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Curves with infinite \(K\)-rational geometric fundamental group. (English) Zbl 0978.14021

Völklein, Helmut (ed.) et al., Aspects of Galois theory. Papers from the conference on Galois theory, Gainesville, FL, USA, October 14-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 256, 85-118 (1999).
From the paper: Let \(K\) be a finitely generated field with separable closure \(K_s\) and absolute Galois group \(G_K\). By a curve \(C/K\) we always understand a smooth geometrically irreducible projective curve. Let \(F(C)\) be its function field and let \(\Pi(C)\) be the Galois group of the maximal unramified extension of \(F(C)\). We have the exact sequence \[ 1\to\Pi_g (C)\to \Pi(C)\to G_K\to 1 \tag{*} \] where \(\Pi_g(C)\) is the geometric (profinite) fundamental group of \(C\times \text{Spec} (K_s)\) (i.e. \(\Pi_g(C)\) is equal to the Galois group of the maximal unramified extension of \(F(C)\otimes K_s)\). This sequence induces a homomorphism \(\rho_C\) from \(G_K\) to \(\text{Out} (\Pi_g(C))\) which is the group of automorphisms modulo inner automorphisms of \(\Pi_g(C)\). It is well known that \(\rho_C\) is an important tool for studying \(C\). For instance, it determines \(C\) up to \(K\)-isomorphisms if the genus of \(C\) is at least 2 and \(K\) is a number field or even a \({\mathfrak p}\)-adic field [see S. Machizuki, Invent. Math. 138, No. 2, 319-423 (1999; Zbl 0935.14019)].
So it is of interest to find quotients \(\overline \Pi(C)\) of \(\Pi(C)\) such that the induced map of \(G_K\) is not the identity but the induced representation \(\overline\rho_C\) becomes trivial. We give a geometric interpretation of such quotients. For this we assume that the sequence (*) is split and choose a section \(s\) which induces a homomorphism \(\sigma\) from \(G_K\) to \(\operatorname{Aut}(\Pi_g (C))\). Let \(U\) be a normal subgroup of \(\Pi(C)\) contained in \(\Pi_g(C)\). The representation \(\rho_C\) becomes trivial modulo \(U\) if and only if the map \(\sigma\bmod U\) has its image inside of \(\text{Inn} (\Pi_g(C)/U)\). Let \(Z\) be the center of \(\Pi_g(C)/U\) and \(\overline\Pi =(\Pi_g(C)/U)/Z\). Our condition on \(U\) implies that there is exactly one group theoretical section \(\overline s\) from \(G_K\) to \(\overline \Pi(C)/U)/Z\) inducing the trivial action on \(\overline\Pi\) and so \(\overline\Pi\) occurs as Galois group of an unramified regular extension of \(F(C)\) in a natural way. Thus, to find center free infinite factors of \(\Pi_g\) on which \(\rho_C\) becomes trivial is equivalent with finding infinite regular Galois coverings of \(C\). Choosing as base point a geometric point of \(C\) we can say that “\(\overline \Pi\) is a factor of the geometric fundamental group of \(C\) over \(K\)”.
Assume that \(C\) has a \(K\)-rational point \(P\) and choose a splitting \(s_P\) of (*) corresponding to \(P\) by identifying \(G_K\) with the decomposition group of an extension of the place corresponding to \(P\) in \(F(C)\) to its separable closure. Now the finite quotients of \(\Pi_g(C)\) on which \(s_P(G_K)\) operates trivially correspond to unramified Galois coverings \(C'\) of \(C\) on which the decomposition group of \(P\) operates trivially. Hence \(P\) has \(K\)-rational extensions to \(C'\) and the choice of \(s_P\) corresponds to the choice of such an extension \(P'\). (If we make another choice, then the corresponding section is replaced by a conjugate in \(\Pi(C).)\) We can regard \(C'\) as étale covering of \(C\) with respect to the base points \(P\) respectively \(P'\). Taking the limit we get the \(K\)-rational geometric fundamental group of \(C\) with base point \(P\): \[ \Pi_g(C,P): =\Pi_g(C)/ \biggl\langle \bigl(s_P (\sigma)-1\bigr) \Pi_g(C) \biggr \rangle_{\sigma \in G_K}. \] We use either quartic coverings of the projective line or quadratic coverings of elliptic curves with enough ramification points to find for every genus \(g\geq 3\) curves with infinite geometric fundamental group defined over \(\mathbb{Q}(i)\) or over \(\mathbb{F}_q(i)\) (where \(i\) is a fourth root of unity) and we find even parametric families of such curves over every ground field containing \(i\). [Remark: We do not have any example of a curve defined over \(\mathbb{Q}\) with infinite \(\mathbb{Q}\)-rational geometric fundamental group.]
We thus have examples of curves of genus \(g\) with an infinite \(K\)-rational geometric fundamental group for every \(g\geq 3\). No such example can exist for curves of genus \(g=1\), as will be explained in the paper. This leaves only the case \(g=2\). Here we find very special curves with an infinite tower of regular unramified Galois coverings but we cannot decide there whether there are rational points in the tower.
For the entire collection see [Zbl 0941.00014].

MSC:

14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
14H30 Coverings of curves, fundamental group
14E20 Coverings in algebraic geometry
14G05 Rational points

Citations:

Zbl 0935.14019
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