Jochen Koenigsmann

On the ‘Section Conjecture’ in anabelian geometry

Abstract Let X  be a smooth projective curve of genus > 1 over a field K  with function field K (X ), let π 1(X ) be the arithmetic fundamental group of X  over K  and let GF   denote the absolute Galois group of a field F . The section conjecture  in Grothendieck’s anabelian geometry says that the sections of the canonical projection π 1(X ) ↠ GK   are (up to conjugation) in one-to-one correspondence with the K-rational points of X, if K  is finitely generated over ℚ. The birational variant conjectures a similar correspondence w.r.t. the sections of the projection GK (X ) ↠ GK . So far these conjectures were a complete mystery except for the obvious results over separably closed fields and some non-trivial results due to Sullivan and Huisman over the reals. The present paper proves—via model theory—the birational section conjecture for all local fields of characteristic 0 (except ℂ), disproves both conjectures e.g. for the fields of all real or p -adic algebraic  numbers, and 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 byproduct 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.

Undecidability in Number Theory

In these lecture notes we give sketches of classical undecidability results in number theory, like Godel’s first Incompleteness Theorem (that the first order theory of the integers in the language of rings is undecidable), Julia Robinson’s extensions of this result to arbitrary number fields and rings of integers in them, as well as to the ring of totally real integers, and Matiyasevich’s negative solution of Hilbert’s 10th problem, i.e., the undecidability of the existential first-order theory of the integers. As Hilbert’s 10th problem is still open for the rationals (i.e., the question whether the existential theory of the field of rational numbers is decidable) we also present a sketch of the fact that there is a universal definition of the ring of integers inside the field of rationals. In terms of complexity this is the simplest definition known so far. If one had an existential definition instead then Hilbert’s 10th problem over the rationals would reduce to that over the integers (and hence be, as expected, unsolvable), but, modulo a widely believed in conjecture in number theory, we also indicate why there should be no such existential definition. We conclude with a list of nice open questions in the area.

Definable Henselian valuations

In this note we investigate the question when a henselian valued field carries a nontrivial ∅-definable henselian valuation (in the language of rings). This is clearly not possible when the field is either separably or real closed, and, by the work of Prestel and Ziegler, there are further examples of henselian valued fields which do not admit a ∅-definable nontrivial henselian valuation. We give conditions on the residue field which ensure the existence of a parameter-free definition. In particular, we show that a henselian valued field admits a nontrivial henselian ∅-definable valuation when the residue field is separably closed or sufficiently nonhenselian, or when the absolute Galois group of the (residue) field is nonuniversal.

Uniformly defining p-henselian valuations☆

Admitting a non-trivial

Products of absolute Galois groups

p

Defining Coarsenings of Valuations

-henselian valuation is a weaker assumption on a field than admitting a non-trivial henselian valuation. Unlike henselianity,

The regular inverse Galois problem over non-large fields

p

PROJECTIVE EXTENSIONS OF FIELDS

-henselianity is an elementary property in the language of rings. We are interested in the question when a field admits a non-trivial 0-definable

From p-rigid elements to valuations (with a Galois-characterization of p-adic fields).

p

Solvable absolute Galois groups are metabelian

-henselian valuation (in the language of rings). We give a classification of elementary classes of fields in which the canonical