Elliptic curves and Hilbert's tenth problem for algebraic function fields over real and p-adic fields

Details Elliptic curves and Hilbert's tenth problem for algebraic function fields over real and p-adic fields Elliptic curves and Hilbert's tenth problem for algebraic function fields over real and p-adic fields

2004-09-07

arxiv.org/abs/math/0409103v3

J. reine angew. Math. 587 (2005), 77-143

Mathematics

Logic

73 pages, LateX2e with hyperref, 1 Postscript figure. v2: Forgotten assumption added in statement of Theorem 1.1 (the result a

Scientific paper

Let k be a field of characteristic zero, V a smooth, positive-dimensional, quasiprojective variety over k, and D a nonempty effective divisor on V. Let K be the function field of V, and A the semilocal ring of D in K. In this paper, we prove the Diophantine undecidability of: (1) A, in all cases; (2) K, when k is (formally) real and V has a real point; (3) K, when k is a subfield of a p-adic field, for some odd prime p. To achieve this, we use Denef's method: from an elliptic curve E over Q, without complex multiplication, one constructs a quadratic twist E' of E over Q(t), which has Mordell-Weil rank one. Most of the paper is devoted to proving (using a theorem of R. Noot) that one can choose f in K, vanishing at D, such that the group E'(K) deduced from the field extension K/Q(f)=Q(t) is equal to E'(Q(t)). Then we mimic the arguments of Denef (for the real case) and of Kim and Roush (for the p-adic case).

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