A Local Version of Szpiro%26apos;s Conjecture

作者：Bennett Michael A^*; Yazdani Soroosh

来源：Experimental Mathematics, 2012, 21(2): 103-116.

DOI：10.1080/10586458.2012.645780

• elliptic curves
• Szpiro%26apos
• s conjecture

Szpiro%26apos;s conjecture asserts the existence of an absolute constant K %26gt; 6 such that if E is an elliptic curve over Q, the minimal discriminant Delta(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{upsilon(p)(Delta(E)) : p vertical bar Delta(E)}. In this paper, we will prove this local version of Szpiro%26apos;s conjecture under the (admittedly strong) additional hypotheses that N(E) is divisible by a %26quot;large%26quot; 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{upsilon(p)(Delta(E)) : p vertical bar Delta(E)} %26gt;= 6 provides an alternative proof of a result of Masser on the sharpness of Szpiro%26apos;s conjecture. We close the paper by reporting on recent computations of examples of curves with large Szpiro ratio.

• 出版日期2012