A Birational Anabelian Reconstruction Theorem for Curves over Algebraically Closed Fields in Arbitrary Characteristic

The aim of Bogomolov's programme is to prove birational anabelian conjectures for function fields $K|k$ of varieties of dimension $\geq 2$ over algebraically closed fields. The present article is concerned with the 1-dimensional case. While it is impossible to recover $K|k$ from its absolute Galois group alone, we prove that it can be recovered from the pair $(\mathrm{Aut}(\overline{K}|k),\mathrm{Aut}(\overline{K}|K))$, consisting of the absolute Galois group of $K$ and the larger group of field automorphisms fixing only the base field.Comment: 15 pages, final version accepted by the journa

Linear and quadratic Chabauty for affine hyperbolic curves

We give sufficient conditions for finiteness of linear and quadratic refined Chabauty-Kim loci of affine hyperbolic curves. We achieve this by constructing depth ≤2 quotients of the fundamental group, following a construction of Balakrishnan-Dogra in the projective case. We also apply Betts' machinery of weight filtrations to give unconditional explicit upper bounds on the number of S-integral points when our conditions are satisfied

Bixbyite-type phases in the system Ta-Zr-O-N

Phase-pure tantalum/zirconium oxide nitrides and nitrides were synthesized by the ammonolysis of amorphous oxide precursors. The nitrogen-rich oxide nitrides with variable anion composition and the nitride TaZrN3 crystallize in the cubic bixbyite-type structure (space group Ia3̅). The nitrogen content of these compounds has a significant influence on the cell parameters, the atomic positions, and the optical band gap. The results extend the already well-studied Ta–Zr–O–N system by new oxide nitrides in addition to the already known baddeleyite- and anosovite-type phases. TaZrN3 can be considered as a thermodynamically stable ternary variant of metastable Ta2N3