Interdefinability of abelian functions

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Title

Interdefinability of abelian functions

Group Mathematical Logic
Supervisor
Description

Recently there has been a great deal of interaction between model theorists and number theorists on topics around `unlikely intersections', see for example [3]. One outcome of this is that there are now various functional transcendence results known for certain covering maps. The original example of this is Ax's functional version [1] of Schanuel's conjecture. This result and its more recent descendants have been used to study interdefinability of Weierstrass elliptic functions [2] and the initial aim of this project is to extend this to abelian functions. This would involve a mixture of model theory, differential algebra and number theory, although these are not all required to get started. It should also lead naturally to further interesting questions in these areas.


[1] Ax, James On Schanuel's conjectures. Ann. of Math. (2) 93 1971 252–268. See  http://www.jstor.org/stable/1970774

[2] Jones, G., Kirby, J. and Servi, T., Local interdefinability of Weierstrass elliptic functions. Journal of the Institute of Mathematics of Jussieu, To appear. See http://dx.doi.org/10.1017/S1474748014000425

[3] Zannier, Umberto, Some problems of unlikely intersections in arithmetic and geometry. Annals of Mathematics Studies, 181. Princeton University Press, Princeton, NJ, 2012

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