## 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)

Zbl 0826.14016
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### References:

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