Hasse norm principle for M₁₁ and J₁ extensions

2022/10/17 by Akinari Hoshi, Hoshi, Akinari, Kazuki Kanai +3
Mathematics · #11E72 #12F20 #13A50 #14E08 #20C10 #20G15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2210.09119

openalex publication_date 2022/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a necessary and sufficient condition for the Hasse norm principle for field extensions K/k when the Galois groups \rm Gal(L/k) of the Galois closure L/k of K/k are isomorphic to the Mathieu group M₁₁ of degree 11 of order 7920 or the Janko group J₁ of order 175560 by determining H¹(k,\rm Pic X)=0 or ℤ/2ℤ for norm one tori T=R⁽¹⁾_K/k(\mathbbG_m) with a smooth k-compactification X and X=X×_kk. The result gives a first step towards understanding the all pictures of the Hasse norm principle for the 26 sporadic simple groups.

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