On the representation of unity by binary cubic forms (Q2701668)

Let \(F(x,y)\) be a binary cubic form with integer coefficients. It was proved about the thirties of the 20th century by \textit{B. N. Delone} (Delaunay) [Math. Z. 31, 1-26 (1929; JFM 55.0720.02)] and independently by \textit{T. Nagell} [Math. Z. 28, 10-29 (1928; JFM 54.0174.02)] that if the discriminant \(D_F\) of \(F\) is negative then the number of solutions \(N_F\) of the Thue equation \(F(x,y)=1\) is at most 5. Moreover they proved that if \(D_F<-44\) then \(N_F \leq 3\). These bounds are sharp in the sense that there exist infinitely many inequivalent cubic forms \(F\) with negative discriminants with \(N_F=3\). NEWLINENEWLINENEWLINEThe same problem for forms with positive discriminants, which is the topic of the present paper, is much more complicated. \textit{C. L. Siegel} [Abh. Preuß. Akad. Wiss. 1929, No. 1 (1929; JFM 56.0180.05)] proved in this case that \(N_F \leq 18\) and \textit{J.-H. Evertse} reduced this bound to 12 [Invent Math. 73, 117-138 (1983; Zbl 0494.10009)]. Moreover he proved \(N_F \leq 10\) provided \(D_F>5 \cdot 10^{10}\). NEWLINENEWLINENEWLINEBased on an extensive computer search the reviewer conjectured that \(N_F \leq 9\), and \(N_F \leq 5\), provided the discriminant of \(F\) is larger than 361 [Number Theory, Proc. 15th Journ. Arith., Ulm 1987, Lect. Notes Math. 1380, 185-196 (1989; Zbl 0677.10012)]. This search was extended by \textit{F. Lippok} [J. Symb. Comput. 15, 297-313 (1993; Zbl 0780.11012)], whose results agreed essentially with the conjecture. NEWLINENEWLINENEWLINEThe main result of the present paper is that \(N_F \leq 10\). From this theorem it is derived that if \(m\) denotes a non-zero integer then the number of solutions of the Thue equation \(F(x,y)=m\) does not exceed \(10 \cdot 3^{\omega(m)}\), where \(\omega(m)\) denotes the number of distinct prime factors of \(m\). It is also proved that Mordell's equation \(y^2 = x^3 +k \) has at most \(10 h_3(-108k)\) solutions in integers \(x,y\), where \(h_3(-108k)\) is the class number of binary cubic forms of discriminant \(k\). NEWLINENEWLINENEWLINEThe proof of the main result consists of two parts. In the first one the author follows the line of the above mentioned paper of Evertse. A careful analysis of this method enables him to prove the theorem for all forms with \(D_F \geq 24000\). NEWLINENEWLINENEWLINEIn the second part the equation \(F(x,y)=1\) is solved for all non-equivalent cubic forms \(F\) with discriminant \(0<D_F\leq 10^6\). To find all representatives a method of \textit{K. Belabas} and \textit{H. Cohen} [AMS/IP Stud. Adv. Math. 7, 191-219 (1998; Zbl 0915.11024)] is used, while for the solution of the resulting 89595 Thue equations the computational number theory systems PARI GP, Version 1.39 and KANT V4, Version 2.0 are used. After this search the author found nothing to contradict the conjecture of the reviewer.