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arXiv 2011.11980math.NT
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关于Weierstrass $σ$ -函数的代数值

On algebraic values of Weierstrass $σ$-functions

Gareth Boxall, Taboka Chalebgwa, Gareth Jones

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摘要：

设$Ω$是复平面上的格，$σ$是相应的Weierstrass $σ$ -函数.假设与标准基本域中的$Ω$相关联的点$τ$具有至多1.9的虚部。假设$Ω$有代数不变量$g_2,g_3$，我们证明了$c d^m (\log H)^n$形式的界对于$σ$的图上高度最多为$H$、度最多为$d$的代数点数成立。为了证明这一点，我们应用Masser和Besson的结果。也许令人惊讶的是，我们能够为整个图建立这样的界限，而不是一些限制。我们证明了一个类似的结果时，而不是$g_2,g_3$，格点是代数。为此，我们自然排除那些$(z,σ(z))$。

英文摘要：

Suppose that $Ω$ is a lattice in the complex plane and let $σ$ be the corresponding Weierstrass $σ$-function. Assume that the point $τ$ associated to $Ω$ in the standard fundamental domain has imaginary part at most 1.9. Assuming that $Ω$ has algebraic invariants $g_2,g_3$ we show that a bound of the form $c d^m (\log H)^n$ holds for the number of algebraic points of height at most $H$ and degree at most $d$ lying on the graph of $σ$. To prove this we apply results by Masser and Besson. What is perhaps surprising is that we are able to establish such a bound for the whole graph, rather than some restriction. We prove a similar result when, instead of $g_2,g_3$, the lattice points are algebraic. For this we naturally exclude those $(z,σ(z))$ for which $z\inΩ$.

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