Yuichiro Hoshi

Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero

Let l be a prime number. In the present paper, we prove that the isomorphism class of an l-monodromically full hy- perbolic curve of genus zero over a nitely generated extension of the eld of rational numbers is completely determined by the ker- nel of the natural pro-l outer Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to S. Mochizuki which asserts that the iso- morphism class of an elliptic curve which does not admit complex multiplication over a number eld is completely determined by the kernels of the natural Galois representations on the various nite quotients of its Tate module.

On Monodromically Full Points of Configuration Spaces of Hyperbolic Curves

We introduce and discuss the notion of monodromically full points of configuration spaces of hyperbolic curves. This notion leads to complements to M. Matsumoto’s result concerning the difference between the kernels of the natural homomorphisms associated to a hyperbolic curve and its point from the Galois group to the automorphism and outer automorphism groups of the geometric fundamental group of the hyperbolic curve. More concretely, we prove that any hyperbolic curve over a number field has many nonexceptional closed points, i.e., closed points which do not satisfy a condition considered by Matsumoto, but that there exist infinitely many hyperbolic curves which admit many exceptional closed points, i.e., closed points which do satisfy the condition considered by Matsumoto. Moreover, we prove a Galois-theoretic characterization of equivalence classes of monodromically full points of configuration spaces, as well as a Galois-theoretic characterization of equivalence classes of quasi-monodromically full points of cores. In a similar vein, we also prove a necessary and sufficient condition for quasi-monodromically full Galois sections of hyperbolic curves to be geometric.

ON THE KERNELS OF THE PRO-l OUTER GALOIS REPRESENTATIONS ASSOCIATED TO HYPERBOLIC CURVES OVER NUMBER FIELDS

In the present paper, we discuss the relationship between the Galois extension corresponding to the kernel of the pro-l outer Galois representation associated to a hyperbolic curve over a number eld and l-moderate points of the hyperbolic curve. In particular, we prove that, for a certain hyperbolic curve, the Galois extension under consideration is generated by the coordinates of the l-moderate points of the hyperbolic curve. This may be regarded as an analogue of the fact that the Galois extension corresponding to the kernel of the l-adic Galois representation associated to an abelian variety is generated by the coordinates of the torsion points of the abelian variety of l-power order. Moreover, we discuss an application of the argument of the present paper to the study of the Fermat equation.

Conditional results on the birational section conjecture over small number fields

In the present paper, we give necessary and sufficient conditions for a birational Galois section of a projective smooth curve over either the field of rational numbers or an imaginary quadratic field to be geometric. As a consequence, we prove that, over such a small number field, to prove the birational section conjecture for projective smooth curves, it suffices to verify that, roughly speaking, for any birational Galois section of the projective line, the local points associated to the birational Galois section avoid distinct three rational points, and, moreover, a certain Galois representation determined by the birational Galois section is unramified at all but finitely many primes. Moreover, as another consequence, we obtain some examples of projective smooth curves for which any prosolvable birational Galois section is geometric.

On the Field-theoreticity of Homomorphisms between the Multiplicative Groups of Number Fields

We discuss the eld-theoreticity of homomorphisms between the multiplicative groups of number elds. We prove that, for instance, for a given isomorphism between the multiplicative groups of number elds, either the isomorphism or its multiplicative inverse arises from an isomorphism of elds if and only if the given isomorphism is SPU-preserving (i.e., roughly speaking, preserves the subgroups of principal units with respect to various nonarchimedean primes).