Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe test function conjecture for parahoric local models
75
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure, and their analogues in equal characteristic.
rate research
Read More
We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over $p$-adic local fields with $pgeq 5$. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.
We study the singularities of integral models of Shimura varieties and moduli stacks of shtukas with parahoric level structure. More generally our results apply to the Pappas-Zhu and Levin mixed characteristic parahoric local models, and to their equal characteristic analogues. For any such local model we prove under minimal assumptions that the entire local model is normal with reduced special fiber and, if $p>2$, it is also Cohen-Macaulay. This proves a conjecture of Pappas and Zhu, and shows that the integral models of Shimura varieties constructed by Kisin and Pappas are Cohen-Macaulay as well.
We elaborate the theory of the stable Bernstein center of a $p$-adic group $G$, and apply it to state a general conjecture on test functions for Shimura varieties due to the author and R. Kottwitz. This conjecture provides a framework by which one might pursue the Langlands-Kottwitz method in a very general situation: not necessarily PEL Shimura varieties with arbitrary level structure at $p$. We give a concrete reinterpretation of the test function conjecture in the context of parahoric level structure. We also use the stable Bernstein center to formulate some of the transfer conjectures (the fundamental lemmas) that would be needed if one attempts to use the test function conjecture to express the local Hasse-Weil zeta function of a Shimura variety in terms of automorphic $L$-functions.
We give upper bounds for the level and the Pythagoras number of function fields over fraction fields of integral Henselian excellent local rings. In particular, we show that the Pythagoras number of $mathbb{R}((x_1,dots,x_n))$ is $leq 2^{n-1}$, which answers positively a question of Choi, Dai, Lam and Reznick.
This is the final version, to appear in Commentarii Mathematici Helvetici.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
