| Project/Area Number |
17H07074
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Single-year Grants |
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| Research Field |
Algebra
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| Research Institution | Gakushuin University |
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Principal Investigator |
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| Project Period (FY) |
2017-08-25 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 楕円曲線 / 保型表現 / モジュラー形式 / ラングランズ対応 / Galois表現 / 保型形式 / 整数論 / Hilbert保型形式 / (Hilbert)モジュラー形式 / Serre予想 |
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| Outline of Final Research Achievements |
This research is about modularity of elliptic curves over a totally real number field. This work generalizes the Shimura--Taniyama conjecture to the totally real case, which played the crucial role in the proof of the Fermat's last theorem.
The result we had before this research was on elliptic curves which are defined over certain "abelian" totally real fields. Because the assumption "abelian" is very strong, the main aim of this research was to weaken this assumption. It turned out during this research that, if we assume the field of definition splits at some primes dividing 13 or 47, we can prove modularity of many elliptic curves over the field without the abelian assumption.
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| Academic Significance and Societal Importance of the Research Achievements |
楕円曲線の保型性は、志村谷山予想の直接の一般化であり、数論において重要な予想であるラングランズプログラムやFontaine-Mazur予想の特殊な場合である。したがって、保型性が保証された楕円曲線の範囲を広げることにより、ラングランズプログラムやFontaine-Mazur予想のひとつの証左になりうる。また、楕円曲線にはL関数と呼ばれる重要な関数（楕円曲線の多くの情報を持った複素関数）が付随するが、楕円曲線の保型性を示すことで初めてL関数が良い解析的性質を持つことが保証されるという直接の利点もある。
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