Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic

Yasuhiro Ishitsuka, Tetsushi Ito, Tatsuya Ohshita

研究成果: Contribution to journalArticle査読

抄録

We use explicit methods to study the 4-torsion points on the Jacobian variety of the Fermat quartic. With the aid of computer algebra systems, we explicitly give a basis of the group of 4-torsion points. We calculate the Galois action, and show that the image of the mod 4 Galois representation is isomorphic to the dihedral group of order 8. As applications, we calculate the Mordell-Weil group of the Jacobian variety of the Fermat quartic over each subfield of the 8th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the 8th cyclotomic field. Thus, we complete Faddeev's work in 1960.

本文言語英語
ページ(範囲)881-905
ページ数25
ジャーナルInternational Journal of Number Theory
16
4
DOI
出版ステータス出版済み - 5 1 2020
外部発表はい

All Science Journal Classification (ASJC) codes

  • 代数と数論

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