The étale cohomology ring of a punctured arithmetic curve

2021/10/04 by Ahlqvist, Eric, Carlson, Magnus
#11R34 #14F20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2110.01597

Abstract

We compute the cohomology ring H^*(U,ℤ/nℤ) for U=X∖ S where X is the spectrum of the ring of integers of a number field K and S is a finite set of finite primes. As a consequence, we obtain an efficient way to compute presentations of Q₂(G_S), where G_S is Galois group of the maximal extension of K unramified outside of a finite set of primes S, for varying K. This includes the following cases (for p any prime dividing n): μ_p(K) \not⊆ K; S does not contain the primes above p; and p=2 with K admitting real archimedean places. We also show how to recover the classical reciprocity law of the Legendre symbol from the graded commutativity of the cup product.

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