An analogue of Hilbert’s tenth problem for fields of meromorphic functions over non-Archimedean valued fields

Let K be a complete and algebraically closed valued field of characteristic 0. We prove that the set of rational integers is positive existentially definable in the field M of meromorphic functions on K in the language Lz of rings augmented by a constant symbol for the independent variable z and by a symbol for the unary relation “the function x takes the value 0 at 0”. Consequently, we prove that the positive existential theory of M in the language Lz is undecidable. In order to obtain these results, we obtain a complete characterization of all analytic projective maps (over K) from an elliptic curve E minus a point to E , for any elliptic curve defined over the field of constants. This research was supported in part by the greek foundation of state scholarship (IKY) and was done at the university of Crete whose hospitality I acknowledge.