)$. We do this by establishing a natural isomorphism with the algebra of theta functions with period i. In addition, we find all algebra structures on $\oplus \Sigma {L^2}({Z / n})$ satisfying the above condition.