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Infinite Analysis Lab (Mochizuki)
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The lab studies arithmetic geometry through anabelian geometry, centered on hyperbolic algebraic curves.
AboutAbout the lab
Exploring arithmetic geometry through curves and fundamental groups
The Mochizuki Lab studies arithmetic geometry.
Its research focuses on the arithmetic geometry of hyperbolic algebraic curves, including elliptic curves and algebraic curves of genus 2 or higher defined over number fields, local fields, and finite fields.
For hyperbolic algebraic curves, the outer action of the absolute Galois group of the base field on the non-abelian fundamental group is a natural starting point for arithmetic geometry.
The lab has worked on topics such as p-adic Teichmüller theory, p-adic anabelian geometry, and the Hodge-Arakelov theory of elliptic curves.
Key terms
- Arithmetic geometry
- A field of mathematics that studies properties of integers and solutions of equations using the language and tools of geometric objects (algebraic varieties)
- Elliptic curve
- A representative curve defined by a cubic equation, and an object that has been studied centrally in number theory
- Fundamental group
- A group formed by collecting the loops on a space, where loops that can be continuously deformed into one another are regarded as the same
- Anabelian geometry
- A theory that investigates how much of the original curve or other geometric object can be reconstructed from non-commutative fundamental groups or Galois groups
ResearchResearch themes
The lab studies the arithmetic geometry of elliptic curves and hyperbolic algebraic curves. Starting from p-adic Teichmüller theory, p-adic anabelian geometry, and the Hodge-Arakelov theory of elliptic curves, its research has expanded to absolute p-adic anabelian geometry and combinatorial anabelian geometry.
*Illustrative imageUnifying theories of anabelian geometry
Since 2000, the focus has shifted to absolute p-adic anabelian geometry and combinatorial anabelian geometry, and to the "interaction" and "fusion" between them.
*Illustrative imageInter-universal Teichmüller theory
A series of papers on "inter-universal Teichmüller theory" was released as preprints in August 2012 and published in 2021.
*Illustrative imageCombinatorial anabelian geometry
In joint research, the lab used the theory of "nodally nondegenerate outer representations" to prove a theorem on the injectivity of outer representations of the absolute Galois group of the base field.
MembersFaculty
- ProfessorShinichi Mochizuki
VoicesStudent voices
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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: “Kyoto University RIMS” KATSURA Roen / CC BY 3.0 (cropped)
- Unifying theories of anabelian geometry: “Fundamental domain of the Bolza surface” Joroco246 / CC BY-SA 4.0 (cropped)
- Inter-universal Teichmüller theory: “Limesmenge einer quasifuchsschen Gruppe” Café Bene / CC BY-SA 3.0 (cropped)
- Combinatorial anabelian geometry: “KQCHP” Greg Egan / CC BY-SA 4.0 (cropped)
- Related lab (Infinite Analysis Research Division (Hoshi) Lab): “Kleinian group limit set on sphere” Adam majewski / CC BY-SA 4.0 (cropped)
- Related lab (Tamagawa Lab): “Klein-map” PSL27 / CC BY-SA 4.0 (cropped)
- Related lab (Infinite Analysis Lab (Naotaka Kajino)): “Pseudo Kleinian OpenCL K 002” PantheraLeo1359531 / CC BY-SA 4.0 (cropped)
- Related lab (Infinite Analysis Lab (Odaka)): “Quartic Fermat curve” Jacob Rus / CC BY-SA 4.0 (cropped)
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