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MAA

Seminar

Model theory of elliptic functions

Macintyre, AJ (Queen Mary, London)
Tuesday 08 February 2005, 16:00-17.00

Seminar Room 2, Newton Institute Gatehouse

Abstract

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.

Bibliography

1.Bianconi, Ricardo. Some Results in the Model Theory of Analytic Functions, Thesis, Oxford ,1990. 2.Bertolin, Cristiana. Periodes de 1-motifs et transcendance (French) [Periods of 1-motives and transcendence], J. Number Theory 97 (2002), no. 2, 204--221.

3. Gabrielov, Andrei. Complements of subanalytic sets and existential formulas for analytic functions. Invent. Math. 125 (1996), no. 1, 1--12. 4. Gabrielov, Andrei; Vorobjov, Nicolai. Complexity of computations with Pfaffian and Noetherian functions. Normal forms, bifurcations and finiteness problems in differential equations, 211--250, NATO Sci. Ser. II Math. Phys. Chem., 137, Kluwer Acad. Pul., Dordrecht, 2004.

5. Macintyre, Angus; Wilkie, A. J. On the decidability of the real exponential field. Kreiseliana, 441--467, A K Peters, Wellesley, MA, 1996. 6. Peterzil, Ya'acov; Starchenko, Sergei. Uniform definability of the Weierstrass p Functions and Generalized Tori of Dimension One, to appear in Selecta Math.

7. Wilkie, A. J. On the theory of the real exponential field. Illinois J. Math. 33 (1989), no. 3, 384--408. 8. Wilkie, A. J. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), no. 4, 1051--1094.

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