Maass Waveforms and Low-Lying Zeros

Alpoge, Levent; Amersi, Nadine; Iyer, Geoffrey; Lazarev, Oleg; Miller, Steven J.; Liyang, Zhang

doi:10.1007/978-3-319-22240-0_2

Cited by 9 publications

(13 citation statements)

References 48 publications

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“…Based on results from [1] and [41], which determined the one-level density for support contained in (−1, 1), we believe the following conjecture.…”

Section: Throughout This Paper T Will Be a Large Positive Odd Integermentioning

confidence: 91%

“…Proof of Lemma 2.2, part (1). We cut the sum above at T log T and below at T log T and apply the previous lemma along with the fact that ||u|| ≍ 1 under our normalizations (see [42]).…”

Section: Lemma 24 Let H T Be As In Theorem 12 Thenmentioning

confidence: 99%

“…Before stating our main result we first describe the weight function used in the one-level density for the family of level 1 Maass forms. The weight function we consider is not as general as other ones investigated (see the arguments for other families of Maass forms in [1]), but leads to a significantly simpler analysis and much greater support. In this sense our work is similar to analyses in other problems where the weight function is chosen to facilitate the application of a summation formula (for example, the use of harmonic weights for the Petersson formula).…”

Section: )mentioning

confidence: 99%

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Low-Lying Zeros of Maass Form L-Functions

Alpoge¹,

Miller²

2014

International Mathematics Research Notices

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The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeros of families of automorphic L-functions agree with the scaling limits of eigenvalue distributions of classical subgroups of the unitary groups U (N ). This conjecture is often tested by way of computing particular statistics, such as the one-level density, which evaluates a test function with compactly supported Fourier transform at normalized zeros near the central point. Iwaniec, Luo, and Sarnak studied the one-level densities of cuspidal newforms of weight k and level N . They showed in the limit as kN → ∞ that these families have one-level densities agreeing with orthogonal type for test functions with Fourier transform supported in (−2, 2). Exceeding (−1, 1) is important as the three orthogonal groups are indistinguishable for support up to (−1, 1) but are distinguishable for any larger support. We study the other family of GL 2 automorphic forms over Q: Maass forms. To facilitate the analysis, we use smooth weight functions in the Kuznetsov formula which, among other restrictions, vanish to order 2M at the origin. For test functions with Fourier transform supported inside −2 + 2 2M +1 , 2 − 2 2M +1 , we unconditionally prove the one-level density of the low-lying zeros of level 1 Maass forms, as the eigenvalues tend to infinity, agrees only with that of the scaling limit of orthogonal matrices.2010 Mathematics Subject Classification. 11M26 (primary), 11M41, 15A52 (secondary).

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“…Based on results from [1] and [41], which determined the one-level density for support contained in (−1, 1), we believe the following conjecture.…”

Section: Throughout This Paper T Will Be a Large Positive Odd Integermentioning

confidence: 91%

“…Proof of Lemma 2.2, part (1). We cut the sum above at T log T and below at T log T and apply the previous lemma along with the fact that ||u|| ≍ 1 under our normalizations (see [42]).…”

Section: Lemma 24 Let H T Be As In Theorem 12 Thenmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

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Low-Lying Zeros of Maass Form L-Functions

Alpoge¹,

Miller²

2014

International Mathematics Research Notices

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“…Later, Iwaniec, Luo and Sarnak [19] gave densities of low-lying zeros of the standard Lfunctions and those of symmetric square L-functions associated with holomorphic elliptic cusp forms both in the weight aspect and the level aspect, assuming GRH of several Lfunctions. Inspired by their study, densities of low-lying zeros of families of automorphic L-functions have been investigated in several settings such as Hilbert modular forms ( [28]), Siegel modular forms of degree 2 ( [24], [25]), and Hecke-Maass forms ( [1], [2] [16], [29], [31], [36]). As of now, the broadest setting for low-lying zeros of automorphic L-functions was setteled by Shin and Templier [40].…”

Section: Introductionmentioning

confidence: 99%

Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

Sugiyama¹

2021

Preprint

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For a totally real number field F and its adèle ring A F , let π vary in the set of irreducible cuspidal automorphic representations of PGL 2 (A F ) corresponding to primitive Hilbert modular forms of a fixed weight. Then, we determine the symmetry type of the one-level density of low-lying zeros of the symmetric power L-functions L(s, Sym r (π)) weighted by special values of symmetric square L-functions L( z+1 2 , Sym 2 (π)) at z ∈ [0, 1] in the level aspect. If 0 < z 1, our weighted density in the level aspect has the same symmetry type as Ricotta and Royer's density of low-lying zeros of symmetric power L-functions for F = Q with harmonic weight. Hence our result is regarded as a zinterpolation of Ricotta and Royer's result. If z = 0, density of low-lying zeros weighted by central values is a different type only when r = 2, and it does not appear in random matrix theory as Katz and Sarnak predicted. Moreover, we propose a conjecture on weighted density of low-lying zeros of L-functions by special L-values.In the latter part, Appendices A, B and C are dedicated to the comparison among several generalizations of Zagier's parameterized trace formula. We prove that the explicit Jacquet-Zagier type trace formula (the ST trace formula) by Tsuzuki and the author recovers all of Zagier's, Takase's and Mizumoto's formulas by specializing several data. Such comparison is not so straightforward and includes non-trivial analytic evaluations.

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“…The five groups have distinguishable 1-level densities. To date, for suitably restricted test functions the statistics of zeros of many natural families of L-functions have been shown to agree with statistics of eigenvalues of matrices from the classical compact groups, including Dirichlet L-functions, elliptic curves, cuspidal newforms, Maass forms, number field L-functions, and symmetric powers of GL 2 automorphic representations [AM,AAILMZ,DM1,FI,Gao,Gü,HM,HR,ILS,KaSa1,KaSa2,Mil1,MilPe,RR,Ro,Rub,ShTe,Ya,Yo2], to name a few, as well as non-simple families formed by Rankin-Selberg convolution [DM2].…”

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confidence: 99%

Surpassing the ratios conjecture in the 1-level density of DirichletL-functions

Fiorilli¹,

Miller²

2015

Algebra Number Theory

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Abstract. We study the 1-level density of low-lying zeros of Dirichlet L-functions in the family of all characters modulo q, with Q/2 < q ≤ Q. For test functions whose Fourier transform is supported in (−3/2, 3/2), we calculate this quantity beyond the square-root cancellation expansion arising from the L-function Ratios Conjecture of Conrey, Farmer and Zirnbauer. We discover the existence of a new lower-order term which is not predicted by this powerful conjecture. This is the first family where the 1-level density is determined well enough to see a term which is not predicted by the Ratios Conjecture, and proves that the exponent of the error term Q

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Maass Waveforms and Low-Lying Zeros

Cited by 9 publications

References 48 publications

Low-Lying Zeros of Maass Form L-Functions

Low-Lying Zeros of Maass Form L-Functions

Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

Surpassing the ratios conjecture in the 1-level density of DirichletL-functions

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