Let λ be a partition, viewed as a Young diagram. We define the hook difference of a cell of λ to be the difference of its leg and arm lengths. Define h 1,1 (λ) to be the number of cells of λ with hook difference one. In [BF], algebraic geometry is used to prove a generating function identity which implies that h 1,1 is equidistributed with a 2 , the largest part of a partition that appears at least twice, over the partitions of a given size. In this paper, we propose a refinement of the theorem of [BF] and prove some partial results using combinatorial methods. We also obtain a new formula for the q-Catalan numbers which naturally leads us to define a new q,t-Catalan number with a simple combinatorial interpretation.

For an abelian variety A over an algebraically closed non-archimedean field of residue characteristic p, we show that there exists a perfectoid space which is the tilde-limit of lim ← −[p] A.Our proof also works for the larger class of abeloid varieties.

Let X be a proper smooth toric variety over a perfectoid field of prime residue characteristic p. We study the perfectoid space X perf which covers X constructed by Scholze, showing that Pic(X perf ) is canonically isomorphic to Pic(X)[p −1 ]. We also compute the cohomology of line bundles on X perf and establish analogs of Demazure and Batyrev-Borisov vanishing. This generalizes the first author's analogous results for projectivoid space.

We extend the analogy between the extended Robba rings of [Formula: see text]-adic Hodge theory and the one-dimensional affinoid algebras of rigid analytic geometry, proving some fundamental properties that are well known in the latter case. In particular, we show that these rings are regular and excellent. The extended Robba rings are of interest as they are used to build the Fargues–Fontaine curve.