Proceedings of the Japan Academy, Series A, Mathematical Sciences

Local fields generated by 3-division points of elliptic curves

Hirotada Naito

Full-text: Open access

Abstract

We determine all the extensions generated by 3-division points of elliptic curves over the fields of $p$-adic numbers. As application, we construct $\mathit{GL}_2(\mathbf{F}_3)$-extensions over the field of rational numbers with given finitely many local conditions.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 78, Number 9 (2002), 173-178.

Dates
First available in Project Euclid: 23 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1148392352

Digital Object Identifier
doi:10.3792/pjaa.78.173

Mathematical Reviews number (MathSciNet)
MR1940375

Zentralblatt MATH identifier
1083.11039

Subjects
Primary: 11F85: $p$-adic theory, local fields [See also 14G20, 22E50] 11G05: Elliptic curves over global fields [See also 14H52] 11G07: Elliptic curves over local fields [See also 14G20, 14H52]

Keywords
Elliptic curves local fields Galois theory

Citation

Naito, Hirotada. Local fields generated by 3-division points of elliptic curves. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 9, 173--178. doi:10.3792/pjaa.78.173. https://projecteuclid.org/euclid.pja/1148392352


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References

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  • Bayer, P., and Rio, A.: Dyadic exercises for octahedral extensions. J. Reine Angew. Math., 517, 1–17 (1999).
  • Fujisaki, G.: A remark on quaternion extensions of the rational $p$-adic field. Proc. Japan Acad., 66A, 257–259 (1990).
  • Koike, M.: Higher reciprocity law, modular forms of weight 1 and elliptic curves. Nagoya Math. J., 98, 109–115 (1985).
  • Lario, J.-C., and Rio, A.: An octahedral-elliptic type equality in $Br_2(k)$. C. R. Acad. Sci. Paris Sér. I Math., 321, 39–44 (1995).
  • Lario, J.-C., and Rio, A.: Elliptic modularity for octahedral Galois representations. Math. Res. Lett., 3, 329–342 (1996).
  • Naito, H.: Dihedral extensions of degree $8$ over the rational $p$-adic fields. Proc. Japan Acad., 71A, 17–18 (1995).
  • Naito, H.: A congruence between the coefficients of the $L$-series which are related to an elliptic curve and the algebraic number field generated by its 3-division points. Mem. Fac. Edu. Kagawa Univ., 37, 43–45 (1987).
  • Naito, H.: Local fields generated by 3-division points of elliptic curves. RIMS Kokyuroku, 971, 153–159 (1996). (in Japanese).
  • Shimura, G.: A reciprocity law in non-solvable extensions. J. Reine Angew. Math., 221, 209–220 (1966).
  • Weil, A.: Exercises dyadiques. Invent. Math., 27, 1–22 (1974).