椭圆曲面上特殊化映射的内射性检验算法
An Algorithm for Checking Injectivity of Specialization Maps from Elliptic Surfaces
摘要：
令$E/\mathbb Q(t)$是椭圆曲线，并且令$t_0 \in \mathbb Q$是有理数，对于该有理数，特化$E_{t_0}$是椭圆曲线。给定一个子群$M$的$E(\mathbb Q(t))$与温和的条件和$t_0 \in \mathbb Q$来自一个相对较大的子集$S_M \subset \mathbb Q$，我们提供了一个算法，可以表明，专业化地图$σ_{t_0} : E(\mathbb Q(t)) \to E_{t_0}(\mathbb Q)$是单射时，限制到$M$。集合$S_M$在某些情况下是有效的可计算的，我们进行这种计算的一些明确的例子，其中$E$是由维尔斯特拉斯方程。
英文摘要：
Let $E/\mathbb Q(t)$ be an elliptic curve and let $t_0 \in \mathbb Q$ be a rational number for which the specialization $E_{t_0}$ is an elliptic curve. Given a subgroup $M$ of $E(\mathbb Q(t))$ with mild conditions and $t_0 \in \mathbb Q$ coming from a relatively large subset $S_M \subset \mathbb Q$, we provide an algorithm that can show that the specialization map $σ_{t_0} : E(\mathbb Q(t)) \to E_{t_0}(\mathbb Q)$ is injective when restricted to $M$. The set $S_M$ is effectively computable in certain cases, and we carry out this computation for some explicit examples where $E$ is given by a Weierstrass equation.
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