173 research outputs found
Recommended from our members
Hadronic decays of the eta/sub c/
No full textResults on hadronic decays of the eta/sub c/ candidate state are presented. A mass value of M = (2978 +- 9) MeV is obtained. The branching fraction for the decay into eta ..pi../sup +/..pi../sup -/ is presented and an upper limit for the decay into ..pi../sup 0/K/sup +/K/sup -/ is given. 6 figures
Defining Transcendentals in Function Fields.
No full textGiven any field K, there is a function field F/K in one variable containing definable transcendentals over K, i,e., elements in F/K first-order definable in the language of fields with parameters from K. Hence, the model-theoretic and the field-theoretic relative algebraic closure of K in F do not coincide. E.g., if K is finite, the model-theoretic algebraic closure of K in the rational function field K(t) is K(t). For the proof, diophantine ∅-definability of K in F is established for any function field F/K in one variable, provided K is large, or K x /(K x)n is finite for some integer n > 1 coprime to char K
On the "Section Conjecture" in anabelian geometry
No full textLet X be a smooth projective curve of genus >1 over a field K which is finitely generated over the rationals. The section conjecture in Grothendieck's anabelian geometry says that the sections of the canonical projection from the arithmetic fundamental group of X onto the absolute Galois group of K are (up to conjugation) in one-to-one correspondence with K-rational points of X. The birational variant conjectures a similar correspondence where the fundamental group is replaced by the absolute Galois group of the function field K(X). The present paper proves the birational section conjecture for all X when K is replaced e.g. by the field of p-adic numbers. It disproves both conjectures for the field of real or p-adic algebraic numbers. And it gives a purely group theoretic characterization of the sections induced by K-rational points of X in the birational setting over almost arbitrary fields. As a biproduct we obtain Galois theoretic criteria for radical solvability of polynomial equations in more than one variable, and for a field to be PAC, to be large, or to be Hilbertian
Contributions to the model theory of henselian fields
No full textThe thesis addresses certain problems in the model theory of henselian fields, with a special focus on decidability. Some new methods are introduced along the way, which are of independent interest. The following results are obtained:
• A transfer theorem for perfectoid fields, saying that a perfectoid field K is decidable relative to its tilt Kb, modulo a subtle (albeit natural) condition on K. As an application, we prove that the fields Qp(p1/p∞) and Qp(ζp∞) admit decidable maximal immediate extensions, thereby obtaining some of the first few decidabilty results for tame fields of mixed characteristic.
• A model-theoretic proof of the Fontaine-Wintenberger theorem, which states that the absolute Galois groups of K and Kb are canonically isomorphic. (joint with F. Jahnke).
• A general existential Ax-Kochen/Ershov principle for tamely ramified fields in all characteristics, conditional to a certain form of resolution of singularities. This extends well-known existential Ax-Kochen/Ershov principles in residue characteristic 0 and also unramified mixed characteristic. It also encompasses the (conditional) existential decidability result of Denef-Schoutens for Fp((t)), which we also strengthen by replacing the assumption of (global) resolution with local uniformization.
• An undecidability result for the asymptotic theory of {K : [K : Qp] < ∞} in the language of valued fields with a cross-section. The proof goes via reduction to positive characteristic, à la Krasner-Kazhdan-Deligne, ultimately adapting Pheidas’ proof of the undecidability of Fp((t)) with a cross-section. This answers a variant of a question of Derakhshan-Macintyre.
• An undecidability result for power series fields k((t)), equipped with a total residue map res : k((t)) ! k, which picks out the constant term of the Laurent series. Becker-Denef-Lipschitz showed that res : k((t)) ! k is definable in the language of rings with a parameter for t, when the base field k is finite. We show that (k((t)),res) is undecidable, whenever k is infinite
Projective extensions of fields
No full textEvery field K admits proper projective extensions, that is, Galois extensions where the Galois group is a non-trivial projective group, unless K is separably closed or K is a Pythagorean formally real field without cyclic extensions of odd degree. As a consequence, it turns out that almost all absolute Galois groups decompose as proper semidirect products. We show that each local field has a unique maximal projective extension, and that the same holds for each global field of positive characteristic. In characteristic 0, we prove that Leopoldt's conjecture for all totally real number fields is equivalent to the statement that, for all totally real number fields, all projective extensions are cyclotomic. So the realizability of any non-procyclic projective group as Galois group over ℚ produces counterexamples to the Leopoldt conjecture. © 2006 London Mathematical Society
The regular inverse Galois problem over non-large fields
No full textBy a celebrated theorem of Harbater and Pop, the regular inverse Galois problem is solvable over any field containing a large field. Using this and the Mordell conjecture for function fields, we construct the first example of a field K over which the regular inverse Galois problem can be shown to be solvable, but such that K does not contain a large field. The paper is complemented by model-theoretic observations on the diophantine nature of the regular inverse Galois problem
First-order definability of affine Campana points in the projective line over a number field
No full textWe offer a -definition for (affine) Campana points over
(where is a number field), which constitute a
set-theoretical filtration between and (-integers),
which are well-known to be universally defined (Koenigsmann 2010, Park 2012,
Eisentraeger & Morrison 2016). We also show that our formulas are uniform with
respect to all possible , are parameter-free as such, and we count the
number of involved quantifiers and offer a bound for the degree of the defining
polynomial
Recovering p-adic valuations from pro-p Galois groups
No full textLet be a field with , where
denotes the maximal pro-2 quotient of the absolute Galois group of a field .
We prove that then admits a (non-trivial) valuation which is
2-henselian and has residue field . Furthermore, is a
minimal positive element in the value group and
. This forms the first positive result on a more
general conjecture about recovering -adic valuations from pro- Galois
groups which we formulate precisely. As an application, we show how this result
can be used to easily obtain number-theoretic information, by giving an
independent proof of a strong version of the birational section conjecture for
smooth, complete curves over , as well as an analogue for
varieties.Comment: Final version, published in the Journal of the London Mathematical
Society (DOI: 10.1112/jlms.12901
Irreducibility of polynomials over global fields is diophantine
No full textGiven a global field and a positive integer , we present a diophantine criterion for a polynomial in one variable of degree over not to have a root in . This strengthens a result by Colliot-Thélène and Van Geel [Compositio Math. 151 (2015), 1965–1980] stating that the set of non-th powers in a number field is diophantine. We also deduce a diophantine criterion for a polynomial over of given degree in a given number of variables to be irreducible. Our approach is based on a generalisation of the quaternion method used by Poonen and Koenigsmann for first-order definitions of in .</jats:p
The p-adic section conjecture for localisations of curves
No full textThe -adic section conjecture predicts that for a smooth, proper, hyperbolic curve over a -adic field , every section of the map of étale fundamental groups is induced by a unique -rational point of . While this conjecture is still open, the birational variant in which is replaced by its generic point is known due to Koenigsmann. Generalising an alternative proof of Pop, we extend this result to certain localisations of at a set of closed points , an intermediate version in between the full section conjecture and its birational variant. As one application, we prove the section conjecture for whenever is a countable set of closed points
- …
