论文标题
关于Weierstrass $σ$ functions的代数值
On algebraic values of Weierstrass $σ$-functions
论文作者
论文摘要
假设$ω$是复杂平面中的晶格，让$σ$为相应的weierstrass $σ$函数。假设标准基本域中与$ω$相关的点$τ$最大为1.9。假设$ω$具有代数不变的$ g_2，g_3 $，我们表明的是，$ c d^m（\ log h）^n $的界限最多可容纳$ h $的代数高度点，最多$ h $，最多$ d $ by of $σ$。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，我们能够为整个图形建立这样的束缚，而不是某些限制。当晶格点为代数时，我们证明了类似的结果。为此，我们自然排除了$ z \inΩ$的$（z，σ（z））$。
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 $z\inΩ$.
