Families of Cubic Thue Equations with Effective Bounds for the Solutions

Abstract

To each non totally real cubic extension K of Q and to each generator α of the cubic field K, we attach a family of cubic Thue equations, indexed by the units of K, and we prove that this family of cubic Thue equations has only a finite number of integer solutions, by giving an effective upper bound for these solutions. StatementsLet us consider an irreductible binary cubic form having rational integers coefficientswith the property that the polynomial F(X, 1) has exactly one real root α and two complex imaginary roots, namely α ′ and α ′ . Hence α ∈ Q, α ′ = α ′ andLet K be the cubic number field Q(α) which we view as a subfield of R. Define σ : K → C to be one of the two complex embeddings, the other one being the conjugate σ . Hence α ′ = σ (α) and α ′ = σ (α). If τ is defined to be the complex conjugation, we have σ = τ • σ and σ • τ = σ .

Solving simultaneously Thue equations in the almost totally imaginary case

“…Given an algebraic number γ of degree ≤ d and norm ≤ m, there exists a unit ε in the field Q(α) such that the conjugates of εγ are bounded from above by a constant times m 1/d . This is a consequence of Lemma A.15 of [3], a result which we want to state; (this is also Lemma 2 of [2]).…”

“…We will use some results in geometry of numbers to establish an equivalence of norms (Lemma 5). Then we state in Lemma 6 what is Lemma 2 of [2]. Finally in Proposition 8 and in Corollaries 9 and 10 we exhibit some lower bounds of linear forms of logarithms of algebraic numbers.…”

“…If the algebraic number field K has a unique real embedding into C, the degree d of K is odd and K has (d − 1)/2 pairs of complex embeddings. In the particular case d = 3, namely for a cubic field K, the hypothesis that there is only one real embedding boils down to requiring that the unit rank of K be 1, and this particular case of Theorem 2 was obtained in [2].…”

A family of Thue equations involving powers of units of the simplest cubic fields

“…It happens that this paper is the fourth one in which we use the effective method arising from Baker's work on linear forms of logarithms for families of Thue equations obtained via a twist of a given equation by units. The first paper [3] was dealing with non totally real cubic fields; the second one [4] was dealing with Thue equations attached to a number field having at most one real embedding. In the third paper [5], for each (irreducible) binary form attached to an algebraic number field, which is not a totally real cubic field, we exhibited an infinite family of equations twisted by units for which Baker's method provides effective bounds for the solutions.…”

“…As in [10], we use λ = λ 0 , λ 1 and λ 2 , but in [8] these elements correspond respectively to λ (3) , λ (1) and λ (2) .…”