Profinite braid groups, Galois representations and complex multiplications. (English) Zbl 0595.12003

Ihara, Yasutaka

Ann. Math. (2) 123, 43-106 (1986).

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 congruences can be seen, for example, by the fact that one corollary of them is a generalization of Iwasawa’s congruence for Jacobi sums. It is also shown that the coefficients of \(F(\rho)\) have special arithmetic significance. In particular, they give rise to 1-cocycles \(\text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\to {\mathbb Z}_{\ell}(m)\), and so are intimately connected with the Galois cohomology groups \(H^ 1({\mathbb Q}, {\mathbb Z}_{\ell}(m))\).

Reviewer: S.Kamienny

MSC:

11R32	Galois theory
14H40	Jacobians, Prym varieties
14K22	Complex multiplication and abelian varieties
14H45	Special algebraic curves and curves of low genus
14H30	Coverings of curves, fundamental group
14G25	Global ground fields in algebraic geometry
11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R34	Galois cohomology

Keywords:

circular units; Jacobian of Fermat curves; Galois representations; pro-\(\ell\) braid groups; étale covering of projective 1-space; representation in outer automorphism group of profinite fundamental; group; absolute Galois group; completed group algebra; Tate module; Jacobi sums; Galois cohomology