Field change for the Cassels–Tate pairing and applications to class groups

Abstract

In previous work, the authors defined a category SModF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SMod}_F$$\\end{document} of finite Galois modules decorated with local conditions for each global field F. In this paper, given an extension K/F of global fields, we define a restriction of scalars functor from SModK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SMod}_K$$\\end{document} to SModF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SMod}_F$$\\end{document} and show that it behaves well with respect to the Cassels–Tate pairing. We apply this work to study the class groups of global fields in the context of the Cohen–Lenstra heuristics.