arXiv subjects
Publications and source records attributed to Florian Pop.
In this note we give generalizations and prove 'minimalistic' refinements of the t-birational Section Conjecture (t-BSC), cf. [Be], by doing both: First, by extending the class of base fields over which the t-BSC holds, and second, by proving refinements of the t-BSC which involve much less, that is minimalistic, Galois theoretical information.
In this work, we introduce a variant of the Grothendieck-Teichm{\"u}ller group, defined in terms of complements of hyperplane arrangements and pro-$\ell$ two-step nilpotent fundamental groups, and prove that it is isomorphic to the absolute Galois group of $\Qbb$.
We resolve the strong Elementary Equivalence versus Isomorphism Problem for finitely generated fields. That is, we show that for every field in this class there is a first-order sentence which characterizes this field within the class up to isomorphism. Our solution is conditional on resolution of singularities in characteristic two and unconditional in all other characteristics.
We consider local-global principles for rational points on varieties, in particular torsors, over one-variable function fields over complete discretely valued fields. There are several notions of such principles, arising either from the valuation theory of the function field, or from the geometry of a regular model of the function field. Our results compare the corresponding obstructions, proving in particular that a local-global principle with respect to valuations implies a local-global principle with respect to a sufficiently fine regular model.
The aim of this short note is to extend results by Denef and Loughran, Skorobogatov, Smeets concerning refinements of a conjecture of Colliot-Thelene. The problem is about giving necessary and sufficient conditions for morphisms of varieties to be surjective on local points for almost all localizations.
We show that the isomorphy type of every finitely generated field $K$ with $\chr(K)\neq2$ is encoded by a \textit{\textbf{single\ha3explicit\ha3axiom}} $\istp K\!$ \textit{\textbf{in\ha3the\ha3language\ha3of\ha3fields}}, i.e., for all finitely generated fields $L$ one has: $\istp K$ holds in $L$ if and only if $K\cong L$ as fields. This extends earlier results by \nmnm{\footnotesize\sc Julia Robinson, Rumely, Poonen, Scanlon}, the author, and others.
Let $Π^c_K\toΠ_K$ be the maximal pro-$\ell$ abelian-by-central, respectively abelian, Galois groups of a function field $K|k$ with $k$ algebraically closed and ${\rm char}\neq\ell$. We show that $K|k$ can be functorially reconstructed by group theoretical recipes from $Π^c_K$ endowed with the set of divisorial inertia ${\rm Inrdiv}(K)\subsetΠ_K$. As applications, one has: (i) A group theoretical recipe to reconstruct $K|k$ from $Π^c_K$, provided either ${\rm Tr.deg}(K|k)>{\rm dim}(k)+1$ or ${\rm tr.deg}(K|k) >{\rm dim}(k)>1$, where ${\rm dim}(k)$ is the Kronecker dimension; (ii) An application to the pro-$\ell\!$ abelian-by-central I/OM (Ihara's question / Oda-Matsumoto conjecture), which in the cases considered here implies the classical I/OM.
We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over ${\mathbb Q}$. More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over ${\mathbb Q}_p(x)$.
In this note we show that a special case of a recent result by Obus-Wewers (used as a black box) together with a deformation argument in characteristic $p$ leads to a proof of the Oort Conjecture in the general case. A boundedness result is given as well.
We establish a valuative version of Grothendieck's section conjecture for curves over p-adic local fields. The image of every section is contained in the decomposition subgroup of a valuation which prolongs the p-adic valuation to the function field of the curve.
We characterize those valued fields for which the image of the valuation ring under every polynomial in several variables contains an element of maximal value, or zero.
We prove a pro-$p$ Hom-form of the birational anabelian conjecture for function fields over sub-$p$-adic fields. Our starting point is the Theorem of Mochizuki in the case of transcendence degree 1.
A field k is called anti-Mordellic if every smooth curve over k with a k-point has infinitely many k-points. We prove that for a function field over an anti-Mordellic field, the subfield of constants is defined by a certain universal first order formula. Under additional hypotheses regarding 2-cohomological dimension we prove that algebraic dependence of an n-tuple of elements in such a function field can be described by a first order formula, for each n. We also give a result that lets one distinguish various classes of fields using first order sentences.
We consider function fields of transcendence degree at least 2 over algebraic closures of finite fields, and describe a functorial way to recover such function fields form their pro-l Galois theory.
Field Arithmetic studies the interplay between arithmetical properties of fields and their absolute Galois groups. Here we studies fields satisfying local global principles for rational points of varieties and profinite groups satisfying local global principles for solvability of embedding problems.
How does the first order language of fields encode birational invariants of varieties?... This question is related to rational points on varieties and effectiveness in algebraic/arithmetic geometry.