On Szpiro’s discriminant conjecture

Abstract: In 1982 Szpiro conjectured that there is a constant c such that if E is an elliptic curve over Q of minimal discriminant \(\Delta\) and conductor N, then \(\vert \Delta\vert\leq N^c\). This conjecture is equivalent to the abc-conjecture. In the talk we give the unconditional result log \(\vert \Delta\vert\leq N^c\). The proof is based on the theory of logarithmic forms. We bound the relative Faltings height of E polynomially in terms of N which leads to the desired estimate.