论文标题
某些椭圆形K3表面的发电机和分裂场
Generators and splitting fields of certain elliptic K3 surfaces
论文作者
论文摘要
令$ k \ subset {\ mathbb c} $为一个数字字段，$ {\ mathcal e} $是椭圆曲线，对椭圆形表面的通用光纤是同构的，该椭圆形表面定义了Phoffiate函数字段$ K（T）$ k（t）$ k（t）$ k（t）$ k（t）$ { $ k $ {\ $ k $ {\ MATHCAL E} $的$ {\ MATHCAL E}（k）$是$ {\ Mathcal E} $的$ {\ Mathcal e} $，对于任何子场$ k $ of $ {\ mathbb c}（t）$，都是有限生成的Abelian group。 $ k（t）$上的$ {\ Mathcal e} $的拆分字段是$ {\ Mathbb C} $的最小子字段$ {\ Mathcal k} $，它是$ k $ and $ k $和$ k $ and $ {\ MATHCAL E}的有限扩展k}（t））$。 在本文中，我们考虑了$ k = {\ mathbb q} $定义的椭圆形$ k3 $表面，并使用weierstrass方程$ {\ mathcal e} _n：\ displayStyle y^2 = x^3 + t^n + 1/t^$ ext $ kn。 $ {\ Mathcal e} _n（{\ Mathcal k_n}（t））$的独立发电机的$ 1 \ leq n \ leq 6 $。
Let $k \subset {\mathbb C}$ be a number field and ${\mathcal E}$ be an elliptic curve that is isomorphic to the generic fiber of an elliptic surface defined over the rational function field $k(t)$ of the projective line ${\mathbb P}^1_k$. The set ${\mathcal E}(K)$ of $K$-rational points of ${\mathcal E}$ is known to be a finitely generated abelian group for any subfield $K$ of ${\mathbb C}(t)$. The splitting field of ${\mathcal E}$ over $k(t)$ is the smallest subfield ${\mathcal K}$ of ${\mathbb C}$ which is a finite extension of $k$ and ${\mathcal E} ({\mathbb C} (t)) ={\mathcal E} ({\mathcal K}(t))$. In this paper, we consider the elliptic $K3$ surfaces defined over $k={\mathbb Q}$ with the generic fiber given by the Weierstrass equation ${\mathcal E}_n: \displaystyle y^2=x^3 + t^n + 1/t^n.$ We will determine the splitting field ${\mathcal K}_n$ and find an explicit set of independent generators for ${\mathcal E}_n ({\mathcal K_n}(t))$ for $1\leq n \leq 6$.
