We study algebraic cycles on moduli spaces ${\mathcal{F}}_h$ of $h$-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 ${\mathcal{F}}_h$. We then study the images of these tautological classes in the cohomology groups of ${\mathcal{F}}_h$ 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${}^{[n]}$-type hyper-Kähler manifolds with $n\le 2$. 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 $L$-functions with mild hypotheses on the size of the Dirichlet coefficients. We verify these hypotheses for all automorphic $L$-functions and (with mild restrictions) the Rankin–Selberg $L$-functions attached to two automorphic representations. The proof relies on a new unconditional log-free zero density estimate for Rankin–Selberg $L$-functions.

We prove the nonvanishing theorem for $3$-folds over an algebraically closed field $k$ of characteristic $p>5$.

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 $L$-functions.