论文标题
爱森斯坦系列的关键点
Critical points of Eisenstein series
论文作者
论文摘要
对于任何偶数整数$ k \ ge 4 $，令$ \ e_k $为归一化的Eisenstein系列重量$ k $，for $ \ sl_2（\ z）$。另外，让$ \ d $成为庞加莱上半平面模型$ \ sl_2（\ z）$的标准基本域的关闭。 F.〜K.〜C.〜Rankin和H. P. F. Swinnerton-Dyer表明，$ \ \ d $中的所有$ \ e_k $的零是模量。在本文中，我们研究了$ \ e_k $的关键点，也就是说，$ \ e_k $的衍生物的零。我们证明它们很简单。我们计算属于$ \ d $的人，证明它们位于$ \ d $的两个垂直边缘，并产生将它们分开的明确间隔。然后，我们计算属于$γ\ d $的那些，对于\ sl_2（\ z）$中的任何$γ\。
For any even integer $k \ge 4$, let $\E_k$ be the normalized Eisenstein series of weight $k$ for $\SL_2(\Z)$. Also let $\D$ be the closure of the standard fundamental domain of the Poincaré upper half plane modulo $\SL_2(\Z)$. F.~K.~C.~Rankin and H. P. F. Swinnerton-Dyer showed that all zeros of $\E_k$ in $\D$ are of modulus one. In this article, we study the critical points of $\E_k$, that is to say the zeros of the derivative of $\E_k$. We show that they are simple. We count those belonging to $\D$, prove that they are located on the two vertical edges of $\D$ and produce explicit intervals that separate them. We then count those belonging to $γ\D$, for any $γ\in \SL_2(\Z)$.
