Geometrically simple counterexamples to a local-global principle for quadratic twists

Two abelian varieties A and B over a number field K are said to be strongly locally quadratic twists if they are quadratic twists at every completion of K. While it was known that this does not imply that A and B are quadratic twists over K, the only known counterexamples (necessarily of dimension >= 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ge 4$$\end{document}) are not geometrically simple. We show that, for every prime p equivalent to 13(mod24)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\equiv 13 \pmod {24}$$\end{document}, there exists a pair of geometrically simple abelian varieties of dimension p-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p-1$$\end{document} over Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document} that are strongly locally quadratic twists but not quadratic twists. The proof is based on Galois cohomology computations and class field theory.