Stratifications of the moduli space of abelian varieties and that of curves

2020 Fiscal Year Final Research Report

• Project/Area Number: 17K05196
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Yokohama National University
• Principal Investigator: Shushi Harashita 横浜国立大学, 大学院環境情報研究院, 准教授 (70396852)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: 代数幾何学 / アーベル多様体 / 代数曲線 / モジュライ空間 / 超特異

Outline of Final Research Achievements

Abelian varieties and algebraic curves treated in this research play central roles in algebraic geometry of mathematics and have many applications in other areas. Socially, there are objects (supersingular curves) that will be used in next-generation information technology such as post-quantum cryptography, and we are intensely studying those objects. As research results, we determine specializations of p-divisible groups (which are attached to abelian varieties and algebraic curves) and enumerate superspecial curves (of various types) of low genera and in small characteristics (using PC computations), and show the existence of a superspecial curve of genus 4 in arbitrary characteristic (theoretical result) and so on.

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

本研究では、アーベル多様体や代数曲線のモジュライ空間に入る階層構造や葉層構造を系統的に研究している。理論的研究と共に、応用面でも利用される超特異アーベル多様体や超特異代数曲線、またそれらのなす空間についても深く調査している。特に、超特異代数曲線やそれらの間の同種写像は、耐量子計算機暗号等にも用いられ、本研究が社会的に意義のある成果を生み出す可能性がある。