Abstract
Let F(x; y) = ax3 + bx2y + cxy2 + dy3 ? Z[x,y] be an irreducible cubic form. In this paper, we investigate arithmetic properties of the common indices of algebraic integers in cubic fields. For each integer k such that v2(k)??0 (mod 3) and 2v2(-2b3 - 27a2d + 9abc) = 3v2(b2 - 3ac), we prove that the cubic Thue equation F(x,y) = k has no solution (x,y) ? Z2. As application, we construct parametrized families of twisted elliptic curves E : ax3 + bx2 + cx + d = ey2 without integer points (x,y).
Highlights
Let f (x, y) ∈ Z[x, y] be a homogeneous irreducible polynomial of degree n ≥ 3 and k be a non-zero integer
We investigate arithmetic properties of the common indices of algebraic integers in cubic fields
For each integer k such that v2(k) ≡ 0 and 2v2(−2b3 − 27a2d + 9abc) = 3v2(b2 − 3ac), we prove that the cubic Thue equation F (x, y) = k has no solution (x, y) ∈ Z2
Summary
Let f (x, y) ∈ Z[x, y] be a homogeneous irreducible polynomial of degree n ≥ 3 and k be a non-zero integer. Wakabayashi [42], using Baker’s method, proved that for any integer n ≥ 1.35 · 1014, the family of parametrized Thue equations x3 − n2xy2 + y3 = 1 has only trivial solutions (x, y) = (0, 1), (1, 0), (1, n2), (n, 1), (−n, 1). The equation ax3 + bx2 + cx + d = ey was studied by Mordell [29, pp.255-261] He proved the following important result: if the polynomial ax3 + bx2 + cx + d has no squared linear factor in x, the equation ax3 + bx2 + cx + d = ey has only a finite number of integer solutions.
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