Linkage of Pfister forms over C(x1,…,xn)\mathbb{C}(x_1,\ldots,x_n)

In this note, we prove the existence of a set of $n$-fold Pfister forms of cardinality $2^n$ over $\mathbb{C}(x_1,\dots,x_n)$ which do not share a common $(n-1)$-fold factor. This gives a negative answer to a question raised by Becher. The main tools are the existence of the dyadic valuation on the complex numbers and recent results on symmetric bilinear over fields of characteristic 2

Four-fold Massey products in Galois cohomology

In this paper, we develop a new necessary and sufficient condition for the vanishing of 4-Massey products of elements in the mod-2 Galois cohomology of a field. This new description allows us to define a splitting variety for 4-Massey products, which is shown in the Appendix to satisfy a local-to-global principle over number fields. As a consequence, we prove that, for a number field, all such 4-Massey products vanish whenever they are defined. This provides new explicit restrictions on the structure of absolute Galois groups of number fields.Comment: Final version: several corrections made throughout the paper; some sections reorganized; will appear in Compositio Mathematic

Flavourful Z′Z' portal for vector-like neutrino Dark Matter and RK(∗)R_{K^{(*)}}

We discuss a flavourful $Z'$ portal model with a coupling to fourth-family singlet Dirac neutrino dark matter. In the absence of mixing, the $Z'$ is fermiophobic, having no couplings to the three chiral families, but does couple to a fourth vector-like family. Due to mixing effects, the $Z'$ gets induced couplings to second family left-handed lepton doublets and third family left-handed quark doublets. This model can simultaneously account for the measured $B$-decay ratios $R_{K}$ and $R_{K^*}$ and for the observed relic abundance of dark matter. We identify the parameter space where this explanation is consistent with existing experimental constraints from dark matter direct and indirect detection, LHC searches, and precision measurements of flavour mixing and neutrino processes

Rayleigh-Taylor instability under an inclined plane

We revisit the canonical Rayleigh-Taylor instability and investigate the case of a thin film of fluid upon the underside of an inclined plane. The presence of a natural flow along the plane competes with the conventional droplet forming instability. In particular, experiments reveal that no drops form for inclinations greater than a critical value. These features are rationalized in the context of the absolute/convective analysis conducted in this article