FAMILIES OF THUE EQUATIONS ASSOCIATED WITH A RANK ONE SUBGROUP OF THE UNIT GROUP OF A NUMBER FIELD

Abstract

Let $K$ be an algebraic number field of degree $d\geqslant 3$, $\unicode[STIX]{x1D70E}_{1},\unicode[STIX]{x1D70E}_{2},\ldots ,\unicode[STIX]{x1D70E}_{d}$ the embeddings of $K$ into $\mathbb{C}$, $\unicode[STIX]{x1D6FC}$ a non-zero element in $K$, $a_{0}\in \mathbb{Z}$, $a_{0}>0$ and

$$\begin{eqnarray}F_{0}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC})Y).\end{eqnarray}$$

Let $\unicode[STIX]{x1D710}$ be a unit in $K$. For $a\in \mathbb{Z}$, we twist the binary form $F_{0}(X,Y)\in \mathbb{Z}[X,Y]$ by the powers $\unicode[STIX]{x1D710}^{a}$ ($a\in \mathbb{Z}$) of $\unicode[STIX]{x1D710}$ by setting

$$\begin{eqnarray}F_{a}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})Y).\end{eqnarray}$$

Given $m>0$, our main result is an effective upper bound for the size of solutions $(x,y,a)\in \mathbb{Z}^{3}$ of the Diophantine inequalities

$$\begin{eqnarray}0<|F_{a}(x,y)|\leqslant m\end{eqnarray}$$

for which $xy\not =0$ and $\mathbb{Q}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})=K$. Our estimate is explicit in terms of its dependence on $m$, the regulator of $K$ and the heights of $F_{0}$ and of $\unicode[STIX]{x1D710}$; it also involves an effectively computable constant depending only on $d$.