## Staff -HOSHI, Yuichiro-

Name

**HOSHI, Yuichiro**
Position
Lecturer

E-Mail
yuichiro (email address: add @kurims.kyoto-u.ac.jp)

Research

Yuichiro Hoshi is working on fundamental groups, from the viewpoint of
``anabelian geometry'', of algebraic varieties related to hyperbolic
curves.

(A) Cuspidalization: He solved affirmatively the pro-l version of the cuspidalization problem for proper hyperbolic curves over finite fields. In this proof, his study on the exactness of the log homotopy sequences played an essential role.

(B) Monodromic Fullness: He has developed the theory of monodromic fullness. The property of being monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves. He obtained a Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero. Moreover, he studied a problem of Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness in the case where a given hyperbolic curve is of genus zero.

(C) Section Conjecture: He gave a counter-example of the pro-p version of the section conjecture for hyperbolic curves over number fields and p-adic local fields, and, moreover, an example of a proper hyperbolic curve over a number field that admits infinitely many conjugacy classes of pro-p Galois sections. He also obtained necessary and sufficient conditions for a birational Galois section of a curve over a small number field to be geometric.

(D) Grothendieck Conjecture: As a study of the Grothendieck conjecture for p-adic local fields, he gave a proof of the fact that, for an open homomorphism between the absolute Galois groups of p-adic local fields, it arises from an extension of fields if and only if it preserves the Hodge-Tate-ness of p-adic representations.

(E) Combinatorial Anabelian Geometry: As a joint work with Shinichi Mochizuki, he has developed the combinatorial anabelian geometry. As consequences of this theory, for instance, they obtained proofs of the faithfulness of the outer Galois representation associated to a hyperbolic curve over a certain field and the geometric version of the Grothendieck conjecture for the universal curve over the moduli stack of curves.

(A) Cuspidalization: He solved affirmatively the pro-l version of the cuspidalization problem for proper hyperbolic curves over finite fields. In this proof, his study on the exactness of the log homotopy sequences played an essential role.

(B) Monodromic Fullness: He has developed the theory of monodromic fullness. The property of being monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves. He obtained a Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero. Moreover, he studied a problem of Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness in the case where a given hyperbolic curve is of genus zero.

(C) Section Conjecture: He gave a counter-example of the pro-p version of the section conjecture for hyperbolic curves over number fields and p-adic local fields, and, moreover, an example of a proper hyperbolic curve over a number field that admits infinitely many conjugacy classes of pro-p Galois sections. He also obtained necessary and sufficient conditions for a birational Galois section of a curve over a small number field to be geometric.

(D) Grothendieck Conjecture: As a study of the Grothendieck conjecture for p-adic local fields, he gave a proof of the fact that, for an open homomorphism between the absolute Galois groups of p-adic local fields, it arises from an extension of fields if and only if it preserves the Hodge-Tate-ness of p-adic representations.

(E) Combinatorial Anabelian Geometry: As a joint work with Shinichi Mochizuki, he has developed the combinatorial anabelian geometry. As consequences of this theory, for instance, they obtained proofs of the faithfulness of the outer Galois representation associated to a hyperbolic curve over a certain field and the geometric version of the Grothendieck conjecture for the universal curve over the moduli stack of curves.