MR 3079286 Reviewed Hoshi Y. On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves: genus zero case. Tohoku Math. J., 65 (2013), 231–242. (Reviewer Francesca Vetro) 14H30 (14H10)

Let \emph{Primes} be the set of all prime numbers, $k$ be a finite extension of the field of rational numbers and $\bar{k}$ be an algebraic closure of $k$. Let $(g, r)$ be a pair of nonnegative integers such that $2g - 2 + r > 0$ and $X$ be a hyperbolic curve of type $(g, r)$ over $k$. The author observes that, for each $l \in \emph{Primes}$, there are two natural outer representations on $\pi^{\{ l\}}_{1} (X \otimes_{k} \bar{k})$: $$\rho_{X / k} ^{\{ l\}}: G_{k} := Gal(\bar{k} / k) \rightarrow Out (\pi^{\{ l\}}_{1} (X \otimes_{k} \bar{k}))$$ and $$ \rho_{g, [r]} ^{\{ l\}}: \pi_{1}(M_{g, [r]}) \rightarrow Out (\pi^{\{ l\}}_{1} (X \otimes_{k} \bar{k})),$$ where $\pi^{\{ l\}}_{1} (X \otimes_{k} \bar{k})$ denotes the maximal pro-$l$ quotient of $\pi_{1} (X \otimes_{k} \bar{k})$ and $M_{g, [r]}$ denotes the moduli stack of hyperbolic curves of type $(g, r)$ over $k$. Let $M_{g, r}$ be the moduli stack of ordered $r$-pointed proper smooth curves of genus $g$ over $k$ and $\Sigma$ be a nonempty subset of \emph{Primes}. $X$ is $\Sigma$-monodromically full (over k) if, for each $l \in \Sigma$, \, $\rho_{g, [r]} ^{\{ l\}} (\pi_{1}(M_{g, r}) ) \subset \rho_{X / k} ^{\{ l\}} (G_{k})$. \,$X$ is quasi-$\Sigma$-monodromically full (over k) if, for each $l \in \Sigma$, \, $\rho_{X / k} ^{\{ l\}} (G_{k})$ is open in $\rho_{g, [r]} ^{\{ l\}} (\pi_{1}(M_{g, [r]}) ) $. Matsumoto and Tamagawa posed the following problem concerning monodromic fullness of hyperbolic curves. \textit{One considers the following three conditions:} \textit{1) X is quasi-\emph{Primes}-monodromically full;} \textit{2) There exists an $l \in \emph{Primes}$ such that $X$ is $l$-monodromically full;} \textit{3)There exists a finite subset $\Sigma$ of Primes such that $X$ is $(\emph{Primes} - \Sigma)$-monodromically full.} \textit{Are these conditions equivalent?} [M. Matsumoto and A. Tamagawa, Mapping-class-group action versus Galois action on profinite fundamental groups, Amer. J. Math., 122 (2000), pp. 1017–1026] In this paper, the author discusses such problem in the case in which $X$ has genus zero.