A basis for the non-archimedean holomorphic theta functions (Q1326944)

Let \(k\) be a complete, non-archimedean, algebraically closed field and \(T = (k^*)^ g/ \Lambda\) an analytic torus. Given a cocycle \(\xi \in Z^ 1 (\Lambda,A)\) \((A\) being the group of nowhere vanishing holomorphic functions on \(G)\) there exists a natural isogeny \(\lambda_{\overline \xi} : T \to \widehat T\) (= dual variety of \(T)\). The author defines an analog of the theta group for abelian varieties for each \(\xi\) and proves that this theta group is a central extension of Ker\((\lambda_{\overline \xi})\) by the multiplicative group. Following the ideas of \textit{D. Mumford} [Invent. Math. 1, 287-354 (1966; Zbl 0219.14024)], the author constructs a basis for the vector space of theta functions in forms of the theta group.