Arithmetic statistics of modular symbols I

Mazur, Rubin, and Stein have recently formulated a series of conjecturesabout 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.