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We study algebraic cycles on moduli spaces of -polarized hyper-Kähler manifolds. Following previous work of Marian, Oprea, and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces, we first define the tautological ring on . We then study the images of these tautological classes in the cohomology groups of and prove that most of them are linear combinations of Noether–Lefschetz cycle classes. In particular, we prove the cohomological version of the tautological conjecture on moduli space of K3-type hyper-Kähler manifolds with . Secondly, we prove the cohomological generalized Franchetta conjecture on a universal family of these hyper-Kähler manifolds.
We describe a new method to obtain weak subconvexity bounds for -functions with mild hypotheses on the size of the Dirichlet coefficients. We verify these hypotheses for all automorphic -functions and (with mild restrictions) the Rankin–Selberg -functions attached to two automorphic representations. The proof relies on a new unconditional log-free zero density estimate for Rankin–Selberg -functions.
We study a local multiplicity problem related to so-called generalized Shalika models. By establishing a local trace formula for these kinds of models, we are able to prove a multiplicity formula for discrete series. As a result, we can show that these multiplicities are, for discrete series, invariant under the local Jacquet–Langlands correspondence and are related to local exterior square -functions.