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The test function conjecture for local models of Weil-restricted groups

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  06 July 2020

Thomas J. Haines  and
Timo Richarz
Thomas J. Haines
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA email tjh@math.umd.edu
Timo Richarz
Affiliation:
Fachbereich Mathematik, TU Darmstadt, Schlossgartenstrasse 7, 64289Darmstadt, Germany email richarz@mathematik.tu-darmstadt.de
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Abstract

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 $p\geqslant 5$. 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.

Keywords

Shimura varietieslocal modelsnearby cycles

MSC classification

Primary: 14G35: Modular and Shimura varieties 11G18: Arithmetic aspects of modular and Shimura varieties 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture)
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 7 , July 2020 , pp. 1348 - 1404
Copyright
© The Authors 2020

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Footnotes

Research of T.J.H. partially supported by NSF DMS-1406787 and by Simons Fellowship 399424, and research of T.R. funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) 394587809.

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