椭圆曲线上有理数点孔的初始化和渐近性检验
Verteilung der Primzahlen und Asymptotik bezüglich der Kongruenzen der rationalen Punkte auf elliptischen Kurven über einem endlichen Körper
摘要：
这项工作有两个主要目的。一方面，我们调查在这项工作中的一个问题H。关于H. Esnault和C. Xu关于局部域上射影平面奇异模型的闭纤维上有理点的个数.从渐近分析的观点来看，这个问题与N的问题十分相似。Koblitz，这反过来又在密码学中有一些有意义的应用。我们不试图解决这些问题，在这项工作中，而是集中在研究的渐近行为与非常基本的技术生成函数。另一方面，我们将生成函数的讨论推广到全局情形，它继承了椭圆曲线的黎曼ζ函数和$L$ -函数的混合性质。在最后，我们将看看一个例子的模形式，在那里我们能够证明一个分析结果，这是类似的结果波利亚黎曼$ξ$ -功能。
英文摘要：
This work has two main purposes. On the one side we investigate in this work a question of H. Esnault on congruence formula in a construction of H. Esnault and C. Xu for the number of rational points on the closed fiber of a singular model of the projective plane over a local field. From the viewpoint of asymptotic analysis, the question is quite familiar with a question of N. Koblitz, which in turn has some meaningful applications in cryptography. We don't try to solve those questions in this work, but rather concentrate on studying asymptotic behaviours with very elementary techniques of generating functions. On the other side we extend the discussion on generating functions to the global situation, which inherit hybrid-properties from the Riemann zeta function and the $L$-function of elliptic curves. At the end we will look at an example of modular forms, where we are able to prove an analytic result, which is similar to a result of Pólya for the Riemann $ξ$-function.
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