Elliptic K3 Surfaces Associated With The product of Two Elliptic Curves: Mordell–weil Lattices and Their Fields Of definition

Abstract

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 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 non-generic 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.

Isogenous components of Jacobian surfaces

“…Note that if we choose other equations of E 1 and E 2 then we get an isomorphic equation for the Kummer surface. Setting t 1 = t 6 6 in the above equation we get an elliptic curve which will be denoted with F (1) E1,E2 and the Néron-Severi model of this elliptic curve over k(t 1 ) is called the Inose surface associated with E 1 and E 2 , see [26] for more details. Lemma 8.…”

“…4 we know that for n = 2, 3 there are infinitely many curves E 1 that are isogenous to E 2 . From [26,Prop. 2.9] we have that if E 1 is isogenous to E 2 and they have complex multiplication, then the rank of F (5) and F (6) is 18.…”

Mordell–Weil lattice of Inose's elliptic K3 surface arising from the product of 3-isogenous elliptic curves

“…Most known examples fall into the case where the degree of isogeny ϕ equals 2, in which case the calculations are straight forward. One particular example of the case deg ϕ = 4 is dealt in [2,Example 9.2]. In this paper we consider a family of the pairs of elliptic curves E 1 and E 2 with an isogeny ϕ : E 1 → E 2 of degree 3 defined over k. We write down a formula of the section of F (1) E1,E2 coming from ϕ defined over the base field k. To do so, we first work with the surface F (6) E1,E2 , which has a simple affine model that can be viewed as a family of cubic curves with a rational point over k. We modify the method in [2] to find sections of F (1) E1,E2 .…”

“…One particular example of the case deg ϕ = 4 is dealt in [2,Example 9.2]. In this paper we consider a family of the pairs of elliptic curves E 1 and E 2 with an isogeny ϕ : E 1 → E 2 of degree 3 defined over k. We write down a formula of the section of F (1) E1,E2 coming from ϕ defined over the base field k. To do so, we first work with the surface F (6) E1,E2 , which has a simple affine model that can be viewed as a family of cubic curves with a rational point over k. We modify the method in [2] to find sections of F (1) E1,E2 . We also give a section of F (2) E1,E2 coming from the isogeny ϕ, and give a basis defined over the field k(E 1 [2], E 2 [2]) when E 1 and E 2 do not have a complex multiplication.…”

“…In this paper, we give a method of writing down generators of the Mordell-Weil lattice of such elliptic surfaces when two elliptic curves are 3-isogenous. In particular, we obtain a basis of the Mordell-Weil lattice for the singular K3 surfaces X [3,3,3] , X [3,2,3] and X [3,0,3] .…”