Hasse norm principle for Heisenberg extensions of degree p³

2025/03/19 by Hoshi, Akinari, Yamasaki, Aiichi · 1 citation
#11E72 #12F20 #13A50 #14E08 #20C10 #20G15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.15408

Abstract

Let k be a global field and p be an odd prime number. We give a necessary and sufficient condition for the Hasse norm principle for separable field extensions K/k, i.e. the determination of the Shafarevich-Tate group Sha(T) of the norm one tori T=R⁽¹⁾_K/k(G_m) of K/k, with [K:k]=p³ or p² when the Galois group of the Galois closure of K/k is the Heisenberg group E_p(p³)≃ (C_p)²\rtimes C_p of order p³, i.e. the extraspecial group of order p³ with exponent p. As a consequence, we get the Tamagawa number τ(T)=p², p or 1 via Ono's formula τ(T)=|H¹(k,\widehatT)|/|Sha(T)|.

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