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Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic. (English) Zbl 07205432
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
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[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
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