论文标题
分布Kloosterman途径到高质量模量
Distribution of Kloosterman paths to high prime power moduli
论文作者
论文摘要
我们考虑将归一化Kloosterman和Modulo的部分总和的多边形路径的分布，一个固定奇数P的越来越高的功率P^n，是Kowalski-Sawin和Ricotta-Royer-Royer-shparlinski的定理的纯度深度模拟。我们发现，kloosterman路径的集合自然会分解为有限的许多不相交的集合，每个集合会在法律上以n-> \ Infty收敛到一个独特的复杂有价值的随机连续函数。我们进一步发现，将这些限制粘合在一起的随机序列在法律上以p-> \ infty收敛，而连接部分kloosterman的路径是在适度的术语重排后获得不同且普遍的限制形状。作为关键算术输入，我们使用固定阶段的P-ADIC方法（包括高度奇异的情况）证明，将任意的许多Kloosterman总和到高质量功率模量的产品总和在参数转移中表现出功率储蓄或功率对齐。
We consider the distribution of polygonal paths joining the partial sums of normalized Kloosterman sums modulo an increasingly high power p^n of a fixed odd prime p, a pure depth-aspect analogue of theorems of Kowalski-Sawin and Ricotta-Royer-Shparlinski. We find that this collection of Kloosterman paths naturally splits into finitely many disjoint ensembles, each of which converges in law as n->\infty to a distinct complex valued random continuous function. We further find that the random series resulting from gluing together these limits for every p converges in law as p->\infty, and that paths joining partial Kloosterman sums acquire a different and universal limiting shape after a modest rearrangement of terms. As the key arithmetic input we prove, using the p-adic method of stationary phase including highly singular cases, that complete sums of products of arbitrarily many Kloosterman sums to high prime power moduli exhibit either power savings or power alignment in shifts of arguments.
