429 research outputs found

### Global Solvably Closed Anabelian Geometry

In this paper, we study the pro-Σ anabelian geometry of hyperbolic curves, where Σ is a nonempty set of prime numbers, over Galois groups of “solvably closed extensions” of number fields — i.e., infinite extensions of number fields which have no nontrivial abelian extensions. The main results of this paper are, in essence, immediate corollaries of the following three ingredients: (a) classical results concerning the structure of Galois groups of number fields; (b) an anabelian result of Uchida concerning Galois groups of solvably closed extensions of number fields; (c) a previous result of the author concerning the pro-Σ anabelian geometry of hyperbolic curves over nonarchimedean local fields.</p

### Resolution of Nonsingularities, Point-theoreticity, and Metric-admissibility for p-adic Hyperbolic Curves

In this paper, we prove that arbitrary hyperbolic curves over p-adic local fields admit resolution of nonsingularities [“RNS”]. This result may be regarded as a generalization of results concerning resolution of nonsingularities obtained by A. Tamagawa and E. Lepage. Moreover, by combining our RNS result with techniques from combinatorial anabelian geometry, we prove that an absolute version of the geometrically pro-Σ Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local fields, where Σ denotes a set of prime numbers of cardinality ≥ 2 that contains p, holds. This settles one of the major open questions in anabelian geometry. Furthermore, we prove --again by applying RNS and combinatorial anabelian geometry-- that the various p-adic versions of the Grothendieck-Teichmüller group that appear in the literature in fact coincide. As a corollary, we conclude that the metrized Grothendieck-Teichmüller group is commensurably terminal in the Grothendieck-Teichmüller group. This settles a longstanding open question in combinatorial anabelian geometry

### Essential dimension and pro-finite group schemes

A. Vistoli observed that, if Grothendieck's section conjecture is true and
$X$ is a smooth hyperbolic curve over a field finitely generated over
$\mathbb{Q}$, then $\underline{\pi}_{1}(X)$ should somehow have essential
dimension $1$. We prove that an infinite, pro-finite \'etale group scheme
always has infinite essential dimension. We introduce a variant of essential
dimension, the fce dimension $\operatorname{fced} G$ of a pro-finite group
scheme $G$, which naturally coincides with $\operatorname{ed} G$ if $G$ is
finite but has a better behaviour in the pro-finite case. Grothendieck's
section conjecture implies $\operatorname{fced}\underline{\pi}_{1}(X)=\dim X=1$
for $X$ as above. We prove that, if $A$ is an abelian variety over a field
finitely generated over $\mathbb{Q}$, then
$\operatorname{fced}\underline{\pi}_{1}(A)=\operatorname{fced} TA=\dim A$.Comment: Simplified proofs and stronger results in the new versio

### On the "Section Conjecture" in anabelian geometry

Let 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.Comment: 21 pages, late

### Anabelian geometry and descent obstructions on moduli spaces

We study the section conjecture of anabelian geometry and the sufficiency of
the finite descent obstruction to the Hasse principle for the moduli spaces of
principally polarized abelian varieties and of curves over number fields. For
the former we show that the section conjecture fails and the finite descent
obstruction holds for a general class of adelic points, assuming several
well-known conjectures. This is done by relating the problem to a local-global
principle for Galois representations. For the latter, we prove some partial
results that indicate that the finite descent obstruction suffices. We also
show how this sufficiency implies the same for all hyperbolic curves.Comment: exposition improve

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