Construction of elliptic curves with high rank via the invariants of the Weyl groups

Tetsuji SHIODA

doi:10.2969/jmsj/04340673

Tetsuji SHIODA

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JOURNAL FREE ACCESS

1991 Volume 43 Issue 4 Pages 673-719

DOI https://doi.org/10.2969/jmsj/04340673

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Published: 1991 Received: July 20, 1990 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : Dedicated to Professor G. Shimura on his 60th birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : Br) C. Bramble, A collineation group isomorphic with the group of the double tangents of the plane quartic, Amer. J. Math., XL (1918), 351-365.
B) N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 et 6, Hermann, Paris, 1968.
CS) J. Conway and N. Sloane, Sphere Packings, Lattices and Groups, Grundlehren Math. Wiss., 290, Springer, 1988.
F) J. Frame, The classes and representations of the groups of 27 lines and 28 bitangents, Annali di Mat. Ser. IV, 32 (1951), 83-119.
K) K. Kodaira, On compact analytic surfaces II-III, Ann. of Math., 71(1963), 563-626; 18, 1-40 (1963); Collected Works, vol. III, Iwanami and Princeton Univ. Press, (1975), 1269-1372.
M) Ju. Manin, Cubic Forms, 2nd ed., North-Holland, 1986.
N1) A. Néron, Propriétés arithmétiques de certaines familles de courbes algébriques, Proc. Intern. Math. Congr. Amsterdam 1954, vol. III, 481-488.
N2) A. Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math. I. H. E. S., 21 (1964).
OS) K. Oguiso and T. Shioda, The Mordell-Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Pauli, 40 (1991), 83-99.
S1) T. Shioda, Mordell-Weil lattices and Galois representation, I, II, III, Proc. Japan Acad., 65A (1989), 267-271, 296-299, 300-303.
S2) F. Shioda, Construction of elliptic curves over Q(t) with high rank: a preview, Proc. Japan Acad., 66A (1990), 57-60.
S3) T. Shioda, Mordell-Weil lattices and sphere packings, to appear in Amer. J. Math..
S4) T. Shioda, Mordell-Weil lattices of type E₈ and deformation of singularities, in Prospects in Complex Geometry, SLN, 1468 (1991), 177-202.
S5) T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli, 39 (1990), 211-240.
S6) T. Shioda, Theory of Mordell-Weil lattices, to appear in Proc. ICM 1990, Kyoto.
S7) T. Shioda, An infinite family of elliptic curves over Q with large rank via Néron's method, to appear in Invent. Math..
Si) J. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1986.
T1) J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Lecture Notes in Math., 476 (1975), 33-52.
T2) J. Tate, Variation of the canonical height of a point depending on a parameter, Amer. J. Math., 105 (1983), 287-294.
W1) A. Weil, Abstract versus classical algebraic geometry, Proc. Intern. Math. Congr. Amsterdam 1954, vol. III, 550-558; Collected Papers, vol. II, Springer-Verlag, 1980, 180-188.
W2) A. Weil, Foundations of Algebraic Geometry, AMS, 1962.

Right : [Br] C. Bramble, A collineation group isomorphic with the group of the double tangents of the plane quartic, Amer. J. Math., XL (1918), 351-365.
[B] N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 et 6, Hermann, Paris, 1968.
[CS] J. Conway and N. Sloane, Sphere Packings, Lattices and Groups, Grundlehren Math. Wiss., 290, Springer, 1988.
[F] J. Frame, The classes and representations of the groups of 27 lines and 28 bitangents, Annali di Mat. Ser. IV, 32 (1951), 83-119.
[K] K. Kodaira, On compact analytic surfaces II-III, Ann. of Math., 77 (1963), 563-626; 78, 1-40 (1963); Collected Works, vol. III, Iwanami and Princeton Univ. Press, (1975), 1269-1372.
[M] Ju. Manin, Cubic Forms, 2nd ed., North-Holland, 1986.
[N1] A. Néron, Propriétés arithmétiques de certaines familles de courbes algébriques, Proc. Intern. Math. Congr. Amsterdam 1954, vol. III, 481-488.
[N2] A. Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math. I. H. E. S., 21 (1964).
[OS] K. Oguiso and T. Shioda, The Mordell-Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Pauli, 40 (1991), 83-99.
[S1] T. Shioda, Mordell-Weil lattices and Galois representation, I, II, III, Proc. Japan Acad., 65A (1989), 267-271, 296-299, 300-303.
[S2] F. Shioda, Construction of elliptic curves over Q(t) with high rank: a preview, Proc. Japan Acad., 66A (1990), 57-60.
[S3] T. Shioda, Mordell-Weil lattices and sphere packings, to appear in Amer. J. Math..
[S4] T. Shioda, Mordell-Weil lattices of type E₈ and deformation of singularities, in Prospects in Complex Geometry, SLN, 1468 (1991), 177-202.
[S5] T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli, 39 (1990), 211-240.
[S6] T. Shioda, Theory of Mordell-Weil lattices, to appear in Proc. ICM 1990, Kyoto.
[S7] T. Shioda, An infinite family of elliptic curves over Q with large rank via Néron's method, to appear in Invent. Math..
[Si] J. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1986.
[T1] J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Lecture Notes in Math., 476 (1975), 33-52.
[T2] J. Tate, Variation of the canonical height of a point depending on a parameter, Amer. J. Math., 105 (1983), 287-294.
[W1] A. Weil, Abstract versus classical algebraic geometry, Proc. Intern. Math. Congr. Amsterdam 1954, vol. III, 550-558; Collected Papers, vol. II, Springer-Verlag, 1980, 180-188.
[W2] A. Weil, Foundations of Algebraic Geometry, AMS, 1962.

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