Model theory of elliptic functions
Seminar Room 1, Newton Institute
The lectures will consider the structure of definitions in the theory of individual Weierstrass elliptic functions, paying as much attention as possible to uniformities as the function, or its associated lattice, varies. The setting will be that of an o-minimal expansion of the real field, and one will interpret therein the elliptic function on a semi-algebraic fundamental parallelogram. This will give o-minimality results, and with more work model- completeness results (related to work of Bianconi). The main novelty will be decidability results for some special elliptic functions, modulo a conjecture of Andre in transcendence theory. The proof will be analogous to that of Wilkie and the author for decidability of the real exponential. Peterzil and Starchenko began the model theory of families of elliptic functions, and showed in particular that this is interpretable in an o-minimal theory, so is certainly not undecidable in any Godelian way. I will outline what one currently knows about the problems of model-completeness and decidability for families of elliptic functions.
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