Variation of the reduction type of elliptic curves under small base change with wild ramification

2003 ◽  
Vol 1 (4) ◽  
pp. 510-560 ◽  
Author(s):  
Masanari Kida
Keyword(s):  
Elliptic Curves ◽  
Base Change ◽  
Wild Ramification ◽  
Small Base

scholarly journals Growth of torsion groups of elliptic curves upon base change

2019 ◽  
Vol 89 (323) ◽  
pp. 1457-1485 ◽  
Author(s):  
Enrique González–Jiménez ◽  
Filip Najman
Keyword(s):  
Elliptic Curves ◽  
Base Change ◽  
Torsion Groups

scholarly journals ELLIPTIC K3 SURFACES ASSOCIATED WITH THE PRODUCT OF TWO ELLIPTIC CURVES: MORDELL–WEIL LATTICES AND THEIR FIELDS OF DEFINITION

2016 ◽  
Vol 228 ◽  
pp. 124-185 ◽  
Author(s):  
ABHINAV KUMAR ◽  
MASATO KUWATA
Keyword(s):  
Elliptic Curves ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
Double Cover ◽  
Base Change ◽  
Elliptic Surfaces ◽  
Low Degree ◽  
Weil Group ◽  
Finite Index Subgroup ◽  
Rational Elliptic Surfaces

To a pair of elliptic curves, one can naturally attach two K3 surfaces: the Kummer surface of their product and a double cover of it, called the Inose surface. They have prominently featured in many interesting constructions in algebraic geometry and number theory. There are several more associated elliptic K3 surfaces, obtained through base change of the Inose surface; these have been previously studied by Masato Kuwata. We give an explicit description of the geometric Mordell–Weil groups of each of these elliptic surfaces in the generic case (when the elliptic curves are non-isogenous). In the nongeneric case, we describe a method to calculate explicitly a finite index subgroup of the Mordell–Weil group, which may be saturated to give the full group. Our methods rely on several interesting group actions, the use of rational elliptic surfaces, as well as connections to the geometry of low degree curves on cubic and quartic surfaces. We apply our techniques to compute the full Mordell–Weil group in several examples of arithmetic interest, arising from isogenous elliptic curves with complex multiplication, for which these K3 surfaces are singular.

Elliptic Curves

1997 ◽  
Author(s):  
Henry McKean ◽  
Victor Moll
Keyword(s):  
Elliptic Curves
Sign in / Sign up

Export Citation Format

Share Document