Profinite braid groups, Galois representations and complex multiplications
type: Item
instance of: scholarly article
title: Profinite braid groups, Galois representations and complex multiplications
zbMATH Open document ID: 0595.12003
DOI: 10.2307/1971352
author: Yasutaka Ihara
published in: Annals of Mathematics. Second Series
publication date: 1986-01-01T00:00:00Z
review text: The author takes an important first step in the study of the absolute Galois group \(\text{Gal}({\overline{\mathbb Q}}/{\mathbb Q})\) and its representation in the outer automorphism group of the profinite fundamental group of \({\mathbb P}^ 1({\overline {\mathbb Q}})\setminus \{0,1,\infty \}\). He considers the Fermat curves of degree \(\ell^ n\) \((n=1,2,...,)\) which form the maximum tower of abelian pro-\(\ell\) étale coverings of \({\mathbb P}^ 1({\overline {\mathbb Q}})\setminus \{0,1,\infty \}\). The Galois group of this tower is the abelianization of the free pro-\(\ell\) group \({\mathcal F}\) of rank 2. The author proves the existence of a unique continuous homomorphism \(F\) from \(\text{Gal}({\overline{\mathbb Q}}/{\mathbb Q}(\mu_{\ell^{\infty}}))\) into the group of units of the completed group algebra \({\mathcal A}={\mathbb Z}_{\ell}[[ {\mathcal F}^{ab}]]\) such that the action of any \(\rho\in \text{Gal}({\overline {\mathbb Q}}/{\mathbb Q}(\mu_{\ell^{\infty}}))\) on the \(\ell\)-adic Tate module of a certain quotient of the Jacobian \(J_ n\) of the Fermat curve of degree \(\ell^ n\) is via \(F(\rho)\) evaluated on certain circular \(\ell^ n\)-units. By identifying \({\mathcal F}'/{\mathcal F}''\) with the projective limit of the \(\ell\)-adic Tate modules of \(J_ n\) (as \(n\to \infty)\), and studying the representation of \(\text{Gal}({\overline{\mathbb Q}}/{\mathbb Q})\) in the pro-\(\ell\) analogue of the braid group on two strings, he also shows that the homomorphism F can be extended to a 1-cocycle \(F: \text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\to {\mathcal A}^*\) such that the action of any \(\rho\in Gal({\overline {\mathbb Q}}/{\mathbb Q})\) on the \(\ell\)-adic Tate module is still as described above. Moreover, identifying \({\mathcal A}\) with \({\mathbb Z}_{\ell}u,v,w/((1+u)(1+v)(1+w)-1)\) the power series \(F(\rho)\) is seen to satisfy very simple and useful congruences modulo \(u, v\), and \(w\). The importance of these congrue…
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14K22
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R34
zbMATH DE Number: 3957239
zbMATH Keywords: circular units
zbMATH Keywords: Jacobian of Fermat curves
zbMATH Keywords: Galois representations
zbMATH Keywords: pro-\(\ell\) braid groups
zbMATH Keywords: étale covering of projective 1-space
zbMATH Keywords: representation in outer automorphism group of profinite fundamental
zbMATH Keywords: group
zbMATH Keywords: absolute Galois group
zbMATH Keywords: completed group algebra
zbMATH Keywords: Tate module
zbMATH Keywords: Jacobi sums
zbMATH Keywords: Galois cohomology
Wikidata QID: Q92945677
reviewed by: Sheldon Kamienny
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971352
OpenAlex ID: W2318095708
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description: scientific article; zbMATH DE number 3957239