On the computational complexity of modular symbols

Abstract

Efficient algorithms are obtained for integrating holomorphic differential one-forms along simple geodesic lines on those compact Riemann surfaces which are given as quotients of the upper half-plane by a congruence subgroup $\Gamma$ of ${\text {SL}}(2,\mathbb {Z})$. We may assume that every geodesic line passes through a cusp which is unique up to $\Gamma$-equivalence. The algorithms we construct run in polynomial time in the height of this cusp.