Experimental Mathematics

A Local Version of Szpiro’s Conjecture

Michael A. Bennett and Soroosh Yazdani

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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.

Article information

Source
Experiment. Math., Volume 21, Issue 2 (2012), 103-116.

Dates
First available in Project Euclid: 31 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.em/1338430824

Mathematical Reviews number (MathSciNet)
MR2931308

Zentralblatt MATH identifier
1294.11086

Subjects
Primary: 11G0S 14H5Z

Keywords
Elliptic curves Szpiro’s conjecture

Citation

Bennett, Michael A.; Yazdani, Soroosh. A Local Version of Szpiro’s Conjecture. Experiment. Math. 21 (2012), no. 2, 103--116. https://projecteuclid.org/euclid.em/1338430824


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