In this talk, we will discuss the following problem posed by Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness of hyperbolic curves.

For a hyperbolic curve X over a number field, are the following three conditions equivalent?

(A) For any prime number l, X is quasi-l-monodromically full.

(B) There exists a prime number l such that X is l-monodromically full.

(C) X is l-monodromically full for all but finitely many prime numbers l.

The property of being (quasi-)monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves, and the above equivalence may be regarded as an analogue for hyperbolic curves of the following result concerning the Galois representation on the Tate module of an elliptic curve over a number field proven by Jean-Pierre Serre.

For an elliptic curve E over a number field, the following four conditions are equivalent:

(0) E does not admit complex multiplication.

(1) For any prime number l, the image of the l-adic Galois representation associated to E is open.

(2) There exists a prime number l such that the l-adic Galois representation associated to E is surjective.

(3) The l-adic Galois representation associated to E is surjective for all but finitely many prime numbers l.

In this talk, I will present some results concerning the above problem in the case where the given hyperbolic curve is of genus zero. In particular, I will give an example of a hyperbolic curve of type (0,4) over a number field which satisfies (C) but does not satisfy (A).