arXiv subjects
Publications and source records attributed to Jakob Stix.
For every $p\geq 2$ we show that each finite $p$-group with an unmixed Beauville structure is part of a surjective infinite projective system of finite $p$-groups with compatible unmixed Beauville structures. This leads to the new notion of an unmixed topological Beauville structure on pro-finite groups. We further construct for $p \geq 5$ a new explicit infinite series of non-abelian $p$-groups that allow unmixed Beauville structures.
We assign functorially a $\mathbb{Z}$-lattice with semisimple Frobenius action to each abelian variety over $\mathbb{F}_p$. This establishes an equivalence of categories that describes abelian varieties over $\mathbb{F}_p$ avoiding $\sqrt{p}$ as an eigenvalue of Frobenius in terms of simple commutative algebra. The result extends the isomorphism classification of Waterhouse and Deligne's equivalence for ordinary abelian varieties.
Unlike in characteristic 0, there are no non-trivial smooth varieties over an algebraically closed field k of characteristic p>0 that are contractible in the sense of etale homotopy theory.
For an abelian variety A over a number field k we discuss the maximal divisibile subgroup of H^1(k,A) and its intersection with the subgroup Sha(A/k). The results are most complete for elliptic curves over Q.
We construct an infinite series of simply transitive irreducible lattices in PGL_2(F_q((t))) \times PGL_2(F_q((t))) by means of a quaternion algebra over F_q(t). The lattices depend on an odd prime power q = p^r and a parameter τ in F_q^* different from 1, and are the fundamental group of a square complex with just one vertex and universal covering T_{q+1} \times T_{q+1}, a product of trees with constant valency q + 1. Our lattices give rise via non-archimedian uniformization to smooth projective surfaces of general type over F_q((t)) with ample canonical class, Chern ratio (c_1)^2/c_2 = 2, trivial Albanese variety and non-reduced Picard scheme. For q = 3, the Zariski-Euler characteristic attains its minimal value χ= 1: the surface is a non-classical fake quadric.
This note explores the consequences of Koenigsmann's model theoretic argument from the proof of the birational p-adic section conjecture for curves in the context of higher dimensional varieties over p-adic local fields.
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.
A torsor under a k-group scheme G on a variety X over a number field k imposes a descent obstruction against the existence of rational points on X. We discuss the finite descent obstruction, that is for all such torsors under finite k-groups G, in view of a local-global interpolation property for sections of the fundamental group short exact sequence of X/k. There are applications to the Brauer-Manin obstruction, to the descent obstruction by torsors under linear groups, and to the birational version of Grothendieck's section conjecture over number fields. In particular, we obtain examples of families of curves over number fields, such that the birational section conjecture is true in a non-trivial way.
We establish Grothendieck's section conjecture for an open subset of the Reichardt-Lind curve, and introduce the notion of a Brauer-Manin obstruction for sections of the fundamental group extension.
This note aims at providing evidence for the section conjecture of anabelian geometry by establishing its behaviour under Weil restriction of scalars. In particular, the etale fundamental group of the Weil restriction is determined by means of a Shapiro Lemma for non-abelian group cohomology.
Rational points in the boundary of a hyperbolic curve over a field with sufficiently nontrivial Kummer theory are the source for an abundance of sections of the fundamental group exact sequence. We follow and refine Nakamura's approach towards these boundary sections. For example, we obtain a weak anabelian theorem for hyperbolic genus 0 curves over quite general fields including for example the maximal abelian extension of the rational numbers.
Period and index of a curve $X/K$ over a $p$-adic local field $K$ such that the fundamental group $π_1(X/K)$ admits a splitting are shown to be powers of $p$. As a consequence, examples of curves over number fields are constructed where having sections is obstructed locally at a $p$-adic place. Hence the section conjecture holds for these curves as there are neither sections nor rational points.
A family of proper smooth curves of genus $\geq 2$, parametrised by an open dense subset $U$ of a normal variety $S$, extends to $S$ if the natural map $π_1(U) \to π_1(S)$ on fundamental groups is an isomorphism. The criterion of this note is actually more precise.
The geometric condition of T. Saito for trivial action of the wild monodromy of a smooth proper curve over the generic point of a trait is transformed to the condition of logarithmic smooth reduction. The proof emphasizes methods and results from logarithmic geometry. It applies to quasi-projective smooth curves with etale boundary divisor.
An investigation of morphisms that coincide topologically is used to generalize to all characteristics and partly reprove Tamagawa's theorem on the Grothendieck conjecture in anabelian geometry for affine hyperbolic curves. The theorem now deals with π^{tame} of curves over a finitely generated field and its effect on the set of isomorphisms. Universal homeomorphisms are formally inverted.