## Duke Mathematical Journal

### A cup product in the Galois cohomology of number fields

#### Abstract

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 $K\sp \mathsf{x}$ 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 $K=\mathbb {Q}(\mu\sb p)$ 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 $\mathbb {Q}(mu\sb p)$ 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 $\mathbf {P}\sp 1(\overline \mathbb {Q})-\{0,1,\infty\}$, and to Greenberg's pseudonullity conjecture.

#### Article information

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

Dates
First available in Project Euclid: 16 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1082138585

Digital Object Identifier
doi:10.1215/S0012-7094-03-12023-2

Mathematical Reviews number (MathSciNet)
MR2019977

Zentralblatt MATH identifier
1047.11106

#### Citation

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

#### References

• \lccJ. Buhler, R. Crandall, R. Ernvall, T. Metsänkylä, and M. A. Shokrollahi, "Irregular primes and cyclotomic invariants to 12 million" in Computational Algebra and Number Theory (Milwaukee, 1996), ed. W. Bosma, J. Symbolic Comput. 31, Academic Press, Oxford, 2001, 89–96.
• \lccW. G. Dwyer and E. M. Friedlander, Algebraic and étale $K$-theory, Trans. Amer. Math. Soc. 292 (1985), 247–280.
• \lccC. Fieker, R. Sharifi, and W. Stein, Magma code for verifying nontriviality of the pairing for $p = 37$, available from \wwwabel.math.harvard.edu/~sharifi/computations.html and \wwwmath.arizona.edu/~wmc
• \lccR. H. Fox, Free differential calculus, I: Derivation in the free group ring, Ann. of Math. (2) 57 (1953), 547–560.
• \lccA. B. Goncharov, The dihedral Lie algebras and Galois symmetries of $\pi_1^{(l)}({{\mathbb{P}}}^1-(\{0,\infty\}\cup \mu_{N}))$, Duke Math. J. 110 (2001), 397–487.
• \lccR. Greenberg, "Iwasawa theory–-past and present" in Class Field Theory: Its Centenary and Prospect (Tokyo, 1998), ed. K. Miyake, Adv. Stud. Pure Math. 30, Math. Soc. Japan, Tokyo, 2001, 335–385.
• \lccR. Hain and M. Matsumoto, Weighted completion of Galois groups and Galois actions on the fundamental group of ${\mathbb P}^1-\{0,1,\infty\}$, to appear in Compositio Math., preprint.
• \lccY. Ihara, Profinite braid groups, Galois representations and complex multiplications, Ann. of Math. (2) 123 (1986), 43–106.
• ––––, "The Galois representation arising from ${\mathbb P}^1-\{0,1,\infty\}$ and Tate twists of even degree" in Galois Groups over $\mathbb Q$ (Berkeley, Calif., 1987), ed. Y. Ihara, K. Ribet, and J.-P. Serre, Math. Sci. Res. Inst. Publ. 16, Springer, New York, 1989, 299–313.
• ––––, "Some arithmetic aspects of Galois actions of the pro-$p$ fundamental group of ${\mathbb P}^1-\{0,1,\infty\}$" in Arithmetic Fundamental Groups and Noncommutative Algebra (Berkeley, Calif., 1999), ed. M. D. Fried and Y. Ihara, Proc. Sympos. Pure Math. 70, Amer. Math. Soc., Providence, 2002, 247–273. \CMP1 935 408
• \lccF. Keune, "On the structure of the $K\sb 2$ of the ring of integers in a number field" in Proceedings of Research Symposium on $K$-Theory and its Applications (Ibadan, Nigeria, 1987), ed. A. O. Kuku and C. Weibel, $K$-Theory 2, Historical Jrl., Ann Arbor, 1989, 625–645.
• \lccM. Kurihara, Ideal class groups of cyclotomic fields and modular forms of level 1, J. Number Theory 45 (1993), 281–294.
• \lccJ. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132.
• \lccA. Lannuzel and T. Nguyen Quang Do, Conjectures de Greenberg et extensions pro-$p$-libres d'un corps de nombres, Manuscripta Math. 102 (2000), 187–209.
• \lccD. Marshall, Greenberg's conjecture and cyclotomic towers, preprint, \wwwma.utexas.edu/users/marshall/research.html
• \lccM. Matsumoto, "On the Galois image in the derivation algebra of $\pi_1$ of the projective line minus three points" in Recent Developments in the Inverse Galois Problem (Seattle, 1993), ed. M. D. Fried, S. S. Abhyankar, W. Feit, Y. Ihara, and H. Völklein, Contemp. Math. 186, Amer. Math. Soc., Providence, 1995, 201–213.
• \lccW. G. McCallum, Greenberg's conjecture and units in multiple ${\mathbb Z}_p$-extensions, Amer. J. Math. 123 (2001), 909–930.
• \lccW. G. McCallum and R. T. Sharifi, Magma routines for computing the table of pairings for $p < 10,000$, available from \wwwabel.math.harvard.edu/~sharifi/computations.html and \wwwmath.arizona.edu/~wmc
• \lccT. Nguyen Quang Do, "Formations de classe et modules d'Iwasawa" in Number Theory (Noordwijkerhout, Netherlands, 1983), ed. H. Jager, Lecture Notes in Math. 1068, Springer, Berlin, 1984, 167–185.
• \lccJ. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of Number Fields, Grundlehren Math. Wiss. 323, Springer, Berlin, 2000.
• \lccR. T. Sharifi, "Relationships between conjectures on the structure of pro-$p$ Galois groups unramified outside $p$" in Arithmetic Fundamental Groups and Noncommutative Algebra (Berkeley, Calif., 1999), ed. M. D. Fried and Y. Ihara, Proc. Sympos. Pure Math. 70, Amer. Math. Soc., Providence, 2002, 275–284. \CMP1 935 409
• \lccC. Soulé, $K$-théorie des anneaux d'entiers de corps de nombres et cohomologie étale, Invent. Math. 55 (1979), 251–295.
• ––––, "On higher $p$-adic regulators" in Algebraic $K$-Theory (Evanston, Ill., 1980), ed. E. M. Friedlander and M. R. Stein, Lecture Notes in Math. 854, Springer, Berlin, 1984, 372–401.
• \lccJ. Tate, Relations between $K_2$ and Galois cohomology, Invent. Math. 36 (1976), 257–274.
• \lccL. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Grad. Texts in Math. 83, Springer, New York, 1997.