We investigate the maximal finite length submodule of the Breuil–Kisin prismatic cohomology of a smooth proper formal scheme over a $p$-adic ring of integers. This submodule governs pathology phenomena in integral $p$-adic cohomology theories. Geometric applications include a control, in low degrees and mild ramifications, of (1) the discrepancy between two naturally associated Albanese varieties in characteristic $p$, and (2) the kernel of the specialization map in $p$-adic étale cohomology. As an arithmetic application, we study the boundary case of the theory due to Fontaine and Laffaille, Fontaine and Messing, and Kato. Also included is an interesting example, generalized from a construction in Bhatt, Morrow and Scholze's work, which illustrates some of our theoretical results being sharp, and negates a question of Breuil.

In this paper, we propose a modified Kudla–Rapoport conjecture for the Krämer model of unitary Rapoport–Zink space at a ramified prime, which is a precise identity relating intersection numbers of special cycles to derivatives of Hermitian local density polynomials. We also introduce the notion of special difference cycles, which has surprisingly simple description. Combining this with induction formulas of Hermitian local density polynomials, we prove the modified Kudla–Rapoport conjecture when $n=3$. Our conjecture, combining with known results at inert and infinite primes, implies the arithmetic Siegel–Weil formula for all non-singular coefficients when the level structure of the corresponding unitary Shimura variety is defined by a self-dual lattice.

We use the tropical geometry approach to compute absolute and relative enumerative invariants of complex surfaces which are $\mathbb {C} P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block–Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invariants, and the quasi-modularity of their generating series.

Let $G$ be a split reductive group over the ring of integers in a $p$-adic field with residue field $\mathbf {F}$. Fix a representation $\overline {\rho }$ of the absolute Galois group of an unramified extension of $\mathbf {Q}_p$, valued in $G(\mathbf {F})$. We study the crystalline deformation ring for $\overline {\rho }$ with a fixed $p$-adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for $G$-valued representations. In particular, we give a root theoretic condition on the $p$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups.