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Coupling of universal monodromy representations of Galois-Teichmüller modular groups. (English) Zbl 0835.14007

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 H. Nakamura, N. Takao and 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.
Reviewer: M.Nakamura (Tokyo)

MSC:

14H10 Families, moduli of curves (algebraic)
14H25 Arithmetic ground fields for curves
11F55 Other groups and their modular and automorphic forms (several variables)

Citations:

Zbl 0826.14016
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References:

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