In order to study the arithmetic structure of elliptic modular groups which
are the fundamental groups of compactified modular curves with cuspidal base
points, these truncated Malcev Lie algebras and their direct sums are
considered as elliptic modular motives. Our main result is a new theory of
Hecke operators on these motives which gives a congruence relation to the
Galois action, and a motivic decomposition to Hecke components on which Hecke
operators act as scalar plus nilpotent matrices. Furthermore, we give a
description of these motives as the spaces of noncommutative modular symbols
with the action of Hecke operators.