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Ochiai Lab
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The lab studies number theory and arithmetic geometry, especially Iwasawa theory.
AboutAbout the lab
Number theory through the lens of Iwasawa theory.
The Ochiai Lab studies number theory and arithmetic geometry.
In its research talks, the lab covers topics such as p-adic L-functions, the Iwasawa main conjecture, and congruences between automorphic representations.
The lab has also written a specialist book on Iwasawa theory, Iwasawa Theory and Its Perspectives (Parts 1 and 2; in Japanese).
For their fourth-year graduation research, students have read textbooks on algebraic number theory, class field theory, and elliptic curves.
Key terms
- Arithmetic geometry
- A field that uses the properties of geometric objects (algebraic varieties) to study properties of integers, such as the integer solutions of equations
- Iwasawa theory
- A theory that considers an infinite tower of number systems stacked one on top of another and studies properties of integers through the behavior of that tower
- p-adic L-function
- A function obtained by reinterpreting an L-function in the world of p-adic numbers, focusing on a prime p
ResearchResearch themes
*Illustrative imageThe Main Conjecture of Iwasawa Theory
In Iwasawa theory, there is a main conjecture stating that a quantity expressing the "distortion" of the world of numbers coincides with a p-adic function constructed from an L-function. The lab treats this conjecture in broad cases using families of automorphic forms and explores paths toward a proof.
*Illustrative imageCongruences between Automorphic Forms
Sometimes the coefficients of different automorphic forms agree in the world of remainders modulo a certain prime, a phenomenon called a congruence. The lab studies how congruences arise, for example by showing that, for the group GSp(4), such congruences can be measured by values of L-functions.
MembersFaculty
- ProfessorTadashi OchiaiPublications ↗
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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: “Q-Eulero” Linas Vepstas / CC BY-SA 3.0 (cropped)
- The Main Conjecture of Iwasawa Theory: “3-adic integers” LightbulbMEOW / CC BY-SA 4.0 (cropped)
- Congruences between Automorphic Forms: “Modular subgroup, fundamental domain” Glakeland / CC0 (cropped)
- Related lab (Yatagawa Lab): “Riemann surface sqrt” Jan Homann / Public domain (cropped)
- Related lab (Kalman Lab): “Trefoil knot build with trebl torus type construction set bricks” Dutchdudedesign / CC BY-SA 4.0 (cropped)
- Related lab (Honda Lab): “Calabi yau” Jbourjai / Public domain (cropped)
- Related lab (Purkait Lab): “J-inv-phase” Linas Vepstas / CC BY-SA 3.0 (cropped)
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