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Algebra (Asakura) Lab
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The lab takes on the Beilinson conjecture, an unsolved problem in arithmetic geometry.
AboutAbout the lab
Tackling a conjecture that links periods and L-functions
Prof. Asakura studies arithmetic geometry using the theory of motives as a guide.
Motives are a vision conceived by Grothendieck in the 1960s; refined by many mathematicians since, they are now an active area of research.
Two of the Millennium Prize Problems, the Hodge conjecture and the Birch and Swinnerton-Dyer conjecture, are also deeply connected to this field.
Centered on the still-unsolved Beilinson conjecture, the research has recently been tackling problems by making active use of p-adic theory.
Key terms
- Arithmetic geometry
- A field that studies the properties of integers and equations using the methods of geometry, which deals with figures
- Motive
- A vision in arithmetic geometry that seeks to extract the properties shared by various geometric objects
- L-function
- A function that packs information about integers or geometric objects into an infinite sum or product. Its special values are considered to carry deep meaning
ResearchResearch themes
Within arithmetic geometry, the lab works with mixed motives and regulators. At the heart of its research is the Beilinson conjecture, which links periods to special values of L-functions.
*Illustrative imageThe Beilinson Conjecture
The conjecture predicts that "period (regulator) = special value of an L-function." Periods arise from algebraic K-theory and algebraic cycles, and an L-function that cannot be seen from those origins appears once the computation is carried out.
*Illustrative imageApproaching the Problem with p-adic Theory
Only very few examples are known in which the Beilinson conjecture has been verified, and even for elliptic curves alone, unsolved problems pile up. The lab has recently been addressing these problems by making active use of p-adic theory.
MembersFaculty
- ProfessorMasanori Asakura
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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: “Barth decic” Rocchini / CC BY-SA 3.0 (cropped)
- The Beilinson Conjecture: “Kummer surface” Claudio Rocchini / CC BY-SA 3.0 (cropped)
- Approaching the Problem with p-adic Theory: “Elliptic curve y^2=x^3-x on finite field Z 71” Technopop.tattoo / CC0 (cropped)
- Related lab (Mathematical Sciences (Masaki) Lab): “Baltic Sea Wave (Cien Water)” Marek Nowocień / CC BY 2.5 (cropped)
- Related lab (Geometry Lab (Inoguchi)): “Hopf Fibration” Niles Johnson / CC BY 4.0 (cropped)
- Related lab (Algebra (Daisuke Matsushita) Lab): “K3 surface” BTotaro / CC BY-SA 4.0 (cropped)
- Related lab (Analysis (Kubo) Lab): “Surface waves” Roger McLassus / CC BY-SA 3.0 (cropped)
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