Paper Info
Title
Minimal Regulators for Rank-2 Subgroups of Rational and K3 Elliptic Surfaces
Abstract
We determine the smallest possible regulator R(P, Q) for a rank-2 subgroup ZP circle plus ZQ of an elliptic curve E over C(t) of discriminant degree 12n for n = 1 (a rational elliptic surface) and n = 2 (a K3 elliptic surface), exhibiting equations for all (E, P, Q) attaining the minimum. The minimum R(P, Q) = 1/36 for a rational elliptic surface was known [Oguiso and Shioda 911, but a formula for (E, P, Q) was not, nor was the fact that this is the minimum for an elliptic curve of discriminant degree 12 over a function field of any genus. For a K3 surface, both the minimal regulator R(P, Q) = 1/100 and the explicit equations are new. We also prove that 1/100 is the minimum for an elliptic curve of discriminant degree 24 over a function field of any genus. The optimal (E, P, Q) are uniquely characterized by having mP and m'Q integral form <= M and m' <= M', where (M, M') = (3, 3) for n = 1 and (M, M') = (6, 3) for n = 2. In each case MM' is maximal. We use the connection with integral points to find explicit equations for the curves. As an application we use the K3 surface to produce, in a new way, the elliptic curves E/Q with nontorsion points of smallest known canonical height. These examples appeared previously in [Elkies 02].
Year
DOI
Venue
2009
10.1080/10586458.2009.10129060
EXPERIMENTAL MATHEMATICS
Keywords
Field
DocType
Elliptic surface,canonical height,elliptic curve,K3 surface,Mordell-Weill lattice
Topology,Supersingular elliptic curve,Elliptic surface,Twists of curves,Sato–Tate conjecture,Mathematical analysis,Hessian form of an elliptic curve,Elliptic curve,Mathematics,Schoof's algorithm,Half-period ratio
Journal
Volume
Issue
ISSN
18.0
4.0
1058-6458
Citations 
PageRank 
References 
0
0.34
0
Authors
1
Name
Order
Citations
PageRank
Sonal Jain11489.29