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

Yasuhiro Ishitsuka, Tetsushi Ito, Tatsuya Ohshita

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)881-905
Number of pages25
JournalInternational Journal of Number Theory
Volume16
Issue number4
DOIs
Publication statusPublished - May 1 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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