Hilbertʹs Tenth Problem for function fields of varieties over number fields and p-adic fields

Let k be a subfield of a p-adic field of odd residue characteristic, and let be the function field of a variety of dimension n 1 over k. Then Hilbertʹs Tenth Problem for is undecidable. In particular, Hilbertʹs Tenth Problem for function fields of varieties over number fields of dimension 1 is undecidable.