The \(p\)-adic finite Fourier transform and theta functions (Q1281195)

Let \(H\) be a finite group. \(\widehat H=\Hom(H,k^*)\) its group of characters over a field \(k\) and \(V(H)\) the vector space of \(k\)-valued functions on \(H\). The finite Fourier transform is a linear isomorphism \(F_A:V(H)\to V(\widehat H)\) which is explicitly computed. The author also states the explicit actions of the Heisenberg group \({\mathcal G}(H\times\widehat H)\) on \(V(H)\) and \(V(\widehat H)\). The last section is devoted to applications to the study of theta functions on an analytic torus.