Hilbert's 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 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. 
Kirsten Eisentraeger