Arithmetic statistics of noncommutative modular symbols

本建议的目的是了解非交换模符号的统计方面，并从代数-几何的观点来解释它们。其中一个核心问题是非交换模符号是否服从正态分布，这对于经典的交换模符号（Petridis—Risager）有一个肯定的答案。这个问题的重要动机一方面来自于模符号的统计性质和深度算术问题之间的密切关系，比如abc猜想，另一方面——这种联系最早是由Dorian Goldfeld描述的。本建议的另一个重要方面是找到模符号统计性质背后的数学结构的代数-几何解释。更准确地说，在Petridis- Risager在交换情况下的开创性工作中，对某个真实解析函数（“Goldfeld’s Eisenstein series”）的详细研究发挥了核心作用。虽然它的解析性质被很好地理解，但这个函数的代数-几何解释尚不清楚。在这种背景下，弗朗西斯·布朗最近发现了一个相当大的实解析模形式的代数-几何解释。本建议的进一步目的是了解Goldfeld的爱森斯坦级数，以及Chinta- Horozov- O'Sullivan的推广与Brown的实解析模形式类之间的联系。

The aim of this proposal is to understand statistical aspects of noncommutative modular symbols, and to interpret them from an algebraic-geometric point of view. One of the central questions is whether noncommutative modular symbols obey a normal distribution, which has a positive answer for classical, commutative modular symbols (Petridis--Risager). Important motivation for this question comes from a close relationship between statistical properties of modular symbols on one hand, and deep arithmetic problems, like the abc-conjecture, on the other - a connection that was first described by Dorian Goldfeld.A further important aspect of this proposal is to find an algebraic-geometric interpretation of the mathematical structures underlying statistical properties of modular symbols. More precisley, in the seminal work of Petridis--Risager in the commutative case, a central role is played by the detailed study of a certain real analytic function ("Goldfeld's Eisenstein series). While its analytic properties are well understood, the algebraic-geometric interpretation of this function is not clear. In this context, Francis Brown very recently discovered that a relatively large class of real analytic modular forms admits an algebraic-geometric interpretation. A further aim of this proposal is to understand the connection between Goldfeld's Eisenstein series, as well as its generalization due to Chinta--Horozov--O'Sullivan, with Brown's class of real analytic modular forms.