Conjecture. (Szpiro, 1981) Let E be an elliptic curve over Q which is a global minimal model with discriminant Δ and conductor N. Then for every ε > 0, there exists k(ε) > 0 such that Δ < k(ε)N^(6+ε). We show that Szpiro’s conjecture above is equivalent to the weak ABC?conjecture. Let A,B,C be coprime integers satisfying A + B + C = 0 and ABC ≠ 0. Set N =Πp|ABC p. Consider the Frey?Hellegouarch curve EA,B : y2 = x(x ? A)(x + B). A minimal model for EA,B has discriminant (ABC)^2 ・ 2^?s and conductor N ・ 2^?t for certain absolutely bounded integers s, t, (see Frey [F1]). Plugging this data into Szpiro's conjecture immediately shows the equivalence.
