A NOTE ON SCHMIDT'S CONJECTURE

Schmidt ['Integer points on curves of genus 1 ', Compos. Math. 81 (1992), 33-59] conjectured that the number of integer points on the elliptic curve defined by the equation y(2) = x(3) + ax(2) + bx + c, with a,b,c is an element of Z, is O-is an element of(max {1,vertical bar a vertical bar ,vertical bar b vertical bar,vertical bar c vertical bar}(is an element of)) for any is an element of > 0. On the other hand, Duke ['Bounds for arithmetic multiplicities', Proc. Int. Congress Mathematicians, Vol. II (1998), 163-172] conjectured that the number of algebraic number fields of given degree and discriminant D is O-is an element of (vertical bar D vertical bar(is an element of)). In this note, we prove that Duke's conjecture for quartic number fields implies Schmidt's conjecture. We also give a short unconditional proof of Schmidt's conjecture for the elliptic curve y(2) = x(3) + ax.