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We introduce the notion of completed F-crystals on the absolute prismatic site of a smooth p-adic formal scheme. We define a functor from the category of completed prismatic F-crystals to that of crystalline étale Zp-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a complete discrete valuation ring with perfect residue field.

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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.

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No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

The aim of the present paper is to suggest that statistical physics provides the correct language to understand the practical behavior of the LLL algorithm, most of which are left unexplained to this day. To this end, we propose sandpile models that imitate LLL with compelling accuracy, and prove for these models some of the most desired statements regarding LLL. We also formulate a few conjectures that formally capture our heuristics and would serve as milestones for further development of the theory.

In this paper we derive refined Petersson/Kuznetsov trace formulae with prescribed local ramifications. The spectral side of these formulae are much shorter than the standard versions. We use them to study the first moment and the subconvexity bound of certain Rankin-Selberg L-function in a hybrid setting.