TY - JOUR

T1 - Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic

AU - Ishitsuka, Yasuhiro

AU - Ito, Tetsushi

AU - Ohshita, Tatsuya

N1 - Funding Information:
The work of the first author was supported by the JSPS KAKENHI Grant Numbers 13J01450 and 16K17572. The work of the second author was supported by the JSPS KAKENHI Grant Numbers 20674001 and 26800013. The work of the third author was supported by JSPS KAKENHI Grant Numbers 26800011 and 18H05233. This work was supported by the Sumitomo Foundation FY2018 Grant for Basic Science Research Projects (Grant Number 180044).
Publisher Copyright:
© 2020 World Scientific Publishing Company.

PY - 2020/5/1

Y1 - 2020/5/1

N2 - 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.

AB - 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.

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U2 - 10.1142/S1793042120500451

DO - 10.1142/S1793042120500451

M3 - Article

AN - SCOPUS:85076620837

SN - 1793-0421

VL - 16

SP - 881

EP - 905

JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 4

ER -