Ax-Kochen-Eršov Theorems for p-adic integrals and motivic integration

Cluckers, Raf; Loeser, François

doi:10.1007/0-8176-4417-2_5

Cited by 25 publications

(37 citation statements)

References 35 publications

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“…(see [Den,Theorem 3.1] for a weaker version and [CL,Theorems 4.4 and 6.9] for a stronger version) Suppose that f : VF n × VF m × VG m → VG is a definable function. Assume that, for every a ∈ F m and λ ∈ Z m , the integral I(a, λ) = F n q f (x,a,λ) dx converges.…”

mentioning

confidence: 99%

Representation growth and rational singularities of the moduli space of local systems

Aizenbud

Avni²

2015

Invent. math.

115

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Let G be a semisimple algebraic group defined over Q p , and let Γ be a compact open subgroup of G(Q p ). We relate the asymptotic representation theory of Γ and the singularities of the moduli space of G-local systems on a smooth projective curve, proving new theorems about both:1. We prove that there is a constant C, independent of G, such that the number of n-dimensional representations of Γ grows slower than n C , confirming a conjecture of Larsen and Lubotzky. In fact, we can take C = 3 · dim(E 8 ) + 1 = 745. We also prove the same bounds for groups over local fields of large enough characteristic.2. We prove that the coarse moduli space of G-local systems on a smooth projective curve of genus at least C/2 + 1 = 374 has rational singularities.For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities.

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confidence: 99%

Representation growth and rational singularities of the moduli space of local systems

Aizenbud

Avni²

2015

Invent. math.

115

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“…For a definable set X , a collection f = ( f F ) F of functions f F : X F → C is called a constructible function if there exist integers N , N , and N , such that f F has the form, for x ∈ X F , for all F ∈ C O , The motivation for such a definition of a constructible function comes from the theory of integration: namely, constructible functions form a rich class of functions which is stable under integration with respect to parameters (as in Theorem 14.4 below). See [28,43] for details.…”

Section: B2 Definable Sets and Constructible Functionsmentioning

confidence: 99%

“…Much of the preliminary and introductory material is quoted freely from [25][26][27][28]43], sometimes without mentioning these ubiquitous citations.…”

Section: Appendix A: By Robert Kottwitzmentioning

confidence: 99%

Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ L -functions

2015

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We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let G be a reductive group over a number field F which admits discrete series representations at infinity. Let L G = G Gal(F/F) be the associated L-group and r : L G → GL(d, C) a continuous homomorphism which is irreducible and does not factor through Gal(F/F). The families under consideration consist of discrete automorphic representations of G(A F ) of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato-Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321-331, 1987) and Serre (J Am Math Soc 10(1):75-102, 1997). As an application we study the distribution of the lowlying zeros of the associated family of L-functions L(s, π, r ), assuming from the principle of functoriality that these L-functions are automorphic. We find that the distribution of the 1-level densities coincides with the distribution of the 1-level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz-Sarnak heuristics. We provide a criterion based on the Frobenius-Schur indicator to determine this symmetry type. If r is not isomorphic to its dual r ∨ then the symmetry type is unitary. Otherwise there is a bilinear form on C d which realizes the isomorphism between r and r ∨ . If the bilinear form is symmetric (resp. alternating) then r is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).

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“…For the reader's convenience we shall start by recalling briefly some definitions, notations and constructions from [8] and [9] that will be used in this article. For an introduction to this circle of ideas we refer to the surveys [6], [2] and [11] and the notes [4], [5] and [7].…”

Section: Motivic Integration and Constructible Motivic Functionsmentioning

confidence: 99%

On the Commutativity of Pull-Back and Push-Forward Functors on Motivic Constructible Functions

Cely¹,

Raibaut²

2019

J. symb. log.

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Ax-Kochen-Eršov Theorems for p-adic integrals and motivic integration

Cited by 25 publications

References 35 publications

Representation growth and rational singularities of the moduli space of local systems

Representation growth and rational singularities of the moduli space of local systems

Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ L -functions

On the Commutativity of Pull-Back and Push-Forward Functors on Motivic Constructible Functions

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