  In the study of the arithmetic structure of elliptic modular groups which are
the fundamental groups of compactified modular curves, these truncated group
algebras and their direct sums are considered to construct 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 their motivic
decomposition. Using our Hecke theory, we show that elliptic modular motives
are the direct sums of pure motives over certain number fields. This fact
implies a kind of algebraicity on iterated Shimura integrals, i.e., multiple
L-values of cusp forms of weight 2, and on the periods of modular Ceresa
cycles.
