# Arithmetic statistics of modular symbols

## TL;DR

This paper proves conjectures about the statistical behavior of modular symbols, including their moments and distribution, using analytic techniques involving Eisenstein series, advancing understanding of their asymptotic properties.

## Contribution

It establishes average growth of moments and the Gaussian distribution of modular symbols, refining previous conjectures with new analytic proofs.

## Key findings

- Proved asymptotic growth of first and second moments of modular symbols.
- Established Gaussian distribution of normalized modular symbols with cusp restrictions.
- Used analytic properties of Eisenstein series twisted by modular symbols.

## Abstract

Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve $L$-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.