Estimates for Kloosterman sums for totally real number fields

Bruggeman, Roelof W.; Miatello, Roberto J.; Pacharoni, I.

doi:10.1515/crll.2001.047

Cited by 13 publications

(35 citation statements)

References 14 publications

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“…It is a well-known problem in number theory to estimate these Fourier coefficients. Sum formulas for these Fourier coefficients which are quite similar to our formulas were obtained in [5][6][7]14]. Our formula is derived from the Kuznecov transform and inversion formula which were proved in [14] for the purpose of estimating these coefficients.…”

Section: Introductionsupporting

confidence: 59%

A Whittaker–Plancherel inversion formula forSL(2,R)

Baruch¹,

Mao²

2006

Journal of Functional Analysis

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We prove a Whittaker-Plancherel inversion formula which gives a Whittaker coefficient of a function on SL(2, R) in terms of certain Bessel coefficients of this function. The Bessel coefficients come from Bessel functions attached to irreducible unitary tempered representations. The Kuznecov transform and Kuznecov inversion formula play a central role in the proof of this Whittaker-Plancherel inversion formula.

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Section: Introductionsupporting

confidence: 59%

A Whittaker–Plancherel inversion formula forSL(2,R)

Baruch¹,

Mao²

2006

Journal of Functional Analysis

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“…A bit of terminology is needed before we can define the Kuznetsov trace formula. We closely follow [BMP1] . Consider the algebraic group G = R K/Q (SL 2 ) over Q obtained by restriction of scalars applied to SL 2 over K. We have…”

Section: Preliminariesmentioning

confidence: 99%

“…Here ν ∈ C, and µ is an element of a lattice in the hyperplane ℜ(x) = 0, x ∈ R 2 . In particular, µ is defined by a[γ] µ = 1 for γ ∈ Γ P , again using similar notation to [BMP1]. The series converges for ν > 1 2 , and has meromorphic continuation in ν with Laplaican eigenvalue 1 4 − (ν + µ 1 ) 2 , 1 4 − (ν + µ 2 ) 2 .…”

Section: Fourier Expansionmentioning

confidence: 99%

“…Each representation Π has the form Π = Π 1 ⊗ Π 2 , with Π j an even unitary irreducible representation of SL 2 (R). Table 1 of [BMP1] lists the possible isomorphism classes for each Π j . For each Π we define a spectral parameter ν Π = (ν Π,1 , ν Π,2 ), with ν Π,j as in the last column of the corresponding table.…”

Section: Representations Associated To Automorphic Forms Overmentioning

confidence: 99%

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Quadratic base change and the analytic continuation of the Asai L-function: A new trace formula approach

Herman¹

2016

American Journal of Mathematics

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Using Langlands's beyond endoscopy idea, we study the Asai L-function associated to a real quadratic field K/Q. We prove that the Asai L-function associated to a cuspidal automorphic representation over K has analytic continuation to the complex plane with at most a simple pole at s = 1. We then show if the L-function has a pole then the representation is a base change from Q.While this result is known using integral representations from the work of Asai and Flicker, the approach here uses novel analytic number techniques and gives a deeper understanding of the geometric side of the relative trace formula. Moreover, the approach in this paper will make it easier to grasp more complicated functoriality via beyond endoscopy.We also describe a connection of Langlands's technique to the distinguishing of representations via periods.

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“…Cependant, la formule est aussi valable pour k 2, comme dans le cas des formes modulaires classiques. On peut le prouver en adaptant un argument de passage à la limite dû à Rankin [21], section 5.7, ou en partant de la formule de Kuznetsov prouvée par [2], en choisissant la fonction test de sorte à isoler les formes de poids k.…”

Section: Annales De L'institut Fourierunclassified

Non annulation des fonctions L des formes modulaires de Hilbert au point central

Trotabas¹

2011

Ann. inst. Fourier

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Birch and Swinnerton-Dyer conjecture allows for sharp estimates on the rank of certain abelian varieties defined over É. in the case of the jacobian of the modular curves, this problem is equivalent to the estimation of the order of vanishing at 1/2 of L-functions of classical modular forms, and was treated, without assuming the Riemann hypothesis, by Kowalski, Michel and VanderKam. The purpose of this paper is to extend this approach in the case of an arbitrary totally real field, which necessitates an appeal of Jacquet-Langlands' theory and the adelization of the problem. To show that the L-function (resp. its derivative) of a positive density of forms does not vanish at 1/2, we follow Selberg's method of mollified moments (Iwaniec, Sarnak, Kowalski, Michel and VanderKam among others applied it successfully in the case of classical modular forms). We generalize the Petersson formula, and use it to estimate the first two harmonic moments, this then allows us to match the same unconditional densities as the ones proved over É by Kowalski, Michel and VanderKam. In this setting, there is an additional term, coming from old forms, to control. Finally we convert our estimates for the harmonic moments into ones for the natural moments.

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Estimates for Kloosterman sums for totally real number fields

Cited by 13 publications

References 14 publications

A Whittaker–Plancherel inversion formula forSL(2,R)

A Whittaker–Plancherel inversion formula forSL(2,R)

Quadratic base change and the analytic continuation of the Asai L-function: A new trace formula approach

Non annulation des fonctions L des formes modulaires de Hilbert au point central

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