Genus of abstract modular curves with level-\(\ell\) structures
Let \(X\) be a smooth geometrically connected curve over a finitely generated field \(k\) of characteristic \(p\ge 0\). Let \(H_\ell\) be a continuous \(\pi_1(X)\)-module of finite rank \(r_\ell\) over \(\mathbb F_\ell\) or a field \(F_\ell\) contained in \(\bar {\mathbb F}_\ell\), and let \(G_\ell\) be the image of the Galois representation \(\rho_\ell: \pi_1(X)\to\mathrm{GL}(H_\ell)\simeq \mathrm{GL}_{r_\ell}(F_\ell)\). An abstract modular curve of level-\(\ell\) structure referred in the title is the étale cover \(X_U\) over \(X\) corresponding to the open subgroup \(\rho_\ell^{-1}(U)\) for a subgroup \(U\subset G_\ell\). Let \(|X|\) denote the set of closed points of \(X\), and for any \(x\in |X|\), denote by \(G_{\ell,x}\) the image of the local Galois representation at \(x\). The aim of the present paper is to describe the local Galois image \(G_{\ell,x}\) as \(x\) varies in \(|X|\). The authors consider a family of \(F_\ell \pi_1(X)\)-modules \(H_\ell\) of rank \(r_\ell\le r\) (\(r\ge 1\) fixed) for \(\ell\) in an infinite set \(L\) of primes \(\neq p\), and fix a set \(\mathcal F_\ell\) (= the ``moduli data'') of subgroups \(U\) of \(G_\ell\) which does not contain the image \(\overline G_\ell\) of the geometric fundamental group \(\pi_1(\overline X)\). The geometric cover \(\overline X_U \to \overline X\) corresponds to the subgroup \(\overline U:=U\cap \overline G_\ell\), and the above condition ensures that the cover is nontrivial. The paper under review mainly studies the problem whether \[ g^{\rho_\ell}_{X_{\mathcal F_\ell}}:=\min \{g(X_U): U\in \mathcal F_\ell\} \to +\infty? \] For a finite subgroup \(G\subset\mathrm{GL}(H_\ell)\), let \(G^+\) denote the (normal) subgroup generated by the elements of order \(\ell\). Let \(\mathcal F_{\ell,+}\) be the set of all subgroups \(U\) of \(G_\ell\) such that \({\overline G}^+_\ell\not \subset \overline U\). and set \(X_+^{\rho_\ell}:=X_{\mathcal F_{\ell,+}}^{\rho_\ell}\). In [Int. Math. Res. Not. 2019, No. 9, 2735--2762 (2019; Zbl 1469.14040)] the authors show that under a tameness condition (T) there exists an open subgroup \(\Pi\subset \pi_1(\overline X)\) such that \(\rho_\ell(\Pi)=\rho_\ell(\Pi)^+\) for \(\ell \gg 0\) (Theorem A). The main theorem of this paper is: Theorem 1.6.1. Assuming \(F_\ell=\mathbb F_\ell\) and under the tameness condition (T) and a perfectness condition (P), the minimal genus \(g_{X_+^{\rho_\ell}}\) goes to infinity. As an arithmetic application, for \(\ell\gg0\) and all but finitely many \(x\in X(k)\) one has \(\overline {G}_\ell^+ \subset {G}_{\ell,x}\). In particular, there exists an integer \(B\ge 1\) such that for \(\ell \gg0\) and all but finitely many \(x\in X(k)\), one has \([G_\ell, G_{\ell,x}]\le B\). The strategy of the proof of Theorem 1.6.1 is to construct a ``universal tensor representation'' in order to separate the groups in \(\mathcal F_{\ell,+}\) by lines (Theorem B). This allows to construct an auxiliary bounded family \(\tilde \rho_\ell:\pi_1(X)\to\mathrm{GL}(\tilde T_\ell)\) of \(\mathbb F_\ell\)-representations \(\tilde T_\ell\) such that every component of \(X_U\) for \(U\in \mathcal F_{\ell,+}\) dominates a component of some cover \(X_{\tilde U}\), where \(\tilde U\) is the stabilizer of a line in \(\tilde T_\ell\). The family of all such groups \(\tilde U\) of \(\tilde G_\ell\) form another collection of abstract modular curves, denoted by \(X_0^{\tilde \rho_\ell}\). As the third main result, the authors show that under the condition (T) and the condition that for every open subgroup \(\Pi \subset \pi_1(\overline X)\) one has \(H_\ell ^\Pi=0\) for \(\ell\gg 0\), the minimal genus \(g_{X^{\tilde \rho_\ell}_0}\) goes to infinity (Theorem C). Theorems B and C imply Theorem 1.6.1. The authors also illustrate how Nori's approximation theory comes into play and how results Larsen and Pink can be applied to work with more general coefficients \(F_\ell\) other than \(\mathbb F_\ell\). The paper is well-written.
scientific article; zbMATH DE number 7075965
scientific article