Staff -HOSHI, Yuichiro-
Name
HOSHI, Yuichiro
HOSHI, Yuichiro
Position
Lecturer
Lecturer
E-Mail
yuichiro (email address: add @kurims.kyoto-u.ac.jp)
yuichiro (email address: add @kurims.kyoto-u.ac.jp)
Research
Yuichiro Hoshi is working on fundamental groups of algebraic varieties,
from the viewpoint of ``anabelian geometry'', 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 Matsumoto and Tamagawa concerning monodromic fullness in the case where a given hyperbolic curve is of genus zero.
(C) Section Conjecture: He has studied Grothendieck's anabelian 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. Moreover, he also gave an example of a proper hyperbolic curve over a number field that admits infinitely many conjugacy classes of pro-p Galois sections. On the other hand, he obtained necessary and sufficient conditions for a birational Galois section of a curve over a small number field to be geometric.
(D) Combinatorial Anabelian Geometry: As a joint work with 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 Matsumoto and Tamagawa concerning monodromic fullness in the case where a given hyperbolic curve is of genus zero.
(C) Section Conjecture: He has studied Grothendieck's anabelian 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. Moreover, he also gave an example of a proper hyperbolic curve over a number field that admits infinitely many conjugacy classes of pro-p Galois sections. On the other hand, he obtained necessary and sufficient conditions for a birational Galois section of a curve over a small number field to be geometric.
(D) Combinatorial Anabelian Geometry: As a joint work with 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.