Generalized Steinberg Relations

2021/09/28 by Efrat, Ido
#12G05 #14H30 #19F15 #55S30 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2109.13519

Abstract

We consider a field F and positive integers n, m, such that m is not divisible by Char(F) and is prime to n!. The absolute Galois group G_F acts on the group \mathbbU_n(ℤ/m) of all (n+1)×(n+1) unipotent upper-triangular matrices over ℤ/m cyclotomically. Given 0,1≠ z∈ F and an arbitrary list w of n Kummer elements (z)_F, (1-z)_F in H¹(G_F,μ_m), we construct in a canonical way a quotient \mathbbU_w of \mathbbU_n(ℤ/m) and a cohomology element ρ^z in H¹(G_F,\mathbbU_w) whose projection to the superdiagonal is the prescribed list. This extends results by Wickelgren, and in the case n=2 recovers the Steinberg relation in Galois cohomology, proved by Tate.

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