## 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) 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 some necessary and sufficient conditions for a birational Galois section of an algebraic curve over a small number field to be geometric.

(2) 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 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.

(3) Grothendieck Conjecture:

He discussed various Grothendieck conjecture-type results concerning hyperbolic polycurves. In particular, he gave a proof of the Grothendieck conjecture for hyperbolic polycurves of dimension less than or equal to four over sub-$p$-adic fields. He also gave, as a study of the Grothendieck conjecture for $p$-adic local fields, 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.

(4) Moderate Points:

The study of moderate points was initiated by Makoto Matsumoto. The notion of moderate points may be regarded as an analogue for hyperbolic curves of the notion of torsion points for abelian varieties. He studied a relationship between the kernel of the outer Galois representation associated to a hyperbolic curve and moderate points of the curve. In particular, he discussed a relationship between some problems concerning such outer Galois representations and the Fermat conjecture. Moreover, he also proved the finiteness of the set of moderate rational points of a once-punctured elliptic curve over a number field, which may be regarded as an analogue of the classical finiteness of the set of torsion rational points of an abelian variety over a number field.

(5) 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 suitable field. They also obtained a proof of the geometric version of the Grothendieck conjecture for the universal curve over the moduli stack of curves.

(1) 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 some necessary and sufficient conditions for a birational Galois section of an algebraic curve over a small number field to be geometric.

(2) 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 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.

(3) Grothendieck Conjecture:

He discussed various Grothendieck conjecture-type results concerning hyperbolic polycurves. In particular, he gave a proof of the Grothendieck conjecture for hyperbolic polycurves of dimension less than or equal to four over sub-$p$-adic fields. He also gave, as a study of the Grothendieck conjecture for $p$-adic local fields, 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.

(4) Moderate Points:

The study of moderate points was initiated by Makoto Matsumoto. The notion of moderate points may be regarded as an analogue for hyperbolic curves of the notion of torsion points for abelian varieties. He studied a relationship between the kernel of the outer Galois representation associated to a hyperbolic curve and moderate points of the curve. In particular, he discussed a relationship between some problems concerning such outer Galois representations and the Fermat conjecture. Moreover, he also proved the finiteness of the set of moderate rational points of a once-punctured elliptic curve over a number field, which may be regarded as an analogue of the classical finiteness of the set of torsion rational points of an abelian variety over a number field.

(5) 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 suitable field. They also obtained a proof of the geometric version of the Grothendieck conjecture for the universal curve over the moduli stack of curves.