Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic
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 8-th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the 8-th cyclotomic field. Thus we complete Faddeev's work in 1960.
Keywords
Cite
@article{arxiv.1804.07109,
title = {Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic},
author = {Yasuhiro Ishitsuka and Tetsushi Ito and Tatsuya Ohshita},
journal= {arXiv preprint arXiv:1804.07109},
year = {2019}
}
Comments
22 pages; minor revision; to appear in International Journal of Number Theory; This arXiv version contains a sample source code for Singular which performs necessary calculation on divisors on the Fermat quartic (see Appendix E)