A Sato–Tate law for GL (3)

Blomer, Valentin; Buttcane, Jack; Raulf, Nicole

doi:10.4171/cmh/337

Cited by 29 publications

(33 citation statements)

References 3 publications

(5 reference statements)

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“…, for some constant > 0, and any fixed > 0 as → ∞. Similar results were independently obtained by Blomer [Blo13] and improved later in [BBR14] and more recently in [BZ20], where an interesting technique is developed to remove the arithmetic weight L .…”

Section: +3supporting

confidence: 73%

“…An orthogonality relation for Maass cusp forms on GL(3, R) was first proved independently by Goldfeld-Kontorovich [GK13] and Blomer [Blo13] in 2013. Further results on orthogonality relations for GL(3, R) were obtained by Blomer-Buttcane-Raulf [BBR14] and Guerreiro [Gue15]. In his 2013 thesis (see [Zho13], [Zho14]) Fan Zhou conjectured a very general orthogonality relation for GL( ) for ≥ 2.…”

Section: Introduction and Main Theoremmentioning

confidence: 95%

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An orthogonality relation for (with an appendix by Bingrong Huang)

Goldfeld

Stade

Woodbury

2021

Forum of Mathematics, Sigma

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Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm {GL}(1)$ ) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm {GL}(2)$ and $\mathrm {GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm {GL}(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.

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Section: +3supporting

confidence: 73%

Section: Introduction and Main Theoremmentioning

confidence: 95%

An orthogonality relation for (with an appendix by Bingrong Huang)

Goldfeld

Stade

Woodbury

2021

Forum of Mathematics, Sigma

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“…If α π (p) ∞ 1 + δ, then λ π (p l ) (l + 1)(l + 2) for some sufficiently large l = l(δ), see [BBR,(24) so that by Cauchy-Schwarz and Theorem 2 the right hand side of (6.1) is bounded by ≪ (N p 2lk ) ε 2 −2k (N 2 + N 1/2 p 2kl ) 1/2 N ≪ (N p 2lk ) ε 2 −2k (N 2 + N 5/4 p kl ).…”

Section: Proofs Of Theorems 2 -5mentioning

confidence: 99%

Applications of the Kuznetsov formula on GL(3) II: the level aspect

2017

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“…Blomer has given a non-optimal version of the spectral large sieve inequalities and Lindelöf on average for the second moment of SL(3) L-functions [3]. Blomer, Raulf, and the author have given results on the distribution of the Fourier coefficients of SL(3) forms, including a Sato-Tate law on average [4]. Kontorovich and Goldfeld obtained Date: 14 November 2014.…”

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confidence: 99%

The Spectral Kuznetsov Formula on $SL(3)$

Buttcane

2016

Trans. Amer. Math. Soc.

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Abstract. The SL(3) Kuznetsov formula exists in several versions, and has been employed with some success to study automorphic forms on SL(3). In each version, the weight functions on the geometric side are given by multiple integrals with complicated oscillating factors; this is the primary obstruction to its use. By describing them as solutions to systems of differential equations, we give power series and Mellin-Barnes integral representations of minimal dimension for these weight functions. This completes the role of harmonic analysis on symmetric spaces on the geometric side of the Kuznetsov formula, so that further study may be done through classical analytic techniques.The classical Kuznetsov formulas serve as a bridge between Fourier coefficients of Maass forms and Kloosterman sums. They are a necessary step in most proofs of subconvexity for Lfunctions, in a variety of equidistribution problems, in proofs of bounds and asymptotics for Maass forms, and in many applications of the circle method to problems in additive number theory. In attempting to generalize the Kuznetsov formula to study Fourier coefficents of SL(3) Maass forms, we encounter difficulties due to the lack of knowledge of the generalized Bessel functions that appear. The goal of this paper is to solve those difficulties for the spectral Kuznetsov formula.There are essentially four versions of the spectral Kuznetsov formula: The first, due to Xiaoqing Li, appears in [9, Thm 11.6.19] and uses an idea of Don Zagier with bi-K invariant test functions and spherical inversion. The second, due to Valentin Blomer, appears in [3] and uses the classical construction with the inner product of two Poincaré series. In [10], Dorian Goldfeld and Alex Kontorovich applied Lebedev-Whittaker inversion to Blomer's construction to obtain an arbitrary test function on the spectrum. Finally, in [7], the author worked out the first part of the integral transform on the geometric side of Li's version to obtain a slightly more usable formula -the main thrust there being an approximation to the geometric Kuznetsov formula.These four versions have been used with varying degrees of success. Li was able to obtain some results on spectral averages of the 1, 1 Fourier coefficients of cusp forms, weighted by their Whittaker functions [12]. Blomer has given a non-optimal version of the spectral large sieve inequalities and Lindelöf on average for the second moment of SL(3) L-functions [3]. Blomer, Raulf, and the author have given results on the distribution of the Fourier coefficients of SL(3) forms, including a Sato-Tate law on average [4].

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A Sato–Tate law for GL (3)

Abstract: We consider statistical properties of Hecke eigenvalues A j .p; 1/ for fixed p as j runs through a basis of Hecke-Maaß cusp forms for the group SL 3 .Z/. We show that almost all of them satisfy the Ramanujan conjecture at p and that their distribution is governed by the Sato-Tate law.

Cited by 29 publications

References 3 publications

An orthogonality relation for (with an appendix by Bingrong Huang)

An orthogonality relation for (with an appendix by Bingrong Huang)

Applications of the Kuznetsov formula on GL(3) II: the level aspect

The Spectral Kuznetsov Formula on $SL(3)$

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