Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the $p$-adic theory of local models

Johannes Anschütz, Ian Gleason, João Lourenço, Timo Richarz

TL;DR

The paper addresses the problem of producing and understanding integral models for local Shimura varieties in the p-adic, v-sheaf context. It develops a specialization-triple framework via π-adic kimberlites to relate generic and special fibers, and proves the Scholze--Weinstein conjecture (existence/uniqueness) for minuscule μ, together with the Haines--Kottwitz test function conjecture in this setting, by analyzing local models as v-sheaf closures inside the Beilinson--Drinfeld Grassmannian and computing their special fibers through hyperbolic localization and nearby cycles. It also compares v-sheaf local models with schematic local models, proving canonical deperfection results and Cohen–Macaulay/Frobenius-split properties under mild hypotheses; these results are extended to the convolution setting to obtain a robust, functorial, group-theoretic framework compatible with earlier constructions (PZ, Lou, FHLR) and with potential applications to p-adic shtukas and Langlands correspondence. The work leverages the six-functor formalism for diamonds, the geometry of affine Grassmannians/flag varieties in the Witt vector setting, and a detailed analysis of G_m-actions on semi-infinite orbits to control nearby cycles and their centrality, culminating in a complete proof of the minuscule case and substantial progress toward the general theory. Overall, the results provide a unifying, functorial approach to local models in p-adic geometry with concrete cohomological applications such as the HK trace formula, and offer a solid bridge between v-sheaf technology and classical schematic moduli problems in the Langlands program.

Abstract

We prove the Scholze--Weinstein conjecture on the existence and uniqueness of local models for local Shimura varieties, as well as the test function conjecture of Haines--Kottwitz in this framework. To this end, we establish a specialization principle for well-behaved $p$-adic kimberlites, show that these include the v-sheaf local models, determine their special fibers using hyperbolic localization for the étale cohomology of small v-stacks, and analyze the resulting specialization morphism using convolution.

On the $p$-adic theory of local models

TL;DR

The paper addresses the problem of producing and understanding integral models for local Shimura varieties in the p-adic, v-sheaf context. It develops a specialization-triple framework via π-adic kimberlites to relate generic and special fibers, and proves the Scholze--Weinstein conjecture (existence/uniqueness) for minuscule μ, together with the Haines--Kottwitz test function conjecture in this setting, by analyzing local models as v-sheaf closures inside the Beilinson--Drinfeld Grassmannian and computing their special fibers through hyperbolic localization and nearby cycles. It also compares v-sheaf local models with schematic local models, proving canonical deperfection results and Cohen–Macaulay/Frobenius-split properties under mild hypotheses; these results are extended to the convolution setting to obtain a robust, functorial, group-theoretic framework compatible with earlier constructions (PZ, Lou, FHLR) and with potential applications to p-adic shtukas and Langlands correspondence. The work leverages the six-functor formalism for diamonds, the geometry of affine Grassmannians/flag varieties in the Witt vector setting, and a detailed analysis of G_m-actions on semi-infinite orbits to control nearby cycles and their centrality, culminating in a complete proof of the minuscule case and substantial progress toward the general theory. Overall, the results provide a unifying, functorial approach to local models in p-adic geometry with concrete cohomological applications such as the HK trace formula, and offer a solid bridge between v-sheaf technology and classical schematic moduli problems in the Langlands program.

Abstract

We prove the Scholze--Weinstein conjecture on the existence and uniqueness of local models for local Shimura varieties, as well as the test function conjecture of Haines--Kottwitz in this framework. To this end, we establish a specialization principle for well-behaved -adic kimberlites, show that these include the v-sheaf local models, determine their special fibers using hyperbolic localization for the étale cohomology of small v-stacks, and analyze the resulting specialization morphism using convolution.
Paper Structure (38 sections, 96 theorems, 263 equations)

This paper contains 38 sections, 96 theorems, 263 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Explanation on perfectoid spaces, diamonds and $v$-sheaves
  4. Strategy of proof
  5. Kimberlites and specialization triples
  6. Special fibers of v-sheaf local models
  7. Special fibers of schematic local models
  8. Specialization maps
  9. Conclusion
  10. Acknowledgements
  11. v-sheaf theory
  12. Closures
  13. The two different diamond functors
  14. $\pi$-adic kimberlites
  15. The affine flag variety
  16. ...and 23 more sections

Key Result

Theorem 1.2

Let $\mu$ be minuscule. Then, the following hold:

Theorems & Definitions (226)

  • Conjecture 1.1: Scholze--Weinstein
  • Theorem 1.2: \ref{['thm_local_model_representable']}, \ref{['cor_special_fiber_local_model_reduced']}
  • Theorem 1.3: \ref{['sec:test-funct-conj-reformulation-implies-test-function-conjecture']}
  • Theorem 1.4: \ref{['prop_fully_faith_triples_kimberlites']}, \ref{['LM_kimberlite']}
  • Theorem 1.5: \ref{['theorem_special_fiber_admissible']}
  • Theorem 1.6: \ref{['BD.attractor.groups.comparison.theorem']}
  • Proposition 1.7: \ref{['prop_closure_relations_semi_infinite_orbits']}, \ref{['equidimensionality.MV.cycles.lemma']}
  • Theorem 1.8: \ref{['prop_ula_special_fiber']}, \ref{['ULA_nearby_cycles']}
  • Theorem 1.10: \ref{['theorem_coherence_allp']}
  • Theorem 1.11: \ref{['lem_comparison_sch_vars_equi_and_mixed']}, FHLR22
  • ...and 216 more