2022/07/13 by Eric Ahlqvist, Ahlqvist, Eric, Magnus Carlson +1
Mathematics · #11R34 #14F20 #14G25 #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2207.06353
openalex publication_date 2022/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
We give formulas for 3-fold Massey products in the étale 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 Gℚ,S which realize an unexpectedly large class group for certain extensions of a quadratic imaginary number field.