Reprint: On the lattice points on curves of genus 11 (1935)

  • Kurt Mahler


Let F(x,y)F(x,y) be a cubic binary form with integer coefficients that is irreducible over the field of rational numbers, and let k0k\neq 0 be an integer. Further, let A(k)A(k) be the number of pairs of integers (x,y)(x,y) satisfying F(x,y)=kF(x,y)=k. Here, Mahler proves that A(k)A(k) is unbounded, and that there are infinitely many integers kk such that A(k)logk4.A(k)\geqslant \sqrt[4]{\log k}.

Reprint of the author's paper [Proc. Lond. Math. Soc. (2) 39, 431--466 (1935; Zbl 0012.15006; JFM 61.0146.02); corrigendum ibid. 40, 558 (1936; JFM 61.1055.02)].