Duke Mathematical Journal

A cup product in the Galois cohomology of number fields

William G. McCallum and Romyar T. Sharifi

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Let $K$ be a number field containing the group $μ_n$ of $n$th roots of unity, and let $S$ be a set of primes of $K$ including all those dividing $n$ and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal $S$-ramified extension of $K$ with coefficients in $μ_n$, which yields a pairing on a subgroup of K x containing the $S$-units. In this general situation, we determine a formula for the cup product of two elements that pair trivially at all local places.

Our primary focus is the case in which K = ( μ p ) for $n=p$, an odd prime, and $S$ consists of the unique prime above $p$ in $K$. We describe a formula for this cup product in the case that one element is a pth root of unity. We explain a conjectural calculation of the restriction of the cup product to $p$-units for all $p≤10,000$ and conjecture its surjectivity for all $p$ satisfying Vandiver's conjecture. We prove this for the smallest irregular prime $p=37$ via a computation related to the Galois module structure of $p$-units in the unramified extension of $K$ of degree $p$.

We describe a number of applications: to a product map in $K$-theory, to the structure of $S$-class groups in Kummer extensions of $K$, to relations in the Galois group of the maximal pro-$p$ extension of ( μ p ) unramified outside $p$, to relations in the graded $ℤ_p$-Lie algebra associated to the representation of the absolute Galois group of $\mathbb{Q}$ in the outer automorphism group of the pro-$p$ fundamental group of P 1 ( ¯ ) { 0 , 1 , } , and to Greenberg's pseudonullity conjecture.

Article information

Duke Math. J., Volume 120, Number 2 (2003), 269-310.

First available in Project Euclid: 16 April 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R34: Galois cohomology [See also 12Gxx, 19A31]
Secondary: 11R23: Iwasawa theory 11R29: Class numbers, class groups, discriminants 11R70: $K$-theory of global fields [See also 19Fxx] 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx]


McCallum, William G.; Sharifi, Romyar T. A cup product in the Galois cohomology of number fields. Duke Math. J. 120 (2003), no. 2, 269--310. doi:10.1215/S0012-7094-03-12023-2. https://projecteuclid.org/euclid.dmj/1082138585

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