Arno Fehm

Existential ∅-Definability of Henselian Valuation Rings

Recently, Anscombe and Koenigsmann gave an existential 0-definition of the ring of formal power series F[[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability results for henselian valued fields with finite or pseudo-algebraically closed residue fields.

ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

Characterizing diophantine henselian valuation rings and valuation ideals

We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature. We also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal.

On the quantifier complexity of definable canonical Henselian valuations

We discuss definability in the language of rings without parameters of the unique canonical Henselian valuation of a field. We show that in most cases where the canonical Henselian valuation is definable, it is already definable by a universal-existential or an existential-universal formula.

Uniform definability of henselian valuation rings in the Macintyre language

We discuss definability of henselian valuation rings in the Macintyre language

The existential theory of equicharacteristic henselian valued fields

\mathcal{L}_{\rm Mac}

Elementary geometric local-global principles for fields

, the language of rings expanded by n-th power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly

The elementary theory of large fields of totally S-adic numbers

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Klein Approximation and Hilbertian fields

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Split embedding problems over the open arithmetic disc

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