A Local Version of Szpiro's Conjecture

Szpiro’s Conjecture asserts the existence of an absolute constant K > 6 such that if E is an elliptic curve over Q, the minimal discriminant ∆(E) of E is bounded above in modulus by the K-th power of the conductor N(E) of E. An immediate consequence of this is the existence of an absolute upper bound upon min {vp(∆(E)) : p | ∆(E)}. In this paper, we will prove this local version of Szpiro’s Conjecture under the (admittedly strong) additional hypotheses that N(E) is divisible by a “large” prime p, and that E possesses a nontrivial rational isogeny. We will also formulate a related conjecture which, if true, we prove to be sharp. Our construction of families of curves for which min {vp(∆(E)) : p | ∆(E)} ≥ 6 provides an alternative proof of a result of Masser on the sharpness of Szpiro’s conjecture. We close the paper by reporting on recent computations of examples of curves with large Szpiro ratio.