Elliptic curves with no rational points
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- by Jin Nakagawa and Kuniaki Horie
- Proc. Amer. Math. Soc. 104 (1988), 20-24
- DOI: https://doi.org/10.1090/S0002-9939-1988-0958035-0
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Abstract:
The existence of infinitely many elliptic curves with no rational points except the origin $\infty$ is proved by refining a theorem of Davenport-Heilbronn. The existence of infinitely many quadratic fields with the Iwasawa invariant ${\lambda _3} = 0$ is proved at the same time.References
- A. Aigner, Über die Möglichkeit von ${x^4} + {y^4} = {z^4}$ in quadratischen Körpern, Jber. Deutsch. Math. Verein. 43 (1934), 226-229.
- J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), no. 3, 223–251. MR 463176, DOI 10.1007/BF01402975
- H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields, Bull. London Math. Soc. 1 (1969), 345–348. MR 254010, DOI 10.1112/blms/1.3.345
- H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405–420. MR 491593, DOI 10.1098/rspa.1971.0075 R. Fueter, Die Diophantische Gleichung ${\xi ^3} + {\eta ^3} + {\varsigma ^3} = 0$, Sitzungsberichte Heidelberg Akad. Wiss. 25. Abh. (1913), 25 pp.
- Rudolf Fueter, Ueber kubische diophantische Gleichungen, Comment. Math. Helv. 2 (1930), no. 1, 69–89 (German). MR 1509405, DOI 10.1007/BF01214451
- Benedict H. Gross and David E. Rohrlich, Some results on the Mordell-Weil group of the Jacobian of the Fermat curve, Invent. Math. 44 (1978), no. 3, 201–224. MR 491708, DOI 10.1007/BF01403161
- P. Hartung, Proof of the existence of infinitely many imaginary quadratic fields whose class number is not divisible by $3$, J. Number Theory 6 (1974), 276–278. MR 352040, DOI 10.1016/0022-314X(74)90022-5
- Kuniaki Horie, A note on basic Iwasawa $\lambda$-invariants of imaginary quadratic fields, Invent. Math. 88 (1987), no. 1, 31–38. MR 877004, DOI 10.1007/BF01405089
- L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
- Trygve Nagell, Sur la résolubilité de l’équation $x^{2}+y^{2}+z^{2}=0$ dans un corps quadratique, Acta Arith. 21 (1972), 35–43 (French). MR 302558, DOI 10.4064/aa-21-1-35-43
- J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (French). MR 646366
Bibliographic Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 20-24
- MSC: Primary 11R45; Secondary 11D25, 11G05, 11R11, 14G25
- DOI: https://doi.org/10.1090/S0002-9939-1988-0958035-0
- MathSciNet review: 958035