Coupling of universal monodromy representations of Galois-Teichmüller modular groups
Let \(M_{g,n}\) be the moduli stack of smooth projective curves of genus \(g\) with \(n\)-marked points, and let \(l\) be a rational prime. The pro-\(l\) universal monodromy representations of Galois-Teichmüller modular groups define canonical pro-\(l\) towers \(\{\mathbb{Q}_{g,n} (m)\}^\infty_{m = 1}\) of fields of definition of certain natural profinite coverings of \(M_{g,n}\). These field towers are shown to be independent of \(n \geq 1\) in a previous paper by \textit{H. Nakamura}, \textit{N. Takao} and \textit{R. Ueno} [Math. Ann. 302, No. 2, 197-213 (1955; Zbl 0826.14016)] and in this paper a fundamental relation with respect to genera: \(\mathbb{Q}_{0,3} (m) \subset \mathbb{Q}_{g,n} (m) \subset \mathbb{Q}_{1,1} (m)\) \((n \geq 1)\) is established. This result leads to applications to certain nonabelian analogs of the Tate conjecture and to a topological problem of estimation of the Johnson homomorphisms.
projective curves of genus \(g\) with \(n\)-marked points
pro-\(l\) towers of fields of definition
nonabelian analogs of the Tate conjecture
scientific article; zbMATH DE number 840216
scientific article