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Some problems on three point ramifications and associated large Galois representations. (English) Zbl 0659.12014
Galois representations and arithmetic algebraic geometry, Proc. Symp., Kyoto/Jap. 1985 and Tokyo/Jap. 1986, Adv. Stud. Pure Math. 12, 173-188 (1987).
[For the entire collection see Zbl 0632.00004.]
Let \({\mathbb{Q}}\) be the field of rational numbers, \({\bar {\mathbb{Q}}}\) its algebraic closure and \(\ell\) a fixed prime number. Then there is a canonical homomorphism from the absolute Galois group \(G_{{\mathbb{Q}}}=Gal({\bar {\mathbb{Q}}}/{\mathbb{Q}})\) to the outer automorphism group of the pro-\(\ell\) fundamental group of the punctured projective line \(Out(\pi_ 1^{\text{pro-}\ell}(P^ 1_{{\mathbb{Q}}}\setminus \{0,1,\infty \}).\)
Let M be the maximum pro-\(\ell\)-extension of the rational function field \(K={\bar {\mathbb{Q}}}(t)\) unramified outside \(t=0,1,\infty\). Then \(G_{{\mathbb{Q}}}\) is isomorphic to the Galois group of the bottom step in the tower of extensions \({\mathbb{Q}}(t)\subset {\bar {\mathbb{Q}}}(t)\subset M\) and the fundamental group mentioned above is isomorphic to the Galois group of the top step; the map \(\phi\) is then constructed by conjugation in Gal(M/\({\mathbb{Q}}(t))\). This paper gives an informal discussion of the basic but difficult problem of the determination of kernel and image of \(\phi\). Some results on approximations of this problem are given. Elliptic curves and modular curves are seen to be of importance for the problems under consideration.
Reviewer: J.Brinkhuis

11R58 Arithmetic theory of algebraic function fields
14H30 Coverings of curves, fundamental group
11R32 Galois theory
14E22 Ramification problems in algebraic geometry
14G25 Global ground fields in algebraic geometry
14H52 Elliptic curves
11F11 Holomorphic modular forms of integral weight