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The \(p\)-adic finite Fourier transform and theta functions. (English) Zbl 0929.14027
Let \(H\) be a finite group. \(\widehat H=\operatorname{Hom}(H,k^*)\) 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.
14K25 Theta functions and abelian varieties
14G20 Local ground fields in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
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