A Local Version of Szpiro’s Conjecture

Abstract

Szpiro’s conjecture asserts the existence of an absolute constant $K \gt 6$ such that if $E$ is an elliptic curve over $\mathbb{Q}$, the minimal discriminant $\Delta(E)$ 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 on $\min\{v_p(\Delta(E )) : p |\Delta(E )\}$. 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 $\min{v_p(\Delta(E)) : p | \Delta(E )} \ge 6$ 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.