## Experimental Mathematics

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

### A Local Version of Szpiro’s Conjecture

Michael A. Bennett and Soroosh Yazdani

#### 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

**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