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Torsion points on curves and common divisors of \(a^k-1\) and \(b^k-1\). (English) Zbl 1057.11018

The authors pose the following conjecture: if \(a,b\) are multiplicative non-zero integers such that \(\gcd(a-1,b-1)=1\), then there are infinitely many integers \(k\) with \(\gcd(a^k-1,b^k-1)=1\). This is related to a recent result by Y. Bugeaud, P. Corvaja and U. Zannier [Math. Z. 243, 79–84 (2003; Zbl 1021.11001)] who proved that gcd\((a^k-1,b^k-1)\) cannot grow too fast in terms of \(k\). More precisely, they proved using Schmidt’s Subspace Theorem that if \(a,b\) are multiplicatively independent positive integers then \(\gcd(a^k-1,b^k-1)\ll_{\varepsilon} e^{\varepsilon k}\) for every \(\varepsilon >0\). The authors of the present paper pose a more general conjecture for matrices which is as follows: let \(A\) be an \(r\times r\) matrix with integer entries such that \(\gcd(A-I)=1\), i.e., the entries of \(A-I\) have \(\gcd 1\). Suppose also that \(A\) has at least two eigenvalues and that two among them are multiplicatively independent. Then there are infinitely many integers \(k>0\) such that \(\gcd(A^k-I)=1\).
The authors show that their conjecture is false for \(2\times 2\)-matrices with two multiplicatively dependent eigenvalues. Further they give evidence for their conjecture by proving the following analogue for polynomials. Let \(A\) be a non-singular \(r\times r\)-matrix with entries in \({\mathbb C}[t]\). Suppose that either \(A\) is not diagonizable over the algebraic closure of \({\mathbb C}(t)\) or that \(A\) has at least two eigenvalues that are multiplicative independent. Then there is a fixed polynomial \(h\not= 0\) such that \(\gcd(A^k-1)\) divides \(h\) for every \(k\geq 0\). Moreover, if \(\gcd(A-I)=1\) then \(\gcd(A^k-1)=1\) for all \(k\) outside a finite union of the shape \(\bigcup d_i{\mathbb N}\) where the \(d_i\) are integers \(\geq 2\). The proof of this result is based on a theorem of Ihara, Serre and Tate, conjectured earlier by Lang, that polynomial equations \(f(x,y)=0\) have only finitely many solutions in roots of unity \(x,y\), unless \(f\) has a divisor of the shape \(X^nY^m-\zeta\) or \(X^m-\zeta Y^n\) where \(\zeta\) is a root of unity.
The authors prove another result supporting their conjecture. Let \(p>2\) be a prime, let \(R\) be the ring of integers of \({\mathbb Q}(\zeta_p)\), choose an integral basis of \(R\), and for \(a\in R\), let \(A(a)\) denote the matrix of the linear map \(x\mapsto ax\) with respect to this basis. Thus, \(A(a)\) is an integer \((p-1)\times (p-1)\)-matrix. Now if \(u\) is a non-real unit of \(R\), then \(\gcd(A(u)^k-I)=1\) for every integer \(k\) not divisible by \(p\). The proof of this result is elementary.

MSC:

11D61 Exponential Diophantine equations
11R58 Arithmetic theory of algebraic function fields

Citations:

Zbl 1021.11001
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