The stable Bernstein center and test functions for Shimura varieties

By Thomas J. Haines. 2013

77 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/92109bd48111c9c92a0e394f14fa4b0e672e7e46

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In Geometry and Topology

1. Xinwen Zhu
2. Peter Scholze
3. Xuhua He
4. Timo Richarz
5. Thomas J. Haines
6. M. Hoff
7. Georgios Pappas
8. Lue Pan

In Mathematical Physics

1. Xinwen Zhu
2. Peter Scholze
3. Xuhua He
4. Timo Richarz
5. Thomas J. Haines
6. M. Hoff
7. Georgios Pappas
8. Lue Pan

Citations in the corpus

• Coherent sheaves on the stack of Langlands parameters (2020, background): “n Proposition 3.1.10 induces the identity map after taking G^-invariants. This algebra is usually called the stable center of G (the quasi-split inner form of G), at least when base changed to C (see [Ha14]). It admits idemponent decompositions indexed by connected components of Loc cG;F;. For a nit”