Families of Thue Equations Associated With a Rank One Subgroup of the Unit Group of a Number Field

Abstract

Abstract. Let K be an algebraic number field of degree d 3, σ 1 , σ 2 , . . . , σ d the embeddings of K into C, α a non-zero element in K , a 0 ∈ Z, a 0 > 0 andLet υ be a unit in K . For a ∈ Z, we twist the binary formGiven m > 0, our main result is an effective upper bound for the size of solutions (x, y, a) ∈ Z 3 of the Diophantine inequalitiesfor which x y = 0 and Q(αυ a ) = K . Our estimate is explicit in terms of its dependence on m, the regulator of K and the heights of F 0 and of υ; it also involves an effectively computable constant depending only on d. §1. Introduction and the main results. Let d 3 be a given integer. We denote by κ 1 , κ 2 , . . . , κ 38 positive effectively computable constants which depend only on d. Let K be a number field of degree d. Denote by σ 1 , σ 2 , . . . , σ d the embeddings of K into C and by R the regulator of K . Let α ∈ K , α = 0, and let a 0 ∈ Z, a 0 > 0, be such that the coefficients of the polynomialare in Z. Let υ be a unit in K , not a root of unity. For a ∈ Z, define the polynomial f a (X ) in Z[X ] and the binary form

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

“…Therefore, let us suppose max{n, a} sufficiently large, so we are free to use the notation O of Section 4. Our paper [6] takes care of the case where n is bounded by an absolute constant. This allows us to suppose n ≥ 1.…”

“…(iv) For n ≥ 0 with a ≥ 1, we have v a > 2v a−1 except for (n, a) = (0, 1) where (v 1 , v 0 ) = (2, 3) and for (n, a) = (0, 3) where (v 3 , v 2 ) = (11,6) and for (n, a) = (1, 1) where (v 1 , v 0 ) = (3,3).…”

“…(v) For n = 0 with a ≥ 1, we have 0 < (−1) a u a ≤ 1 2 v a , except for a = 2 where (u 2 , v 2 ) = (5,6).…”

“…In this paper, we deal with equations related to infinite families of cyclic cubic fields. In a forthcoming paper [6], we go one step further by considering twists by a power of a totally real unit.…”