Arithmetic statistics of noncommutative modular symbols

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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.