Integrability of oscillatory functions on local fields: Transfer principles

Cluckers, Raf; Gordon, Julia; Halupczok, Immanuel

doi:10.1215/00127094-2713482

Cited by 29 publications

(74 citation statements)

References 29 publications

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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%

“…Finally, the question is reduced to the study of Presburger constructible functions of several Z-variables, which are similar to constructible functions as defined above in Sect. B.2, but without the factors #( p −1 i F (x)), see [25]. Roughly, Presburger constructible functions in x ∈ Z r are sums of products of piecewise linear functions in x and of powers of q F , where the power also depends piecewise linearly on x.…”

Section: B3 Boundedness Of Constructible Functionsmentioning

confidence: 99%

“…Since the number of terms is bounded, one obtains an upper bound of the right form. The reduction to single terms instead of their sum is made precise via the Parametric Rectilinearization (see Theorem 2.1.9 of [25]) and Lemma 2.1.8 of [25]. In summary, the main tools used to obtain these rather strong results with seeming ease are the cell decomposition theorem and the understanding of Presburger constructible functions.…”

Section: B3 Boundedness Of Constructible Functionsmentioning

confidence: 99%

“…This case (1) reduces to the case that the α i and β i j are restrictions of Z-linear functions and that W = s × N for some ≥ 0 and some finite set s depending on s ∈ Z n by Theorem 2.1.9 of [25] applied to X = S × W with S = Z n in the notation of that theorem. If s is a singleton, then the result follows from Lemma 2.1.8 of [25]. For s with at least two elements, one replaces H F by the sum of (H F + 1) 2 over the elements of s and the proof is completed by Theorem 14.4 and induction on r .…”

Section: B3 Boundedness Of 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%

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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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Section: B2 Definable Sets and Constructible Functionsmentioning

confidence: 99%

Section: B3 Boundedness Of Constructible Functionsmentioning

confidence: 99%

Section: B3 Boundedness Of Constructible Functionsmentioning

confidence: 99%

Section: B3 Boundedness Of 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%

See 3 more Smart Citations

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

2015

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

Cluckers

Loeser²

Progress in Mathematics

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We give the p-adic and F q ((t)) analogue of the real van der Corput Lemma, where the real condition of sufficient smoothness for the phase is replaced by the condition that the phase is a convergent power series. This van der Corput style result allows us, in analogy to the real situation, to study singular Fourier transforms on suitably curved (analytic) manifolds and opens the way for further applications. As one such further application we give the restriction theorem for Fourier transforms of L p functions to suitably curved analytic manifolds over non-archimedean local fields, similar to the real restriction result by E. Stein and C. Fefferman.Key words and phrases. -van der Corput Lemma, p-adic oscillatory integrals of the first kind, singular Fourier transforms, p-adic restriction theorems, p-adic analytic manifolds of finite type.

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Transfer Principles for Bounds of Motivic Exponential Functions

Cluckers¹,

Gordon²,

Halupczok³

2016

Simons Symposia

Self Cite

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We study transfer principles for upper bounds of motivic exponential functions and for linear combinations of such functions, directly generalizing the transfer principles from [7] and [13, Appendix B]. These functions come from rather general oscillatory integrals on local fields, and can be used to describe e.g. Fourier transforms of orbital integrals. One of our techniques consists in reducing to simpler functions where the oscillation only comes from the residue field.2000 Mathematics Subject Classification. Primary 14E18; Secondary 22E50, 40J99.

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Integrability of oscillatory functions on local fields: Transfer principles

Cited by 29 publications

References 29 publications

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

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

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

Transfer Principles for Bounds of Motivic Exponential Functions

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