Profinite braid groups, Galois representations and complex multiplications
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 \(H^ 1({\mathbb Q}, {\mathbb Z}_{\ell}(m))\).
representation in outer automorphism group of profinite fundamental
scientific article; zbMATH DE number 3957239