论文标题
semiabelian品种和猜想的Silverman最大的共同除数结果
Greatest Common Divisor results on semiabelian varieties and a Conjecture of Silverman
论文作者
论文摘要
划分序列是整数$ \ {d_n \} $的序列，因此$ d_m $ divides $ d_n $如果$ m $ divides $ n $。 Bugeaud，Corvaja，Zannier等的结果表明，与乘法组的亚组相对应的两个分裂序列的GCD以受控的方式生长。 Silverman猜想在许多代数组中应该出现类似的行为。我们扩展了Ghioca-Hsia-tucker和Silverman的椭圆曲线结果，并证明了Silverman对Abelian和Split Seprian Semiabelian品种功能领域的猜想的类似物以及对此结果的一些概括。我们采用了来自不太可能的交集理论的工具，以及与Abelian计划的一部分相关的所谓Betti地图的属性。
A divisibility sequence is a sequence of integers $\{d_n\}$ such that $d_m$ divides $d_n$ if $m$ divides $n$. Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to subgroups of the multiplicative group grows in a controlled way. Silverman conjectured that a similar behaviour should appear in many algebraic groups. We extend results by Ghioca-Hsia-Tucker and Silverman for elliptic curves and prove an analogue of Silverman's conjecture over function fields for abelian and split semiabelian varieties and some generalizations of this result. We employ tools coming from the theory of unlikely intersections as well as properties of the so-called Betti map associated to a section of an abelian scheme.
