Let be a prime number. In this paper, we prove that the isomorphism class of an -monodromically full hyperbolic curve of genus zero over a finitely generated extension of the field of rational numbers is completely determined by the kernel of the natural pro- outer Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to Mochizuki which asserts that the isomorphism class of an elliptic curve which does not admit complex multiplication over a number field is completely determined by the kernels of the natural Galois representations on the various finite quotients of its Tate module.
"Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero." Nagoya Math. J. 203 47 - 100, September 2011. https://doi.org/10.1215/00277630-1331863