Development of an algorithm for the calculation of the Tate-Shafarevich groups and applicaiton

• Project/Area Number: 22740024
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Tsuda College
• Principal Investigator: MATSUNO KAZUO 津田塾大学, 学芸学部, 准教授 (40332936)
• Project Period (FY): 2010-04-01 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2013: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 整数論 / 楕円曲線 / Tate-Shafarevich群 / 代数学 / 岩澤理論

Research Abstract

We develop an algorithm for the calculation of the Tate-Shafarevich groups of elliptic curves defined over number fields. We also make a numerical verification of a conjecture by Greenberg for 2-adic Iwasawa mu-invariants of elliptic curves and 2-adic Iwasawa main conjecture for elliptic curves by using a relation between mu-invariants of elliptic curves and partly ramified Iwasawa modules associated with certain number fields. Moreoer, we find some examples related to the problem about the rank of the Mordell-Weil group of elliptic curves over number fields.

Research Products

• [Journal Article] 代数体上の楕円曲線の計算とMagma, Magma で広がる数学の世界 (2010)
  • Author(s): 松野 一夫
  • Journal Title: Math-for-Industry COE Lecture Note Volume: Vol.29 Pages: 134-144
• [Presentation] 楕円曲線のMordell-Weil 群について (2013)
  • Author(s): 松野 一夫
  • Organizer: 北陸数論研究集会
  • Place of Presentation: 金沢大学サテライトプラザ
  • Year and Date: 2013-12-27
• [Presentation] 2 等分点をもつ楕円曲線の岩澤μ不変量の計算 (2012)
  • Author(s): 松野 一夫
  • Organizer: 九州代数的整数論2012
  • Place of Presentation: 九州大学
  • Year and Date: 2012-02-22
• [Presentation] 2等分点をもつ楕円曲線の岩澤μ不変量の計算 (2012)
  • Author(s): 松野一夫
  • Organizer: 九州代数的整数論2012
  • Place of Presentation: 九州大学
  • Year and Date: 2012-02-22
• [Presentation] 岩澤λ不変量のRiemann-Hurwitz 公式とその周辺 (2010)
  • Author(s): 松野 一夫
  • Organizer: 北陸数論研究集会
  • Place of Presentation: 金沢大学サテライトプラザ
  • Year and Date: 2010-12-27
• [Presentation] 岩澤λ不変量のRiemann-Hurwitz公式とその周辺 (2010)
  • Author(s): 松野一夫
  • Organizer: 北陸数論研究集会
  • Place of Presentation: 金沢大学サテライトプラザ
  • Year and Date: 2010-12-27
• [Presentation] 代数体上の楕円曲線の計算とMagma (2010)
  • Author(s): 松野 一夫
  • Organizer: Magmaで広がる数学の世界
  • Place of Presentation: 九州大学
  • Year and Date: 2010-10-10
• [Presentation] 楕円曲線のMordell-Weil群について
  • Author(s): 松野 一夫
  • Organizer: 北陸数論研究集会
  • Place of Presentation: 金沢大学サテライトプラザ