HECKE ACTIONS ON LOOPS AND PERIODS OF ITERATED SHIMURA INTEGRALS

We show that the classical Hecke correspondences TN act on the free abelian group generated by the conjugacy classes of the modular group SL2.Z/and the conjugacy classes of its profinite completion. We show that this action induces a dual action on the ring of class functions of a certain relative unipotent completion of the modular group. This ring contains all iterated integrals of modular forms that are constant on conjugacy classes. It possesses a natural mixed Hodge structure and, after tensoring with Q`, a natural action of the absolute Galois group. Each Hecke correspondence preserves this mixed Hodge structure and commutes with the action of the absolute Galois group. Unlike in the classical case, where Hecke correspondences are acting on modular forms, the algebra generated by these generalized Hecke operators is not commutative.

Duke Scholars

Author Richard Hain Mathematics