Local spectral equidistribution for Siegel modular forms and applications

Kowalski, Emmanuel; Saha, Abhishek; Tsimerman, Jacob

doi:10.1112/s0010437x11007391

Cited by 37 publications

(91 citation statements)

References 42 publications

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“…(ii) This result is a generalization of Proposition 3.3 in Kowalski-Saha-Tsimerman [7]. They applied the estimate to show an equidistribution result for L-functions associated to Siegel cusp forms of genus 2 and growing weight k. So it is expected that our result can be used to prove a similar result for growing level N.…”

Section: Introduction and The Statement Of Main Resultsmentioning

confidence: 65%

On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups

Chida¹,

Katsurada

Matsumoto

2013

Abh. Math. Semin. Univ. Hambg.

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Abstract. We prove a formula of Petersson's type for Fourier coefficients of Siegel cusp forms of degree 2 with respect to congruence subgroups, and as a corollary, show upper bound estimates of individual Fourier coefficient. The method in this paper is essentially a generalization of Kitaoka's previous work which studied the full modular case, but some modification is necessary to obtain estimates which are sharp with respect to the level aspect.

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Section: Introduction and The Statement Of Main Resultsmentioning

confidence: 65%

On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups

Chida¹,

Katsurada

Matsumoto

2013

Abh. Math. Semin. Univ. Hambg.

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“…In a large number of cases, and with high accuracy, the distribution of zeros of automorphic L-functions coincide with the distribution of eigenvalues of random matrices. See [37,85] for numerical investigations and conjectures and see [40,49,50,53,68,82,84] and the references therein for theoretical results.…”

Section: Introductionmentioning

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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“…with the subspace S 2, k replaced by a subspace generated by "suitable" lifts from lower degree, including Ikeda lifts [14]. This will be proved using a result in [20] recalled below. Theorem 1.2 essentially follows from Proposition 7.4 and some standard analytic arguments.…”

Section: Proof Of Theorem 12mentioning

confidence: 96%

“…In this context, we observe that using the main theorem in [20], one obtains as a corollary (cf. Corollary 7.3) that for all sufficiently large k, there exists an eigenform in S 2, k whose first Fourier-Jacobi coefficient is non-zero.…”

Section: Introductionmentioning

confidence: 94%