We pose the problem to determine explicit defining equations of various elliptic fibrations on a given K3 surface, and study the case of the Kummer surfaces of the product of two elliptic curves.
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.
Abstract. Let L(E/Q, s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the vanishing and non-vanishing of the central values L (E, 1, χ) of the twisted L-function as χ ranges over Dirichlet characters of given order.
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