Laurent Moret-Bailly
Elliptic curves and Hilbert’s tenth problem for algebraic function fields over real and p-adic fields
Let k
be a field of characteristic zero, V? a smooth,
positive-dimensional, quasiprojective variety over k, and Q? a
nonempty divisor on V. Let K? be the function field of
V, and A ? K? the semilocal ring of Q.
We prove the Diophantine undecidability of: (1) A, in all cases; (2) K, when k? is 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 ?, without complex multiplication, one constructs a quadratic twist ? of
E over ?(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 ? in K, vanishing at Q, such that the
group ?(K ) deduced from the field extension ?(t ) ?? ?(? ) ? K? is equal to
?(?(t )). Then we mimic the arguments of Denef (for the real case) and
of Kim and Roush (for the p -adic case).
Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter
Print ISSN: 0075-4102
Volume: 2005, 10/2005
Seiten: 77 - 143
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