2023/09/28 by Akinari Hoshi, Hoshi, Akinari, Aiichi Yamasaki +1
Mathematics · #11E72 #12F20 #13A50 #14E08 #20C10 #20G15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2309.16187
openalex publication_date 2023/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a complete answer to the rationality problem (up to stable k-equivalence) for norm one tori T=R(1)K/k(\mathbbGm) of K/k whose Galois closures L/k are A5≃ \rm PSL2(\mathbbF4) and \rm PSL2(\mathbbF8) extensions. In particular, we prove that T is stably k-rational for G=\rm Gal(L/k)≃ \rm PSL2(\mathbbF8), H=\rm Gal(L/K)≃ (C2)3 and H≃ (C2)3\rtimes C7 where Cn is the cyclic group of order n by using GAP computations with the aid of PARI/GP. Based on the result, we conjecture that T is stably k-rational for G≃ \rm PSL2(\mathbbF2d), (C2)d≤ H≤ (C2)d\rtimes C2d-1. Some other cases G≃ An, Sn, \rm GLn(\mathbbFpd), \rm SLn(\mathbbFpd), \rm PGLn(\mathbbFpd), \rm PSLn(\mathbbFpd) and H\lneq G are also investigated for small n and pd.