Ishitsuka, Yasuhiro; Ito, Tetsushi; Ohshita, Tatsuya Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic. (English) Zbl 1491.11065 Int. J. Number Theory 16, No. 4, 881-905 (2020). Summary: 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. Cited in 1 Document MSC: 11G25 Varieties over finite and local fields 11D41 Higher degree equations; Fermat’s equation 11F80 Galois representations 11G10 Abelian varieties of dimension \(> 1\) 14H50 Plane and space curves 14K15 Arithmetic ground fields for abelian varieties 14K30 Picard schemes, higher Jacobians Keywords:Fermat quartic; Jacobian varieties; rational points; Galois representation Software:divisors.lib; Maxima; SINGULAR; SageMath PDFBibTeX XMLCite \textit{Y. Ishitsuka} et al., Int. J. Number Theory 16, No. 4, 881--905 (2020; Zbl 1491.11065) Full Text: DOI arXiv References: [1] Aigner, A., Uber die Möglichkeit von \(x^4+ y^4= z^4\) in quadratischen Körpern, Jber. Deutsch. Math.-Verein.43 (1934), 226-229. · Zbl 0008.29502 [2] J. Boehm, L. Kastner, B. Lorenz, H. Schönemann and Y. Ren, A Singular 4-1-1 library for computing divisors and P-Divisors, divisors.lib (2017). [3] Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron Models, , Vol. 21 (Springer-Verlag, Berlin, 1990). · Zbl 0705.14001 [4] W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, A computer algebra system for polynomial computations, Singular (Version 4-1-1) (2018); http://www.singular.uni-kl.de. [5] D. K. Faddeev, Group of divisor classes on the curve defined by the equation \(x^4+ y^4=1\), Soviet Math. Dokl.1 (1960) 1149-1151; Dokl. Akad. Nauk SSSR134 (1960) 776-777. (Russian original). · Zbl 0100.03401 [6] Howe, E. W., The Weil pairing and the Hilbert symbol, Math. Ann.305(2) (1996) 387-392. · Zbl 0854.11031 [7] Ishitsuka, Y., Ito, T. and Ohshita, T., On algorithms to obtain linear determinantal representations of smooth plane curves of higher degree, JSIAM Lett.11 (2019) 9-12. · Zbl 1409.14098 [8] Ito, T., On the Manin-Mumford conjecture for abelian varieties with a prime of supersingular reduction, Proc. Amer. Math. Soc.134(10) (2006) 2857-2860. · Zbl 1100.14036 [9] Kenku, M. A., Rational \(2^n\)-torsion points on elliptic curves defined over quadratic fields, J. London Math. Soc. (2)11(1) (1975) 93-98. · Zbl 0313.14002 [10] Kiming, I. and Rustom, N., Dihedral group, \(4\)-torsion on an elliptic curve, and a peculiar eigenform modulo \(4\), SIGMA14 (2018) 057. · Zbl 1422.11103 [11] M. J. Klassen, Algebraic points of low degree on curves of low rank, Thesis (Ph.D.)-The University of Arizona (1993) 51 pp.; http://hdl.handle.net/10150/186403. [12] Klassen, M. J. and Schaefer, E. F., Arithmetic and geometry of the curve \(y^3+1= x^4\), Acta Arith.74(3) (1996) 241-257. · Zbl 0838.14018 [13] Maxima.sourceforge.net. Maxima, a Computer Algebra System (Version 5.41.0) (2015); http://maxima.sourceforge.net/. [14] Mordell, L. J., The Diophantine equation \(x^4+ y^4=1\) in algebraic number fields, Acta Arith.14 (1968) 347-355. · Zbl 0191.04904 [15] Mumford, D., Abelian Varieties, , Vol. 5 (Oxford University Press, London, 1970). · Zbl 0223.14022 [16] B. Poonen, Rational Points on Varieties, Graduate Studies in Mathematics, Vol. 186 (American Mathematical Society, Providence, RI, 2017). · Zbl 1387.14004 [17] Rohrlich, D. E., Points at infinity on the Fermat curves, Invent. Math.39(2) (1977) 95-127. · Zbl 0357.14010 [18] SageMath, The Sage Developers (2017), The Sage Mathematics Software System (Version 8.1); http://www.sagemath.org. [19] Shimura, M., Defining equations of modular curves \(X_0(N)\), Tokyo J. Math.18(2) (1995) 443-456. · Zbl 0865.11052 [20] Tu, F.-T. and Yang, Y., Defining equations of \(X_0( 2^{2 n})\), Osaka J. Math.46(1) (2009) 105-113. · Zbl 1210.11069 [21] Tzermias, P., The group of automorphisms of the Fermat curve, J. Number Theory53(1) (1995) 173-178. · Zbl 0853.14015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.