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arXiv:1012.0094v1[math.NT]1Dec2010 Periods of quadratic twists of elliptic curves

Summary: arXiv:1012.0094v1[math.NT]1Dec2010
Periods of quadratic twists of elliptic curves
Vivek Pal
with an appendix by Amod Agashe
In this paper we prove a relation between the period of an elliptic curve
and the period of its real and imaginary quadratic twists. This relation is
often misstated in the literature.
1 Introduction
One of the central conjectures in Number Theory is the Birch and Swinnerton-
Dyer Conjecture, which predicts how one can obtain arithmetic information
from the L-function. A simpler questions is to ask:
(*) if an elliptic curve satisfies the Birch and Swinnerton-Dyer Con-
jecture then will its (quadratic) twist also satisfy the Birch and
Swinnerton-Dyer Conjecture.
Part two of the Birch and Swinnerton-Dyer Conjecture involves many ellip-
tic curve invariants, namely the order of the Tate-Shafarevich group, the period
and the order of the torsion subgroup among other important invariants. In this
paper we relate the period of an elliptic curve with the period of its quadratic
twists. A relation between the orders of the torsion subgroups has already been


Source: Agashe, Amod - Department of Mathematics, Florida State University


Collections: Mathematics