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A “class group” obstruction for the equation \(Cy^d=F(x,z)\). (English) Zbl 1204.11063
In this paper, Diophantine equations of the form \[ Cy^d = F(x,z)\tag{1} \] are investigated, where \(F \in \mathbb Z[x,z]\) is an irreducible, primitive form of degree \(n \geq 2\), and \(C \neq 0\), \(d\geq2\) are integers. Let \(K = \mathbb Q (\theta)\) be the algebraic number field, given by \(F(\theta, 1)=0\). The author [C. R., Math., Acad. Sci. Paris 336, No. 1, 7–10 (2003; Zbl 1038.11073)] associates to the form \(F\) an order \(\mathbb Z_F\) within the ring of integers of \(K\), an ideal \(\mathfrak b\) of \(\mathbb Z_F\), and defines the class of \(F\) to be the class of \(\mathfrak b\) within the class group \(Cl(\mathbb Z_F)\). Each proper solution \((x_0, y_0, z_0)\) of (ref {1}) gives rise to an ideal \(\mathfrak D = \mathfrak b (x_0- \theta z_0)\) of \(\mathbb Z_F\) with very special properties, as described in Theorem 3.
From this, necessary conditions for the existence of proper solutions of (ref {1}) are derived, in more detail, there must exist some ideal \(\mathfrak c\) of \(\mathbb Z_F\) with norm equal to \(C\) and divisible only by degree \(1\) prime ideals, and the classes of \(\mathfrak c\) and of \(F\) should only differ by a \(d\)-th power inside \(Cl(\mathbb Z_F)\). For \(n=2\) this reduces to previous results of Cassels and Darmon-Granville.
11D41 Higher degree equations; Fermat’s equation
11R29 Class numbers, class groups, discriminants
ecdata; PARI/GP; Magma
Full Text: DOI Numdam EuDML
[1] B.J. Birch, H.P.F. Swinnerton-Dyer, Notes on elliptic curves. J. Reine Angew. Math. 212 (1963), 7-25. · Zbl 0118.27601
[2] J.W.S. Cassels, Rational Quadratic Forms. L.M.S. Monographs, Academic Press, 1978. · Zbl 0395.10029
[3] J.E. Cremona, Algorithms for Modular Elliptic Curves. Cambridge University Press, 1997, Second Edition. · Zbl 0872.14041
[4] J.E. Cremona, Elliptic Curve Data. http://modular.math.washington.edu/cremona/INDEX.html · JFM 01.0253.01
[5] H. Cohen, Number Theory. Vol I : Tools and Diophantine Equations. GTM 239, Springer Verlag, 2007. · Zbl 1119.11001
[6] H. Darmon, A. Granville, On the equations \(z^m=F(x,y)\) and \(Ax^p+By^q=Cz^r\). Bull. London Math. Soc 27 (1995), 513-543. · Zbl 0838.11023
[7] I. Delcorso, R. Dvornicich, D. Simon, The decomposition of primes in nonmaximal orders. Acta Arithmetica 120 (2005), 231-244. · Zbl 1163.11342
[8] F. G. Frobenius, Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1896), 689-703; Gesammelte Abhandlungen II, 719-733. · JFM 27.0091.04
[9] E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, 2te Unveränderte Auflage (1954), Leipzig, Akademische Verlagsgesellschaft (1923). · Zbl 0057.27301
[10] S. Lang, Algebraic number theory, GTM 110, second edition. New York, Springer-Verlag, 1994. · Zbl 0811.11001
[11] MAGMA Computational Algebra System, http://magma.maths.usyd.edu.au/magma/
[12] J. R. Merriman, S. Siksek, N. P. Smart, Explicit 4-descent on an elliptic curve. Acta Arithmetica 77 (1996), 385-404. · Zbl 0873.11036
[13] pari/gp, The Pari group (K. Belabas, H. Cohen,...). http://pari.math.u-bordeaux.fr/ · Zbl 1441.11003
[14] I. Reiner, Maximal Orders. L.M.S. Monographs, Academic Press, 1975. · Zbl 0305.16001
[15] D. Simon, La classe invariante d’une forme binaire. Comptes Rendus Mathématiques 336, Issue 1 , (2003) 7-10. · Zbl 1038.11073
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