关于低次L-数据的注记
Notes on Low Degree L-Data
摘要：
本文是作者2015年11月在京都大学数理科学研究所举行的“解析数论及相关领域”会议上的演讲的扩展版。我们对“$L$ -data”感兴趣，这是Andrew Booker在2013年引入的$L$ -函数的公理化框架。与每个$L$ -datum相关联，有一个称为度的实数不变量。推测度$d$是整数。此外，如果$d\in\mathbb{N}$，那么对于某个数字字段$F$，人们期望$L$数据是$GL_n(\mathbb{A}_F)$自守表示的数据。事实上，如果是$F=\mathbb{Q}$，那么就是$n=d$。布克在他的开创性论文中对$0\leq d<5/3$证明了这种说法是正确的，在这些注释中，我们考虑将他的方法扩展到$0\leq d<2$。这是同时概括布克的结果和结果和技术的Kaczorowski-佩雷利在塞尔伯格类（最有名的日期）。此外，我们考虑的应用程序的自守$L$ -函数的零点。在这些笔记中，我们回顾布克的结果，并宣布新的出现在其他地方不久。
英文摘要：
These notes are an extended version of a talk given by the author at the conference "Analytic Number Theory and Related Areas", held at Research Institute for Mathematical Sciences, Kyoto University in November 2015. We are interested in "$L$-data", an axiomatic framework for $L$-functions introduced by Andrew Booker in 2013. Associated to each $L$-datum, one has a real number invariant known as the degree. Conjecturally the degree $d$ is an integer. Moreover, if $d\in\mathbb{N}$ then one expects that the $L$-datum is that of a $GL_n(\mathbb{A}_F)$-automorphic representation, for some number field $F$. In fact, if $F=\mathbb{Q}$, then $n=d$. This statement was shown to be true for $0\leq d<5/3$ by Booker in his pioneering paper, and in these notes we consider an extension of his methods to $0\leq d<2$. This is simultaneously a generalisation of Booker's result and the results and techniques of Kaczorowski--Perelli in the Selberg class (the best known to date). Furthermore, we consider applications to zeros of automorphic $L$-functions. In these notes we review Booker's results and announce new ones to appear elsewhere shortly.
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