A local version of Szpiro's conjecture

摘要

Szpiro's conjecture asserts the existence of an absolute constant K > 6 such that if E is an elliptic curve over ?, the minimal discriminant Δ(E) of E is bounded above in modulus by the Kth power of the conductor N(E) of E. An immediate consequence of this is the existence of an absolute upper bound on min{v _p(Δ(E)): p | Δ(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 that if true, we prove to be sharp. Our construction of families of curves for which min{v _p(Δ (E)): p | Δ(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.