We extend our previous theory of étale parallel transport to a larger class of slope zero vector bundles on $p$-adic curves. The new class is stable under pullback by ramified coverings. We also construct $p$-adic representations of a central extension of the fundamental group for certain bundles of non-zero slope.

We study the Picard variety of the Fano surface of nodal and mildly cuspidal cubic threefolds in arbitrary characteristic by relating divisors on the Fano surface to divisors on the symmetric product of a curve of genus 4.

The variety of all smooth hypersurfaces of given degree and dimension has the Fermat hypersurface as a natural base point. In order to study the period map for such varieties, we first determine the integral polarized Hodge structure of the primitive cohomology of a Fermat hypersurface (as a module over the automorphism group of the hypersurface). We then focus on the degree 3 case and show that the period map for cubic fourfolds as analyzed by R. Laza and the author gives complete information about the period map for cubic hypersurfaces of lower dimension. In particular, we thus recover the results of Allcock–Carlson–Toledo on the cubic surface case.

We construct a canonical compactification $SQ_{g,K}^{\mathrm{toric}}$ of the moduli space $A_{g,K}$ of abelian varieties over $\mathbf{Z}[\zeta_N, 1/N]$ by adding certain reduced singular varieties along the boundary of $A_{g,K}$, where $K$ is a symplectic finite abelian group, $N$ is the maximal order of elements of $K$, and $\zeta_N$ is a primitive $N$-th root of unity. In [18] a canonical compactification $SQ_{g,K}$ of $A_{g,K}$ was constructed by adding possibly non-reduced GIT-stable (Kempf-stable) degenerate abelian schemes. We prove that there is a canonical bijective finite birational morphism sq : $SQ_{g,K}^{\mathrm{toric}} \to SQ_{g,K}$. In particular, the normalizations of $SQ_{g,K}^{\mathrm{toric}}$ and $SQ_{g,K}$ are isomorphic.

We consider the question that the spectrum and arithmetic of locally symmetric spaces defined by congruent arithmetical lattices should mutually determine each other. We frame these questions in the context of automorphic representations.

Associated to reductive groups and their maximal parabolic subgroups are genuine zeta functions. Naturally related to Riemann's zeta function and governed by symmetries, including that of Weyl, these zeta functions are expected to satisfy the Riemann Hypothesis.

Stability plays a central role in arithmetic. In this article, we explain some basic ideas and present certain constructions for our studies. There are two aspects: namely, general Class Field Theories for Riemann surfaces using semi-stable parabolic bundles and for $p$-adic number fields using what we call semi-stable filtered $(\varphi, N;\omega)$-modules; and non-abelian zeta functions for function fields over finite fields using semi-stable bundles and for number fields using semi-stable lattices.

We give a purely local proof, in the depth 0 case, of the result by Harris–Taylor which asserts that the local Langlands correspondence for $GL_n$ is realized in the vanishing cycle cohomology of the deformation spaces of one-dimensional formal modules of height $n$. Our proof is given by establishing the direct geometric link with the Deligne–Lusztig theory for $GL_n (\mathbb{F}_q)$.

We construct an action of a Lie algebra on the homology groups of moduli spaces of stable sheaves on $K3$ surfaces under some technical conditions. This is a generalization of Nakajima's construction of $\mathfrak{sl}_2$-action on the homology groups [N6]. In particular, for an $A$, $D$, $E$-type configulation of $(-2)$-curves, we shall give a collection of moduli spaces such that the associated Lie algebra acts on their homology groups.

Let $k$ be a perfect field of characteristic $p \geq 3$. We classify $p$-divisible groups over regular local rings of the form $$ W(k) [[t_1, \dots, t_r, u]]/(u^e + pb_{e-1} u^{e-1} + \dots + pb_1 u + pb_0), $$ where $b_0, \dots, b_{e-1} \in W(k) [[t, \dots, t_r]]$ and $b_0$ is an invertible element. This classification was in the case $r = 0$ conjectured by Breuil and proved by Kisin.