Hecke Actions on Loops and Periods of Iterated Shimura Integrals

Attend a one-hour conference talk by R. Hain from Duke University on "Hecke Actions on Loops and Periods of Iterated Shimura Integrals" as part of the "Periods, Shafarevich Maps and Applications" conference at the University of Miami. Explore the concept of iterated Shimura integrals as elements of the coordinate ring of the relative unipotent completion of SL_2(Z), viewed as the fundamental group of the modular curve. Examine Francis Brown's proposal regarding the coordinate ring of relative completions of SL_2(Z) and its potential to generate the tannakian category of mixed modular motives. Delve into the action of classical Hecke operators on the free abelian group generated by conjugacy classes of SL_2(Z) and their dual action on elements of the coordinate ring constant on conjugacy classes. Investigate how this Hecke action commutes with the natural Galois action and preserves mixed Hodge structures. Discover the surprising non-commutativity of certain Hecke operators, such as T_p and T_{p^2} for prime p, contrasting with the classical case and resulting in a non-commutative Hecke algebra.