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Research on the arithmetic of algebraic curves and jacobian varieties

Research Project

Project/Area Number 09640075
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionWaseda University

Principal Investigator

HASHIMOTO Kiichiro  Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (90143370)

Co-Investigator(Kenkyū-buntansha) HASEGAWA Yuji  Waseda Univ. School of Sci. and Eng., (JSPS Fellow), 特別研究員 (30287982)
ADACHI Norio  Waseda Univ. School of Sci. and Eng., Professor, 理工学部, 教授 (60063731)
KOMATSU Keiichi  Waseda Univ. School of Sci. and Eng., Professor, 理工学部, 教授 (80092550)
KAGAWA Takaaki  Waseda Univ. School of Sci. and Eng., Assistant, 理工学部, 助手(平10) (90298175)
OZAKI Manabu  Waseda Univ. School of Sci. and Eng., Assistant, 理工学部, 助手(平9) (80287961)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1999: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1998: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1997: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordselliptic curves / modular curves / Taniyama-Simura Conjecture / jacobian variety / modular forms / abelian varieties / algebraic curvers / Q-curves / modularity / アーベル曲面 / Q-曲線 / テータ級数 / 保型形式
Research Abstract

In 1994 Wiles and Taylor have settled the proof of Taniyama-Shimura conjecture for (semistable) elliptic curves over Q. This, with its application to the proof of Fermat's Last Theorem, was one of the greatest achievment in this century. In our previous research, we extended the result of Wiles-Taylor proving the modularity of certain abelian varieties over Q, including Q-curves over number fields, and jacobians of QM-curves of GL (2) -type. The aim of the present research has been to provide as many as possible the concrete examples of algebraic curves over Q, for which our modularity criterion for their jacobian can be applied, as well as to investigate various arithmetic properties of such curves. Some of our main results are :
・ We obtained some families of genus 2 curves over Q whose jacobian varieties are of GL (2) -type, and checked their modularity numerically and theoretically.
・ Conversely, for each cusp f (z) of weight 2 whose Fourier coefficients generate a quadratic field K, we tried to find an algebraic curve over QィイD4-ィエD4 shose jacobian variety is isogenous to the Shimura's abelian surface AィイD2fィエD2 attached to f. We have settled this problem in all known cases for K = Q(ィイD8-5ィエD8), Q (ィイD8-1ィエD8). There are 11 such f.
・ We constructed the most general family with 7 free parameters, of genus 2 curves over Q which form a double cuver of a family of elliptic curves. Among them we found a generic family of the covering C (j) → E (j) where E (j) is the Tate's model of elliptic curve with j (E (j) ) = j. Then the simple factor of JacC (j) is shown to be a Q-curve over quadratic field Q (ィイD8j-12ィイD13ィエD1ィエD8).

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (37 results)

All Other

All Publications (37 results)

  • [Publications] Ki-ichiro Hashimoto: "Q-curves of degree 5 and jacobian surfaces of GL_2-type"Manuscripta Mathematica. 98. 165-182 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ki-ichiro Hashimoto: "Modularity conjecture for Q-curves and QM-curves"International J. Math.. 10-7. 1011-1036 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ki-ichiro Hashimoto: "On the Sato-Tate Conjecture for QM-curves of Genus Two"Mathematics of Computation. 68-228. 1649-1662 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ki-ichiro Hashimoto: "Inverse Galois Problem for Dihedral Groups"Number Theory and its Applications(Kluwer). 2. 165-181 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ki-ichiro Hashimoto: "Linear relations of theta series attached to Eichler orders of quaternion algebras"Contemporary Mathematics(AMS). 249. 262-302 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Yuji Hasegawa: "Hyperelliptic quotients of modular curves Xo(N)"Tokyo Journal of Mathematics. 22. 105-125 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Yuji Hasegawa: "Hyperelliptio modular curves Xo^<^*>(N)"ACTA ARITHMETICA. 81. 369-385 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Yuji Hasegawa: "Q-curves over quadratic fields"manuscript mathematics. 94. 347-369 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Yuji Hasegawa: "Trigonal modular curves"ACTA ARITHMETICA. 81. 129-140 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kiichiro Hashimoto: "Q-curves of degree 5 and jacobian surfaces of GL2-type"Manuscripta Math. 98. 165-182 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kiichiro Hashimoto, Yuji HasegawaFumiyuki Momose: "Modularity conjecture for Q-curves and QM-curves"International J. Math. vol.10-7. 1011-1036 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kiichiro Hashimoto Hiroshi Tsunogi: "On the Sato-Tate Conjecture for QM-curves of Genus Two"Mathematics of Computation. 68 no.228. 1649-1662 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kiichiro Hashimoto Katsuya Miyake: "Inverse Galois Problem for Dihedral Groups"Number Theory and its Applications Kluwer. 165-181 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kiichiro Hashimoto: "Linear relations of theta series attached to Eichler orders of quaternion algebras"Contemporary Math 249 (AMS). 262-302

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Yuji Hasegawa: "Hyperelliptic modular curves XOィイD1^★ィエD1(N)"ACTA ARITHMETICA. 81. 369-385 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Yuji Hasegawa: "Q-curves over quadratic fields"manuscripta mathematica. 94. 347-364 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Yuji Hasegawa Mahoro Shimura: "Trigonal modular curves"ACTA ARITHMETICA. 88. 129-140 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Yuji Hasegawa: "Hyperelliptic quotients of modular curves X_O (N)"Tokyo Journal of Mathematics. 22. 105-125 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ki-ichiro hashimoto: "Q-curves of degree 5 and jacobian surfaces of GL_2-type"Manuscripta Mathematica. 98. 165-182 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Ki-ichiro Hashimoto: "Modularity conjecture for Q-curves and QM-curves"International J.Math.. 10-7. 1011-1036 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Ki-ichiro Hashimoto: "On the Sato-Tate Conjecture for QM-curves of Genus Two"Mathematics of Computation. 68-228. 1649-1662 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Ki-ichiro Hashimoto: "Inverse Galois Problem for Dihedral Groups"Number Theory and its Applications (Kluwer). 2. 165-181 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Ki-ichiro Hashimoto: "Linear relations of theta series attached to Eichler orders of quaternion algebras"Contemporary Mathematics (AMS). 249. 262-302 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Ki-ichiro Hashimoto: "Modularity conjecture for Q-curves and QM-curves" International Journal of Math.(予定). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Ki-ichiro Hashimoto: "On the Sato-Tate Conjecture for QM-curves of genus two" Mathematics of Computations. (予定). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Ki-ichiro Hashimoto: "Q-curves of degree 5 and jacobian survaces of GL_2-type" Manuscripta Mathematica. (予定). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Ki-ichiro Hashimoto: "Inverse Galois Problem for Dihedral Groups" Technical Report Adv.Research Inst.Waseda. 98-4. 1-17 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Keiichi Komatsu: "On the Z_3-extension of a certain cubic cyclic field" Proceedings of the Japan Academy. 74-10. 165-166 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Keiichi Komatsu: "On the Iwasawa λ-invariants of quaternion extensions" Acta Arithmetica. 137-3. 219-221 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 橋本喜一朗・百瀬文之(他): "楕円曲線とそのArithmetic Moduli(第6回整数論サマースクール報告集)" 橋本喜一朗(早稲田大学)・百瀬文之(中央大学), 205 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 橋本喜一郎: "Modular conjecture for Q-curves and QM-curves (with Y.Hasegawa,F.Momose)" to appear in International J.of Math.

    • Related Report
      1997 Annual Research Report
  • [Publications] 橋本喜一郎: "On the Sato-Tate Conjecture for QM-curves of genus two" to appear in Mathematics of Computations.

    • Related Report
      1997 Annual Research Report
  • [Publications] 橋本喜一郎: "Q-curves of degree 5 and jacobian surfaces of GL_2-type," Technical Report No.97-8, Advanced Research Inst for Sci.& Eng.,Waseda Univ.97-8. 1-20 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M.Kida and T.Kagawa: "Nonexistence of elliptic curves with good reduction everywhere over real quadratic fields," J.Number Theory. 66. 201-210 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Kagawa: "Determination of elliptic curves with everywhere good reduction over Q (√<37>)," Acta Arith.83. 253-269 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Komatsu: "Construction of normal basis by special values at Hilbert modular functions," Proc.Japan Acad.,. 73-3. 42-44 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 橋本喜一郎(他16名): "「数学教育とコンピュータ」(早稲田大学教育総合研究室叢書3)" 学文社(守屋悦朗 編), 254 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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