Masato Kuwata

Vanishing and non-vanishing Dirichlet twists of L-functions of elliptic curves

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,\chi) of the twisted L-function as \chi ranges over Dirichlet characters of given order.

The canonical height and elliptic surfaces

Abstract Let π: S → P 1 be an elliptic surface over the complex numbers. Let E be the generic fiber of S. Then E is a curve of genus 1 which we assume has a rational point over K = C(u). By the Mordell-Weil theorem, one knows that E(K) is finitely generated. In this paper the author uses the canonical height pairing to help in determining E(K). In particular, suppose that the rank of E(K) is r. Then the author is interested in determining whether r sections form a basis. As an illustration, the author examines the equation A4+B4=C4+D4

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

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.

Inose’s construction and elliptic 3 surfaces with Mordell-Weil rank 15 revisited

We describe two constructions of elliptic K3 surfaces starting from the Kummer surface of the Jacobian of a genus 2 curve. These parallel the base-change constructions of Kuwata for the Kummer surface of a product of two elliptic curves. One of these also involves the analogue of an Inose fibration. We use these methods to provide explicit examples of elliptic K3 surfaces over the rationals of geometric Mordell-Weil rank 15.

Points Defined over Cyclic Quartic Extensions on an Elliptic Curve and Generalized Kummer Surfaces

Let E be an elliptic curve over a number field k. By the Mordell-Weil theorem the group E(K) of K-rational points on E, where K/k is a finite extension of k,is a finitely generated abelian group. We fix E/k once and for all, and we study the behavior of the rank of the group E(K) as K varies through a certain family. We are particularly interested in the family F k (G) of all Galois extensions K/k whose Galois group Gal(K/k) is isomorphic to a prescribed finite group G. In this article we focus on the case \( G = \mathbb{Z}/4\mathbb{Z} \)

Ramified primes in the field of definition for the Mordell-Weil group of an elliptic surface

Let 7r: X -* C be an elliptic surface defined over a number field k . We consider the field K in which all the sections are defined. Assuming that the j-invariant is nonconstant, K is again a number field. We describe the primes of possible ramification of the extension K/k in terms of the configuration of the points of bad fibers in C. Aside from few possible exceptions, K/k is unramified outside of the primes of bad reduction of C and the primes p for which two or more points of bad fibers become identical modulo p .