On the Piatetski-Shapiro construction for integral models of Shimura varieties
Pol van Hoften, Jack Sempliner
TL;DR
This work develops and exploits the Piatetski-Shapiro construction to compare Shimura varieties and their p-adic integral models across a totally real Galois extension. By embedding the induced datum into a Hodge-type framework and using a rich 2-categorical and cohomological toolkit, the authors establish fixed-point theorems: Gamma-fixed points on the Shimura varieties and on integral models correspond (under precise hypotheses) to the original G-datum structures, with robust 2-Cartesian and isomorphism results at both generic and integral levels. A key innovation is the use of Igusa stacks to bridge fixed points between the generic fiber and integral models, together with a systematic treatment of local models, shtukas, and the Pappas–Rapoport axioms. The results enable descent-style comparisons and pave the way for constructing new exotic Hecke correspondences and advancing the Langlands program at non-quasi-split primes. Overall, the paper provides a coherent framework relating fixed points, Gamma-actions, and integral models for Shimura varieties via a network of 2-categorical, cohomological, and shtuka-theoretic techniques, with broad implications for arithmetic geometry and automorphic forms.
Abstract
We study the Piatetski-Shapiro construction, which takes a totally real field F and a Shimura datum (G,X) and produces a new Shimura datum (H,Y). If F is Galois, then the Galois group Gamma of F acts on (H,Y), and we show that the Gamma-fixed points of the Shimura varieties for (H,Y) recover the Shimura varieties for (G,X) under some hypotheses. For Shimura varieties of Hodge type with parahoric level, we show that the same is true for the p-adic integral models constructed by Pappas--Rapoport. We also study the Gamma-fixed points of the Igusa stacks of Zhang for (H,Y) and prove optimal results.