Generators and splitting fields of certain elliptic K 3 surfaces

Abstract

Let k ⊂ ℂ be a number field and ℰ be an elliptic curve defined over k ( t ) that is isomorphic to the generic fiber of an elliptic surface π : 𝒮 ℰ → ℙ k 1 . For any extension 𝒦 ⊆ ℂ of k , the set ℰ ( 𝒦 ( t ) ) of 𝒦 ( t ) -rational points of ℰ is known to be a finitely generated abelian group. The splitting field of ℰ defined over k ( t ) is the smallest finite extension 𝒦 ⊂ ℂ of k such that ℰ ( ℂ ( t ) ) ≅ ℰ ( 𝒦 ( t ) ) . In this paper, we consider the elliptic K 3 surfaces defined over k = ℚ with the generic fiber given by the Weierstrass equation ℰ n : y 2 = x 3 + t n + 1 / t n , 1 ≤ n ≤ 6 , and determine the splitting field 𝒦 n , and find an explicit set of linearly independent generators for ℰ n ( 𝒦 𝓃 ( t ) ) for 1 ≤ n ≤ 6 .