Sum Formula for SL 2 over a Number Field and Selberg Type Estimate for Exceptional Eigenvalues

Bruggeman, Roelof W.; Miatello, Roberto J.

doi:10.1007/s000390050069

Cited by 29 publications

(53 citation statements)

References 9 publications

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“…The same method led us in [5] to a Selberg type estimate for the coordinates of the eigenvalues. We apply the sum formula to a convenient test function and we estimate the right hand side of the sum formula.…”

Section: Estimation Of the Measurementioning

confidence: 97%

“…Note that d r k qY n Q 0 for all weights q that occur in an irreducible unitary representation with spectral parameter n. This d r k is related to the function d r in (4) of [5], but not exactly equal. A computation based on (11) shows that the equations a k rY 6Y q c r k 6d r k qY n 6 Y 17…”

Section: Automorphic Formsmentioning

confidence: 99%

“…In [5], we restricted ourselves to trivial K-types. In that situation, the weight is zero at all in®nite places, so the automorphic representations have no factors of discrete series type (characterized by n j e 1 2 Z 0 ).…”

Section: Estimation Of the Measurementioning

confidence: 99%

“…In order to do an exact inversion, it will now be necessary to use a version of the sum formula that includes the contribution of non-trivial K-types. (This is not guaranteed for the Kuznetsov formula in, for instance, [5]. This will allow to get an arbitrary smooth function of compact support ask in (4).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Estimates for Kloosterman sums for totally real number fields

Bruggeman¹,

Miatello²,

Pacharoni³

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Section: Estimation Of the Measurementioning

confidence: 97%

Section: Automorphic Formsmentioning

confidence: 99%

Section: Estimation Of the Measurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Estimates for Kloosterman sums for totally real number fields

Bruggeman¹,

Miatello²,

Pacharoni³

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…C'est ce résultat, écrit sous une autre forme, qui est au coeur de la proposition 9.4 de [1], et par conséquent de la formule de Kuznetsov. Dans notre situation, en section 5, on pourrait conclure plus élémentairement (pour F = Q, la formule de Petersson se passe de celui-ci), mais il est important de remarquer qu'il est central dans toutes les formes "difficiles" des formules de traces de Petersson/Kuznetsov.…”

Section: Représentations Et Fonctions De Besselunclassified

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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Density results for automorphic forms on Hilbert modular groups

Bruggeman

Miatello

2003

Geometric And Functional Analysis

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Abstract. We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for SL 2 over a totally real number field F, with discrete subgroup of Hecke type Γ 0 (I) for a non-zero ideal I in the ring of integers of F. The weights are products of Fourier coefficients. This implies in particular the existence of infinitely many cuspidal automorphic representations with multieigenvalues in various regions growing to infinity. For instance, in the quadratic case, the regions include floating boxes, floating balls, sectors, slanted strips (see §1.2.4-1.2.13) and products of prescribed small intervals for all but one of the infinite places of F. The main tool in the derivation is a sum formula of Kuznetsov type ([3], Theorem 2.1).

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Sum Formula for SL 2 over a Number Field and Selberg Type Estimate for Exceptional Eigenvalues

Cited by 29 publications

References 9 publications

Estimates for Kloosterman sums for totally real number fields

Estimates for Kloosterman sums for totally real number fields

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

Density results for automorphic forms on Hilbert modular groups

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