Dynamics of continued fractions and distribution of modular symbols

Abstract

We formulate a dynamical approach to the study of distribution of modular symbols, motivated by the work of Baladi–Vallée. We introduce the modular partition functions of continued fractions and observe that the modular symbols are special cases of modular partition functions. We prove the limit Gaussian distribution and residual equidistribution for modular partition functions as random variables on the set of rationals whose denominators are up to a fixed positive integer, by studying the spectral properties of the transfer operator associated to the underlying dynamics. The approach leads to a few applications. We show an average version of a conjecture of Mazur–Rubin on statistics for modular symbols of rational elliptic curves. We further observe that the equidistribution of mod $p$ values of modular symbols leads to a mod $p$ non-vanishing result for special modular $L$-values twisted by Dirichlet characters.