Statistics for low-lying zeros of symmetric power L-functions in the level aspect

Ricotta, Guillaume; Royer, Emmanuel

doi:10.1515/form.2011.035

Cited by 33 publications

(33 citation statements)

References 25 publications

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“…That is, as the conductor tends to infinity, the zero statistics approach the scaling limit for large matrix size of the corresponding statistic for the eigenvalues of matrices from SO(2N) or SO(2N + 1). (Similar agreement with random matrix theory is shown for many other families of L-functions, see for example [DM06,FI03,Gül05,HR03,HM07,ILS00,ÖS99,RRb,Roy01,Rub01].) The test functions involved in these calculations have a limited range of support, but nonetheless the evidence is compelling.…”

Section: Introductionsupporting

confidence: 73%

Lower order terms for the one-level density of elliptic curve L-functions

Huynh

Keating

Snaith

2009

Journal of Number Theory

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It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve L-functions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the orthogonal group. For test functions with restricted support, this is known to be the true for the one-and two-level densities of zeros within the families studied to date. However, for finite conductor Miller's experimental data reveal an interesting discrepancy from these limiting results. Here we use the L-functions ratios conjectures to calculate the 1-level density for the family of even quadratic twists of an elliptic curve L-function for large but finite conductor. This gives a formula for the leading and lower order terms up to an error term that is conjectured to be significantly smaller. The lower order terms explain many of the features of the zero statistics for relatively small conductor and model the very slow convergence to the infinite conductor limit. However, our main observation is that they do not capture the behaviour of zeros in the important region very close to the critical point and so do not explain Miller's discrepancy. This therefore implies that a more accurate model for statistics near to this point needs to be developed.

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Section: Introductionsupporting

confidence: 73%

Lower order terms for the one-level density of elliptic curve L-functions

Huynh

Keating

Snaith

2009

Journal of Number Theory

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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: 92%

An orthogonal test of the L -functions Ratios conjecture

Miller

2009

Proceedings of the London Mathematical Society

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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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“…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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Statistics for low-lying zeros of symmetric power L-functions in the level aspect

Cited by 33 publications

References 25 publications

Lower order terms for the one-level density of elliptic curve L-functions

Lower order terms for the one-level density of elliptic curve L-functions

An orthogonal test of the L -functions Ratios conjecture

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

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