Title: Tautological classes with twisted coefficients part I
Abstract: Let M_g, for g \geq 2, be the moduli space of smooth curves of genus g. We explain how to associate to any irreducible algebraic representation of Sp(2g) a relative Chow motive V_\lambda over M_g, and how to define a tautological subgroup R(M_g, V_\lambda) inside CH(M_g, V_\lambda). Computing R(M_g, V_\lambda) for all \lambda is equivalent to computing the tautological rings of all fibered powers of the universal curve over M_g simultaneously. We are able to completely determine R(M_g, V_\lambda) for all \lambda when g is at most 4. A particular consequence is that the tautological rings of all fibered powers of the universal curve over Mg satisfy Poincaré duality in these genera. This was previously known only in genus 2. We also obtain results about conjectural failures of Poincaré duality for g \geq 5; specifically, we can show that if certain cycles related to modified diagonals on products of very general curves are nonzero in Chow, then Poincaré duality fails in the tautological ring. (Joint with Qizheng Yin.)