Solutions to infinite families of complex cubic Thue equations. (English) Zbl 0780.11014

Thomas, Emery

J. Reine Angew. Math. 441, 17-32 (1993).

Let \(\Phi(x,y)\in\mathbb{Z}[x,y]\) be an irreducible cubic form with negative discriminant and consider the equation (1) \(\Phi(x,y)=1\) in \(x,y\in\mathbb{Z}\). T. Nagell [Math. Z. 28, 10-29 (1928; JFM 54.0174.02)] proved, that equation (1) has (apart from a few exceptions) at most three solutions. Restricting ourselves to reversible forms \(\Phi(x,y)=x^ 3- bx^ 2 y+cxy^ 2- y^ 3\) (with \(b,c\in\mathbb{Z}\)) Nagell’s result means that if \(\text{disc}(\Phi)<-44\) then equation (1) has at most one nontrivial solution in addition to the trivial solutions (1,0), \((0,-1)\). Simple examples show, that there are infinite families of reversible forms admitting a nontrivial solution, but until now it was not known if there are reversible forms with no nontrivial solutions.
In the present paper the author constructs infinite families of reversible cubic forms with negative discriminant, having only the two trivial solutions. The proof of the main statement combines Baker’s method with a very accurate analysis of the equation in question.

Reviewer: I.Gaál (Debrecen)

MSC:

11D25	Cubic and quartic Diophantine equations
11J86	Linear forms in logarithms; Baker’s method

Keywords:

cubic Thue equations; infinite families of reversible cubic forms with negative discriminant; Baker’s method

Citations:

JFM 54.0174.02