東工大 数論・幾何学セミナー：Liang-Chung Hsia 氏

　In this talk, we'll discuss the common divisors among sequences arising from cyclic groups generated by rational points $P_i\in E_i(K)$ of elliptic curves $E_i$ ($i=1, 2$) defined over the function field $K$ of a smooth projective curve $C$ over $\bar{\mathbb Q}$. More precisely, let $\mathcal{E}_i\rightarrow C$ be the elliptic surfaces over $C$ with generic fiber $E_i$ and let $\sigma_{P_i}, \sigma_{Q_i}$ be sections (corresponding to points $P_{i}, Q_{i}$ of the generic fibers) of $\mathcal{E}_i$ (for $i=1,2$). The question that we're concerned with is whether or not there are infinitely many $t \in C(\Qbar)$ such that for some integers $m_{1,t},m_{2,t}$ we have $[m_{i,t}](\sigma_{{P_{i}}}(t))=\sigma_{Q_{i}}(t)$ on $\mathcal{E}_i$ (for $i=1,2$). We provide an answer to this question. A special case of our result answers a conjecture made by Silverman.

This is a joint work with Dragos Ghioca and Tom Tucker.