## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Local fields generated by 3-division points of elliptic curves

#### 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

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

#### 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

#### References

• 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). \beginthebibliography99
• 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).