On the Cohomology of Certain Non-Compact Shimura Varieties
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A01=Sophie Morel
Accuracy and precision
Adjoint
Algebraic closure
Archimedean property
Author_Sophie Morel
Automorphism
Base change
Base change map
Calculation
Category=PBMW
Clay Mathematics Institute
Coefficient
Compact element
Compact space
Comparison theorem
Conjecture
Connected space
Connectedness
Constant term
Corollary
Duality (mathematics)
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Existential quantification
Exterior algebra
Finite field
Finite set
Fundamental lemma (Langlands program)
Galois group
General linear group
Haar measure
Hecke algebra
Homomorphism
L-function
Logarithm
Mathematical induction
Mathematician
Maximal compact subgroup
Maximal ideal
Morphism
Neighbourhood (mathematics)
Open set
Parabolic induction
Permutation
Prime number
Ramanujan-Petersson conjecture
Reductive group
Ring (mathematics)
Scientific notation
Shimura variety
Simply connected space
Special case
Subalgebra
Subgroup
Subquotient
Symplectic group
Theorem
Trace formula
Unitary group
Weyl group
Product details
- ISBN 9780691142937
- Weight: 340g
- Dimensions: 152 x 235mm
- Publication Date: 24 Jan 2010
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Paperback
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This book studies the intersection cohomology of the Shimura varieties associated to unitary groups of any rank over Q. In general, these varieties are not compact. The intersection cohomology of the Shimura variety associated to a reductive group G carries commuting actions of the absolute Galois group of the reflex field and of the group G(Af) of finite adelic points of G. The second action can be studied on the set of complex points of the Shimura variety. In this book, Sophie Morel identifies the Galois action - at good places - on the G(Af)-isotypical components of the cohomology. Morel uses the method developed by Langlands, Ihara, and Kottwitz, which is to compare the Grothendieck-Lefschetz fixed point formula and the Arthur-Selberg trace formula. The first problem, that of applying the fixed point formula to the intersection cohomology, is geometric in nature and is the object of the first chapter, which builds on Morel's previous work. She then turns to the group-theoretical problem of comparing these results with the trace formula, when G is a unitary group over Q. Applications are then given.
In particular, the Galois representation on a G(Af)-isotypical component of the cohomology is identified at almost all places, modulo a non-explicit multiplicity. Morel also gives some results on base change from unitary groups to general linear groups.
Sophie Morel is a member in the School of Mathematics at the Institute for Advanced Study in Princeton and a research fellow at the Clay Mathematics Institute.
