## 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.

(1) Cuspidalization Problem:

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.

(2) Combinatorial Anabelian Geometry:

As a joint work with Shinichi Mochizuki, he has developed the combinatorial anabelian geometry. By means of this theory, they proved, for instance, the faithfulness of the outer Galois representation associated to a hyperbolic curve over a certain field. They also obtained a proof of the geometric version of the Grothendieck conjecture for the universal curve over the moduli stack of curves.

(3) Monodromic Fullness:

He has developed the theory of monodromic fullness of hyperbolic curves. 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 solved negatively a problem of Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness in the case where a given hyperbolic curve is of genus zero.

(4) 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.

(5) Reconstruction Results on Arithmetic 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. He also proved that, for an isomorphism between the multiplicative groups of number fields, it holds that either the given isomorphism or its multiplicative inverse arises from an isomorphism of fields if and only if the given isomorphism preserves the subgroups of principal units with respect to various nonarchimedean primes.

(1) Cuspidalization Problem:

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.

(2) Combinatorial Anabelian Geometry:

As a joint work with Shinichi Mochizuki, he has developed the combinatorial anabelian geometry. By means of this theory, they proved, for instance, the faithfulness of the outer Galois representation associated to a hyperbolic curve over a certain field. They also obtained a proof of the geometric version of the Grothendieck conjecture for the universal curve over the moduli stack of curves.

(3) Monodromic Fullness:

He has developed the theory of monodromic fullness of hyperbolic curves. 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 solved negatively a problem of Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness in the case where a given hyperbolic curve is of genus zero.

(4) 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.

(5) Reconstruction Results on Arithmetic 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. He also proved that, for an isomorphism between the multiplicative groups of number fields, it holds that either the given isomorphism or its multiplicative inverse arises from an isomorphism of fields if and only if the given isomorphism preserves the subgroups of principal units with respect to various nonarchimedean primes.