We study the geometry of the p‑adic analogues of the complex analytic period spaces first introduced by Griffiths. More precisely, we prove the Fargues–Rapoport conjecture for p‑adic period domains: for a reductive group G over a p‑adic field and a minuscule cocharacter μ of G, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G,μ) is fully Hodge–Newton decomposable.

The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model. The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rødland. A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties.

On any smooth algebraic variety over a p-adic local field, we construct a tensor functor from the category of de Rham p-adic ́etale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a p-adic analogue of Deligne’s classical Riemann–Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of residues; hence the title of the paper. As an application, we show that this p-adic Riemann–Hilbert functor is compatible with the classical one over all Shimura varieties, for local systems attached to representations of the associated reductive algebraic groups.

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In this paper, we prove that the degree of regularity of square systems, a subfamily of the HFE systems, over a prime finite field of odd characteristic q is exactly q and, therefore, prove that inverting square systems algebraically using Gröbner basis algorithm is exponential, when q = \Omega(n), where n is the number of variables of the system.