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Arithmetic Geometry and Algebraic Geometry Lab
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The lab studies rational solutions of equations using techniques from higher-dimensional algebraic geometry.
AboutAbout the lab
Studying rational solutions of equations with algebraic geometry
Prof. Tanimoto studies algebraic geometry and arithmetic geometry.
For equations with rational coefficients, the field that investigates solutions whose coordinates are rational numbers is called Diophantine geometry, and it has been studied since the time of ancient Greece.
In modern mathematics, rational solutions are regarded as points on the geometric object (algebraic variety) defined by the equation, and are studied with the methods of algebraic geometry.
The professor has applied methods from higher-dimensional algebraic geometry, which have developed in recent years, to this problem and has achieved results.
Key terms
- algebraic variety
- A geometric object formed by collecting the points that satisfy a set of polynomial equations
- rational point
- A point on an algebraic variety whose coordinates are all rational numbers (numbers that can be written as fractions)
- moduli space
- A space formed by collecting all objects of a certain kind and regarding each one as a point
ResearchResearch themes
The lab studies rational solutions of equations with rational coefficients by regarding them as points on algebraic varieties (rational points). Since the professor's student days, the work has centered on Manin's conjecture, which concerns counting rational points on Fano varieties.
*Illustrative imageApplications of higher-dimensional algebraic geometry to arithmetic geometry
Manin's conjecture predicts that the asymptotic formula for the function counting rational points on a Fano variety can be expressed in terms of invariants of the variety. The lab investigates the birational-geometric background of this conjecture using methods from higher-dimensional algebraic geometry.
*Illustrative imageModuli spaces of curves on Fano varieties
Conversely, the lab applies ideas from arithmetic geometry to the study of the moduli space formed by all the curves lying on a Fano variety.
*Illustrative imageManin's conjecture over global function fields
The lab studies Manin's conjecture over global function fields by combining algebro-geometric and topological methods.
MembersFaculty
- ProfessorSho Tanimoto
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: “Algebraic–Geometry1” Jean Constant / CC BY-SA 4.0 (cropped)
- Applications of higher-dimensional algebraic geometry to arithmetic geometry: “Hexastix Glass Art” Artofglassmuseum / CC BY-SA 4.0 (cropped)
- Moduli spaces of curves on Fano varieties: “Calabi-Yau” Lunch / CC BY-SA 2.5 (cropped)
- Manin's conjecture over global function fields: “Stainless steel Möbius strip sculpture” Fumikas Sagisavas / CC0 (cropped)
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（藤原 一宏） ): “Clay tablet, mathematical, geometric-algebraic, similar to the Euclidean geometry. From Tell Harmal, Iraq. BCE. Iraq Museum” Osama Shukir Muhammed Amin FRCP(Glasg) / CC BY-SA 4.0 (cropped) - Related lab (組合せ論・
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