イベントカレンダー

# 東工大 数論・幾何学セミナー：Liang-Chung Hsia 氏

2018年1月26日（金）

16:00～17:00

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Liang-Chung Hsia 氏 (National Taiwan Normal University)

On common divisors of sequences over function fields
アブストラクト

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.