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Infinite Analysis Research Division (Hoshi) Lab
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The lab studies the arithmetic geometry of hyperbolic algebraic curves from the viewpoint of anabelian geometry.
AboutAbout the lab
Exploring the arithmetic geometry of hyperbolic algebraic curves
The Hoshi Lab studies the arithmetic geometry of hyperbolic algebraic curves from perspectives such as anabelian geometry and p-adic Teichmüller theory.
It has worked on the section conjecture, anabelian geometry for algebraic curves and higher-dimensional algebraic varieties, combinatorial anabelian geometry, and more.
Inter-universal Teichmüller theory and the geometry of hyperbolic algebraic curves in positive characteristic are also among its research subjects.
Recently, the lab has been advancing research on the Galois groups of solvably closed Galois extensions of number fields and on the Grothendieck–Teichmüller group.
Key terms
- Anabelian geometry
- A field that studies how much of the shape of an object such as an algebraic curve can be recovered from a group called the arithmetic fundamental group
- Hyperbolic algebraic curve
- An algebraic curve whose genus and number of removed points are sufficiently large. Over the complex numbers, it is based on the hyperbolic plane.
- Section conjecture
- A conjecture asking whether the "sections" of the exact sequence of fundamental groups of a curve are limited to those coming from rational points on the curve
ResearchResearch themes
The lab studies the arithmetic geometry of hyperbolic algebraic curves, mainly from the perspectives of anabelian geometry and p-adic Teichmüller theory. Its past work includes the section conjecture, the anabelian conjecture, and combinatorial anabelian geometry.
*Illustrative imageResearch on the section conjecture
For hyperbolic algebraic curves over number fields and p-adic local fields, the lab constructed counterexamples to the pro-p version of the section conjecture. It also studies the geometricity of birational Galois sections for algebraic curves over the rational number field and imaginary quadratic fields.
*Illustrative imageAnabelian geometry of algebraic curves
The lab studied anabelian pro-ℓ cuspidalizations of projective hyperbolic algebraic curves over finite fields. It resolved negatively the problem of ℓ-independence of the quasi-monodromically full property, which was raised by Makoto Matsumoto and Akio Tamagawa.
*Illustrative imageAnabelian geometry of higher-dimensional algebraic varieties
The lab resolved the anabelian conjecture for hyperbolic polycurves of dimension at most 4 and for hyperbolic polycurves of strictly decreasing type.
*Illustrative imageCombinatorial anabelian geometry
In joint research with Arata Minamide and Shinichi Mochizuki, the lab established purely combinatorial and purely topological-group-theoretic descriptions of the Grothendieck–Teichmüller group.
MembersFaculty
- Associate ProfessorYuichiro Hoshi
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Kleinian group limit set on sphere” Adam majewski / CC BY-SA 4.0 (cropped)
- Research on the section conjecture: “Fermat curve” 不詳 / Public domain (cropped)
- Anabelian geometry of algebraic curves: “Hyperboloid-tiling-klein” PSL27 / CC BY-SA 4.0 (cropped)
- Anabelian geometry of higher-dimensional algebraic varieties: “16-cell skew octagram” Dc.samizdat / CC BY-SA 4.0 (cropped)
- Combinatorial anabelian geometry: “Topological recursion illustration” Eynard / CC BY-SA 4.0 (cropped)
- Related lab (無限解析研究部門
（望月 新一） ): “Kyoto University RIMS” KATSURA Roen / CC BY 3.0 (cropped) - Related lab (無限解析研究部門
（玉川 安騎男） ): “Klein-map” PSL27 / CC BY-SA 4.0 (cropped) - Related lab (無限解析研究部門
（梶野 直孝） ): “Pseudo Kleinian OpenCL K 002” PantheraLeo1359531 / CC BY-SA 4.0 (cropped) - Related lab (無限解析研究部門
（並河 良典） ): “Consani-Scholten” Ursulageorges / CC BY 4.0 (cropped)
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