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We construct an Enriques surface X over Q with empty etale-Brauer set (and hence no rational points) for which there is no algebraic Brauer-Manin obstruction to the Hasse principle. In addition, if there is a transcendental obstruction on X, then we obtain a K3 surface that has a transcendental obstruction to the Hasse principle.
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It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least $3$ over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thel`ene that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least $3$. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.
Let k be a global field of characteristic not 2. We prove a local-global principle for the existence of self-dual normal bases, and more generally for the isomorphism of G-trace forms, of G-Galois algebras over k.
Let $L$ be a finite extension of $mathbb{F}_q(t)$. We calculate the proportion of polynomials of degree $d$ in $mathbb{F}_q[t]$ that are everywhere locally norms from $L/mathbb{F}_q(t)$ which fail to be global norms from $L/mathbb{F}_q(t)$.
Let $K/k$ be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for $K/k$ and the defect of weak approximation for the norm one torus $R^1_{K/k} mathbb{G}_m$. We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of $K/k$ has symmetric or alternating Galois group.
We construct a stacky curve of genus $1/2$ (i.e., Euler characteristic $1$) over $mathbb{Z}$ that has an $mathbb{R}$-point and a $mathbb{Z}_p$-point for every prime $p$ but no $mathbb{Z}$-point. This is best possible: we also prove that any stacky curve of genus less than $1/2$ over a ring of $S$-integers of a global field satisfies the local-global principle for integral points.
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