# On the $p$-adic theory of local models

@inproceedings{Anschutz2022OnT, title={On the \$p\$-adic theory of local models}, author={Johannes Anschutz and Ian Gleason and Joao N. P. Lourencco and Timo Richarz}, year={2022} }

We prove the Scholze--Weinstein conjecture on the existence and uniqueness of local models of local Shimura varieties and the test function conjecture of Haines--Kottwitz in this setting. In order to achieve this, we establish the 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 \'etale cohomology of small v-stacks and analyze the resulting specialization…

## 14 Citations

### Tubular neighborhoods of local models

- Mathematics
- 2022

. We show that the v-sheaf local models of [SW20] are unibranch. In particular, this proves that the scheme-theoretic local models deﬁned in [AGLR22] are always normal with reduced special ﬁber,…

### The connected components of affine Deligne--Lusztig varieties

- Mathematics
- 2022

. We compute the connected components of arbitrary parahoric level aﬃne Deligne–Lusztig varieties for quasisplit reductive groups, by relating them to the connected components of inﬁnite level moduli…

### $F$-zips with additional structure on splitting models of Shimura varieties

- Mathematics
- 2022

We construct universal G-zips on good reductions of the Pappas-Rapoport splitting models for PEL-type Shimura varieties. We study the induced Ekedahl-Oort stratification, which sheds new light on the…

### On integral local Shimura varieties

- Mathematics
- 2022

. We give a construction of integral local Shimura varieties which are formal schemes that generalize the well-known integral models of the Drinfeld p -adic upper half spaces. The construction…

### A G ] 1 2 A pr 2 02 2 TUBULAR NEIGHBORHOODS OF LOCAL MODELS

- Mathematics
- 2022

. We show that the v-sheaf local models of [SW20] are unibranch. In particular, this proves that the scheme-theoretic local models deﬁned in [AGLR22] are always normal with reduced special ﬁber,…

### On the geometric connected components of moduli spaces of p-adic shtukas and local Shimura varieties

- Mathematics
- 2021

We study connected components of local Shimura varieties. Given local shtuka datum (G, b, μ), with G unramified over Qp and (b, μ) HN-irreducible, we determine π0(ShtG,b,[μ],∞×Cp) with its G(Qp)×…

### Geometrization of the Satake transform for mod $p$ Hecke algebras

- Mathematics
- 2022

. We geometrize the mod p Satake isomorphism of Herzig and Henniart– Vign´eras using Witt vector aﬃne ﬂag varieties for reductive groups in mixed characteristic. We deduce this as a special case of a…

### The EKOR-Stratification on the Siegel Modular Stack

- Mathematics
- 2022

. This paper is about the arithmetic geometry of the reduction modulo p of Shimura varieties with parahoric level structure. We realize the EKOR-stratiﬁcation on the special ﬁber of the Siegel…

### On the connectedness of $p$-adic period domains

- Mathematics
- 2022

. We prove that all p -adic period domains (and their non-minuscule analogues) are geometrically connected. This answers a ques-tion of Hartl [Har13] and has consequences to the geometry of Shimura…

### Specialization maps for Scholze's category of diamonds

- Mathematics
- 2020

We introduce the specialization map in Scholze’s theory of diamonds. We consider v-sheaves that “behave like formal schemes” and call them kimberlites. We attach to them: a reduced special fiber, an…