An undecidability result for the asymptotic theory of $p$-adic fields

Abstract: Fix a prime

p

. We prove that the set of sentences true in all but finitely many finite extensions of

m a t h b b Q_{p}

is undecidable in the language of valued fields with a cross-section. The proof goes via reduction to characteristic

p

, adapting Pheidas' proof of the undecidability of

m a t h b b F_{p} (! (t)!)

with a predicate for powers of

t

. This answers a variant of a question of Derakhshan-Macintyre.

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