On the representation of unity by binary cubic forms

Abstract:

If $F(x,y)$ is a binary cubic form with integer coefficients such that $F(x,1)$ has at least two distinct complex roots, then the equation $F(x,y) = 1$ possesses at most ten solutions in integers $x$ and $y$, nine if $F$ has a nontrivial automorphism group. If, further, $F(x,y)$ is reducible over $\mathbb {Z}[x,y]$, then this equation has at most $2$ solutions, unless $F(x,y)$ is equivalent under $GL_2(\mathbb {Z})$-action to either $x (x^2-xy-y^2)$ or $x (x^2-2y^2)$. The proofs of these results rely upon the method of Thue-Siegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations $F(x,y)=1$ for $F$ cubic and irreducible of positive discriminant $D_F \leq 10^6$. As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form $F(x,y)=m$ and to Mordell’s equation $y^2=x^3+k$, where $m$ and $k$ are nonzero integers.