The generalized Miller–Morita–Mumford classes of a manifold bundle with fiber M depend only on the underlying τM-fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for τM-fibrations, Baut(τM), and its cohomology ring, i.e., the ring of characteristic classes of τM-fibrations. For a bundle ξ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal ξ-fibration with holonomy in a given connected monoid, together with explicit cocycle representatives for the characteristic classes of the canonical bundle over its total space. This yields tools for computing the rational cohomology ring of Baut(ξ) as well as the subring generated by the generalized Miller–Morita–Mumford classes. To illustrate, we carry out sample computations for spheres and complex projective spaces. We discuss applications to tautological rings of simply connected manifolds and to the problem of deciding whether a given τM-fibration comes from a manifold bundle.
We present a general conjecture on congruences between Hecke eigenvalues of parabolically induced and cuspidal automorphic representations of split reductive groups, modulo divisors of critical values of certain L-functions. We examine the consequences in several special cases and use the Bloch–Kato conjecture to further motivate a belief in the congruences.
We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of degree three, such as dimension formulas and traces of Hecke operators. We also use it to predict several lifts from genus 1 to genus 3, as well as lifts from and new congruences of Harder type.
This paper provides explicit closed formulas in terms of tautological classes for the cycle classes of the height and Artin invariant strata in families of K3 surfaces. The proof is uniform for all strata and uses a flag space as the computations in Ekedahl and van der Geer (Algebra, arithmetic and geometry, progress in mathematics, vol. 269-270, Birkhauser, Basel, 2010) for the Ekedahl-Oort strata for families of abelian varieties, but employs a Pieri formula to determine the push down to the base space.