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Tamagawa Lab
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The lab studies number theory and arithmetic geometry, centered on the fundamental groups of algebraic curves.
AboutAbout the lab
Exploring number theory through the fundamental groups of curves.
The Tamagawa Lab studies number theory and arithmetic geometry.
From the viewpoint of number theory, it investigates coverings and fundamental groups of algebraic curves and their moduli spaces.
At the Research Institute for Mathematical Sciences, research in this area is actively pursued, centered on anabelian geometry in the broad sense.
For the anabelian geometry of curves, the lab has obtained results over finite fields, over fields finitely generated over the rational number field, and over algebraically closed fields of positive characteristic.
Key terms
- Arithmetic geometry
- A field of mathematics that studies the properties of integers and of solutions to equations using the language and tools of geometric objects (algebraic varieties)
- Fundamental group
- A group formed by collecting the loops that go once around a geometric object, where loops that can be continuously deformed into one another are regarded as the same
- Anabelian geometry
- A theory that studies how much of the original object, such as a curve, can be recovered from its non-commutative fundamental group or Galois group
- Galois representation
- A representation of the Galois group, which expresses the symmetry of the solutions of an equation, as a group of matrices
ResearchResearch themes
The lab studies arithmetic geometry concerning coverings and fundamental groups of algebraic varieties, especially algebraic curves and their moduli spaces. Centered on anabelian geometry, it has obtained results through joint research on Galois representations of fundamental groups and on the arithmetic of abelian varieties.
*Illustrative imageAnabelian Geometry of Curves over Finite Fields
In connection with the section conjecture, the lab introduced the discrete Selmer group and the discrete Shafarevich-Tate group, which are new arithmetic-geometric invariants of abelian varieties.
*Illustrative imageBehavior of the Images of Systems of l-adic Representations
In joint research, the lab studied the images of the Galois representations obtained by restricting l-adic representations of the arithmetic fundamental group of a curve to the decomposition groups of closed points. It has also proved results such as a Tannakian Chebotarev density theorem.
*Illustrative imageSemisimplicity of Geometric Monodromy
In joint research, the lab obtained a strong result on the mod l semisimplicity of geometric monodromy, corresponding to a mod l version of Deligne's theorem.
*Illustrative imageIhara's Problem and Abelian Varieties
The lab conjectured that there are only finitely many pairs consisting of an isomorphism class of abelian varieties satisfying certain conditions and a prime l, and has obtained affirmative solutions in several cases. It also studies the l-power torsion points of Jacobian varieties of l-power cyclic covers of the projective line.
MembersFaculty
- ProfessorAkio Tamagawa
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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: “Klein-map” PSL27 / CC BY-SA 4.0 (cropped)
- Anabelian Geometry of Curves over Finite Fields: “Pants in the plane” Jean Raimbault / CC BY-SA 4.0 (cropped)
- Behavior of the Images of Systems of l-adic Representations: “Cayley graph of Alternating Group A7” Kathryn Cramer / CC BY-SA 4.0 (cropped)
- Semisimplicity of Geometric Monodromy: “Riemann sqrt” Wlongqi / CC BY-SA 3.0 (cropped)
- Ihara's Problem and Abelian Varieties: “Tetrahedroid” Salix alba / CC BY-SA 4.0 (cropped)
- Related lab (無限解析研究部門
（星 裕一郎） ): “Kleinian group limit set on sphere” Adam majewski / 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
（並河 良典） ): “Consani-Scholten” Ursulageorges / CC BY 4.0 (cropped) - Related lab (無限解析研究部門
（尾髙 悠志） ): “Quartic Fermat curve” Jacob Rus / CC BY-SA 4.0 (cropped)
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