# zbMATH — the first resource for mathematics

$$p$$-adic logarithmic forms and group varieties. II. (English) Zbl 0928.11031
Let $$K$$ be a number field of degree $$d$$, $$\alpha_{1},\ldots,\alpha_{n}$$ non zero elements in $$K$$, $$b_{1},\ldots,b_{n}$$ rational integers such that the number $$\Xi=\alpha_{1}^{b_{1}}\cdots\alpha_{n}^{b_{n}} -1$$ is not zero. Let $$\mathfrak p$$ be a prime ideal in the ring of integers of $$K$$ above a prime number $$p$$. The goal is to establish an explicit upper bound for $$\text{ord}_{\mathfrak p}(\Xi)$$, namely the exponent to which $$\mathfrak p$$ divides the principal fractional ideal $$(\Xi)$$ in $$K$$. Denote by $$h$$ the absolute logarithmic height and define $$h_{1},\ldots,h_{n}$$ by $h_{j}=\max\{h(\alpha_{j}),\log p\} \qquad (1\leq j\leq n).$ Let $$B\geq 3$$ be a positive integer such that $$| b_{i}| \leq B$$ for $$1\leq i\leq n$$. As a special case of his main estimate, the author deduces $\text{ord}_{\mathfrak p}(\Xi)< C(n,d,{\mathfrak p})h_{1}\cdots h_{n}\log B,$ where $C(n,d,{\mathfrak p})=19\bigl(20\sqrt{n+1}d\bigr)^{2(n+1)} e_{\mathfrak p}^{n-1}\cdot {{\mathfrak p}^{{f_{\mathfrak p}}}\over (f_{\mathfrak p}\log {\mathfrak p})^{2}} \cdot \log(e^{5}nd),$ $$e_{\mathfrak p}$$ is the ramification index and $$f_{\mathfrak p}$$ the residue class degree of $${\mathfrak p}$$.
This is a sharpening of the author’s previous result [J. Reine Angew. Math. 502, 29-92 (1998; Zbl 0912.11029)]. The refinement is due to the fact that the author succeeds to adapt to the $$p$$-adic case an argument of E. M. Matveev [Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 4, 81-136 (1998), Engl. translation in Izv. Math. 62, No. 4, 723-772 (1998; Zbl 0923.11107)].
Such estimates have a wide range of applications; therefore one should stress that for practical purposes (for instance for solving diophantine equations), as a general rule, it is much more efficient to apply the more elaborate statements in the paper under review than the special above mentioned easy case. For instance a much smaller value for $$C(n,d,\mathfrak p)$$ is achieved when $[K(\alpha_{0}^{1/q},\alpha_{1}^{1/q},\ldots, \alpha_{n}^{1/q}):K]=q^{n+1},$ where $q=\begin{cases} 2 & \text{if } p>2 ,\\ 3 & \text{if } p=2, \end{cases}$ and $$\alpha_{0}$$ is a generator of the torsion subgroup of $$K^{\times}$$.
In a forthcoming joint paper of the author with C. L. Stewart, they improve their result related to the $$abc$$ conjecture, replacing an exponent $$2/3$$ by $$1/3$$ [Math. Ann. 291, No. 2, 225-230 (1991; Zbl 0761.11030)].

##### MSC:
 11J86 Linear forms in logarithms; Baker’s method 11J61 Approximation in non-Archimedean valuations
Full Text: