Constructible exponential functions, motivic Fourier transform and transfer principle

Cluckers, Raf; Loeser, François

doi:10.4007/annals.2010.171.1011

Cited by 81 publications

(156 citation statements)

References 26 publications

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“…The form of our expansions in Theorems 5.3 and 5.12 is consistent with the results in [11] and [13], where a similar form of such expansions is derived for p big enough, in a more general setting, but without information on the powers of |z| K and ln |z| K .…”

Section: Expansions For Oscillatory Integralssupporting

confidence: 88%

“…In the context of meromorphic functions however, one typically has expansions as in (1.2), even when the support of Φ is arbitrarily small! In [13], see also [11], R. Cluckers and F. Loeser obtained similar expansions to those presented in Theorem 5.12, for p big enough, in a more general nonArchimedean setting, but without information on the powers of |z| K and ln |z| K . We also note that the existence of two types of asymptotic expansions for p-adic oscillatory integrals attached to Laurent polynomials, satisfying certain very restrictive conditions, was established by E. León-Cardenal and the second author in [34].…”

supporting

confidence: 56%

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Zeta functions and oscillatory integrals for meromorphic functions

Veys

Zúñiga–Galindo

2017

Advances in Mathematics

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Abstract. In the 70's Igusa developed a uniform theory for local zeta functions and oscillatory integrals attached to polynomials with coefficients in a local field of characteristic zero. In the present article this theory is extended to the case of rational functions, or, more generally, meromorphic functions f /g, with coefficients in a local field of characteristic zero. This generalization is far from being straightforward due to the fact that several new geometric phenomena appear. Also, the oscillatory integrals have two different asymptotic expansions: the 'usual' one when the norm of the parameter tends to infinity, and another one when the norm of the parameter tends to zero. The first asymptotic expansion is controlled by the poles (with negative real parts) of all the twisted local zeta functions associated to the meromorphic functions f /g − c, for certain special values c. The second expansion is controlled by the poles (with positive real parts) of all the twisted local zeta functions associated to f /g.

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Section: Expansions For Oscillatory Integralssupporting

confidence: 88%

supporting

confidence: 56%

Zeta functions and oscillatory integrals for meromorphic functions

Veys

Zúñiga–Galindo

2017

Advances in Mathematics

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“…To this aim we have to use the field‐independent descriptions provided by model theory and uniform integration, cf. . More precisely, we introduce a class of distributions given by uniform, field‐independent descriptions, called of

{scriptC}^{\exp}

‐class.…”

Section: Introductionmentioning

confidence: 99%

Distributions and wave front sets in the uniform non‐archimedean setting

Cluckers

Halupczok²,

Loeser

et al. 2018

Trans. London Math. Soc.

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We study some constructions on distributions in a uniform p‐adic context, and also in large positive characteristic, using model theoretic methods. We introduce a class of distributions which we call distributions of scriptC exp ‐class and which is based on the notion of scriptC exp ‐class functions from Cluckers and Halupczok [J. Ecole Polytechnique (JEP) 5 (2018) 45–78]. This class of distributions is stable under Fourier transformation and has various forms of uniform behavior across non‐archimedean local fields. We study wave front sets, pull‐backs and push‐forwards of distributions of this class. In particular, we show that the wave front set is always equal to the complement of the zero locus of a scriptC exp ‐class function. We first revise and generalize some of the results of Heifetz that he developed in the p‐adic context by analogy to results about real wave front sets by Hörmander. In the final section, we study sizes of neighborhoods of local constancy of Schwartz–Bruhat functions and their push‐forwards in relation to discriminants.

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“…Let us recall one of the results of [25], the first part of which generalizes a result of [30], and which shows that the class of constructible functions is a natural class to work with for the purposes of integration. Note that although the theorem is stated for the affine measure on F n , it also holds for measures given by definable differential forms, by working with charts as is done in [29, §15].…”

Section: B2 Definable Sets and Constructible Functionsmentioning

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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Constructible exponential functions, motivic Fourier transform and transfer principle

Cited by 81 publications

References 26 publications

Zeta functions and oscillatory integrals for meromorphic functions

Zeta functions and oscillatory integrals for meromorphic functions

Distributions and wave front sets in the uniform non‐archimedean setting

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

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