Hilbert's Tenth Problem for function fields of varieties over number fields and p-adic fields
arXiv:math/0610132
Abstract
Let k be a subfield of a p-adic field of odd residue characteristic, and let L be the function field of a variety of dimension n >= 1 over k. Then Hilbert's Tenth Problem for L is undecidable. In particular, Hilbert's Tenth Problem for function fields of varieties over number fields of dimension >= 1 is undecidable.
19 pages; to appear in Journal of Algebra