The low lying zeros of a GL(4) and a GL(6) family of $L$-functions

Duéñez, Eduardo; Miller, Steven J.

doi:10.1112/s0010437x0600220x

Cited by 52 publications

(66 citation statements)

References 34 publications

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“…We refer the reader to the previously mentioned surveys for the details. In these and many other cases (see [9,85,87,[100][101][102][103][104][105][106][107][108][109] for a representative sampling of results), we can show for suitably restricted φ that the 1-level density agrees with the scaling limit of one of the mentioned classical compact groups.…”

Section: Cuspidal Newforms: Letmentioning

confidence: 90%

Nuclei, Primes and the Random Matrix Connection

Firk

Miller

2009

Symmetry

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In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mathematics -random matrix theory -provides the connection between the two fields. We assume no detailed knowledge of number theory, nuclear physics, or random matrix theory; all that is required is some familiarity with linear algebra and probability theory, as well as some results from complex analysis. Our goal is to provide the inquisitive reader with a sound overview of the subjects, placing them in their historical context in a way that is not traditionally given in the popular and technical surveys.

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Section: Cuspidal Newforms: Letmentioning

confidence: 90%

Nuclei, Primes and the Random Matrix Connection

Firk

Miller

2009

Symmetry

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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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“…3 Katz and Sarnak [KaSa1,KaSa2] conjectured that as the conductors tend to infinity, the 1-level density agrees with the scaling limit of a classical compact group. There are now many cases where, for suitably restricted test functions, we can show agreement between the main terms and the conjectures; see, for example [DM1,FI,Gao,Gü,HR,HM,ILS,KaSa2,Mil1,OS,RR,Ro,Rub,Yo2]. Now that the main terms have been successfully matched in numerous cases, it is natural to try to analyze the lower order terms.…”

mentioning

confidence: 99%

An orthogonal test of the L-functions Ratios Conjecture, II

Miller

Montague

2011

Acta Arith.

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ABSTRACT. We test the predictions of the L-functions Ratios Conjecture for the family of cuspidal newforms of weight k and level N , with either k fixed and N → ∞ through the primes or N = 1 and k → ∞. We study the main and lower order terms in the 1-level density. We provide evidence for the Ratios Conjecture by computing and confirming its predictions up to a power savings in the family's cardinality, at least for test functions whose Fourier transforms are supported in (−2, 2). We do this both for the weighted and unweighted 1-level density (where in the weighted case we use the Petersson weights), thus showing that either formulation may be used. These two 1-level densities differ by a term of size 1/ log(k 2 N ). Finally, we show that there is another way of extending the sums arising in the Ratios Conjecture, leading to a different answer (although the answer is such a lower order term that it is hopeless to observe which is correct).

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The low lying zeros of a GL(4) and a GL(6) family of $L$-functions

Cited by 52 publications

References 34 publications

Nuclei, Primes and the Random Matrix Connection

Nuclei, Primes and the Random Matrix Connection

Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ L -functions

An orthogonal test of the L-functions Ratios Conjecture, II

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