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arXiv · 2207.06353

Massey products in the \'etale cohomology of number fields

Eric Ahlqvist·Magnus Carlson
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Abstract

We give formulas for 3-fold Massey products in the \'etale cohomology of the ring of integers of a number field and use these to find the first known examples of imaginary quadratic fields with class group of $p$-rank two possessing an infinite $p$-class field tower, where $p$ is an odd prime. Furthermore, a necessary and sufficient condition, in terms of class groups of $p$-extensions, for the vanishing of 3-fold Massey products is given. As a consequence, we give an elementary and sufficient condition for the infinitude of class field towers of imaginary quadratic fields. We also disprove McLeman's $(3,3)$-conjecture. Lastly, we relate the vanishing of Massey products to the existence of Galois representations of $G_{\mathbb{Q},S}$ which realize an unexpectedly large class group for certain extensions of a quadratic imaginary number field.

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Eric Ahlqvist, Magnus Carlson. 2022-07-13. Massey products in the \'etale cohomology of number fields. https://doi.org/10.1515/crelle-2025-0006

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