TITLE:
On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields
AUTHORS:
Shun’ichi Yokoyama
KEYWORDS:
Elliptic Curves over Number Fields; Mordell-Weil Group; Two-Descent
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.2 No.4,
December
31,
2012
ABSTRACT: We prove the existence and nonexistence of elliptic curves having good reduction everywhere over certain real quadratic fields Q(m) for m≤200. These results of computations give best-possible data including structures of Mordell-Weil groups over some real quadratic fields via two-descent. We also prove similar results for the case of certain cubic fields. Especially, we give the first example of elliptic curve having everywhere good reduction over a pure cubic field using our method.