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Uniform definability of the Weierstrass \(\wp\) functions and generalized tori of dimension one. (English) Zbl 1071.03022
The authors’ main theme is the generalization, to any algebraically closed field, of the classical result about the equivalence of complex tori, \(\mathbb{C}/\Lambda\), and non-singular cubic curves via the Weierstrass \(\wp\) functions. When the field is non-Archimedean, the quotient space construction does not work, and they invent the construction ‘gluing along a homeomorphism’. They carefully analyze the equivalence of tori, and of tori and curbs. A corollary to this analysis is: Let \(R\) be a non-Archimedean real closed field and \(K= R(\sqrt{-1})\), then there are some \(K\)-tori that are not \(K\)-biholomorphic to any \(K\)-curbs, and not algebraic even in any \({\mathcal R}\), o-minimal expansion of \(R\). A second theme of the authors is definability. If \({\mathcal R}\) is elementarily equivalent to \(\mathbb{R}_{\text{an.exp}}\), then any smooth cubic \(K\)-curb is definably \(K\)-biholomorphic to a \(K\)-torus. A third theme is the exponential function. For instance, the authors show: If the function \(\wp(\tau,z)\), \(-{1\over 2}\leq \text{Re}(\tau)<{1\over 2}\) and \(|\tau|\geq 1\), is definable in an o-minimal expansion of \(\mathbb{R}_{\text{an}}\), then so is the real exponential function.

03C64 Model theory of ordered structures; o-minimality
03C98 Applications of model theory
30G06 Non-Archimedean function theory
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