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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.
MSC:
 14K25 Theta functions and abelian varieties 14G20 Local ground fields in algebraic geometry 14C20 Divisors, linear systems, invertible sheaves
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