Iterated Shimura integrals are iterated integrals of classical modular forms. They are elements of the coordinate ring of the relative unipotent completion of $SL_2(Z)$, which we regard as the fundamental group of the modular curve. Francis Brown has proposed that the coordinate ring of the appropriate relative completions of $SL_2(Z)$ generate the (conjectural) tannakian category of mixed modular motives --- the category of mixed motives generated by the motives of classical modular forms.

The goal of this talk is to explain how the classical Hecke operators act on the free abelian group generated by the conjugacy classes of $SL_2(Z)$ and, dually, on those elements of the coordinate ring of the relative completion of $SL_2(Z)$ that are constant on conjugacy classes. This Hecke action commutes with the natural Galois action and each Hecke operator is a morphism of mixed Hodge structure. One surprising fact is that, while the Hecke operators $T_N$ and $T_M$ (acting on conjugacy classes) commute when $N$ and $M$ are relatively prime, $T_p$ and $T_{p^2}$ do not commute when $p$ is a prime. Consequently, the corresponding Hecke algebra is not commutative, in contrast with the classical case.