n-Level Density of the Low-lying Zeros of Quadratic Dirichlet L-Functions

Gao, Peng

doi:10.1093/imrn/rns261

Cited by 35 publications

(64 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Rubinstein [96] established (17) for any r 1 as long as the support Φ ⊂ ξ : r j=1 |ξ j | < 1 . Later Gao [46] proved that the limit on the l.h.s. of (17) exists for support Φ ⊂ ξ : r j=1 |ξ j | < 2 but attempts to prove that his answer agrees with the r.h.s.…”

Section: Examplesmentioning

confidence: 99%

Families of L-Functions and Their Symmetry

Sarnak

Shin

Templier

2016

Families of Automorphic Forms and the Trace Formula

View full text Add to dashboard Cite

Abstract. In [100] the first-named author gave a working definition of a family of automorphic L-functions. Since then there have been a number of works [32], [118], [73] [49], [72] and especially [108] by the second and third-named authors which make it possible to give a conjectural answer for the symmetry type of a family and in particular the universality class predicted in [68] for the distribution of the zeros near s = 1 2 . In this note we carry this out after introducing some basic invariants associated to a family. Definition of families and ConjecturesThe zoo of automorphic cusp forms π on G = GL n over Q correspond bijectively to their standard completed L-functions Λ(s, π) and they constitute a countable set containing species of different types. For example there are self-dual forms, ones corresponding to finite Galois representations, to Hasse-Weil zeta functions of varieties defined over Q, to Maass forms, etc. From a number of points of view (including the nontrivial problem of isolating special forms) one is led to study such Λ(s, π)'s in families in which the π's have similar characteristics. Some applications demand the understanding of the behavior of the L-functions as π varies over a family. Other applications involve questions about an individual L-function. In practice a family is investigated as it arises.For example the density theorems of Bombieri [13] and Vinogradov [113] are concerned with showing that in a suitable sense most Dirichlet L-functions have few violations of the Riemann hypothesis, and as such it is a powerful substitute for the latter. Other examples are the GL 2 subconvexity results which are proved by deforming the given form in a family (see [60] and [85] for accounts). In the analogous function field setting the notion of a family of zeta functions is well defined, coming from the notion of a family of varieties defined over a base. Here too the power of deforming in a family in order to understand individual members is amply demonstrated in the work of Deligne [30]. In the number field setting there is no formal definition of a family F of L-functions.Our aim is to give a working definition for the formation of a family which will correspond to parametrized subsets of A(G), the set of isobaric automorphic representations on G(A). As far as we can tell these include almost all families that have been studied. For the most part our families can be investigated using the trace formula, monodromy groups in arithmetic geometry and the geometry of numbers, and these lead to a determination of the distribution of the zeros near s = 1 2 of members of the family. For the high zeros of a given Λ(s, π), it was shown in [98] that the local scaled spacing statistics follows the universal GUE

show abstract

Section: Examplesmentioning

confidence: 99%

Families of L-Functions and Their Symmetry

Sarnak

Shin

Templier

2016

Families of Automorphic Forms and the Trace Formula

View full text Add to dashboard Cite

show abstract

“…In this section we show that we can use (1.16) to match the n-level density of zeros from quadratic Dirichlet L-functions derived by Gao [13] to the n-level density of eigenvalues of matrices from U Sp(2N ).…”

Section: Supportmentioning

confidence: 99%

“…We start by writing out Gao's result (Theorem II.1 of [12], with minor changes of notation) without the restriction that the zeros must be distinct. This can also be compared with Theorem 7.2 in the paper [9], which is identical to Gao's expression.…”

Section: Supportmentioning

confidence: 99%

“…This can also be compared with Theorem 7.2 in the paper [9], which is identical to Gao's expression. Both [13] and [9] use the family of Dirichlet L-functions L(s, χ 8d ) (with odd, positive, square-free d) as it simplifies the workings somewhat, but this does not make a material difference to the end result and does not affect the comparison with random matrix theory as the family is still expected to have zeros showing the behaviour of eigenvalues from U Sp(2N ).…”

Section: Supportmentioning

confidence: 99%

“…Gao [13] succeeded in writing down the n-level densities when the support is extended to n i=1 |u i | < 2, but was only able to verify that these agree, at leading order, with random matrix theory for n ≤ 3. Levinson and Miller [20] extended this to n ≤ 7.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Symplectic n-level densities with restricted support

Mason

Snaith²

2016

Random Matrices: Theory Appl.

View full text Add to dashboard Cite

In this paper we demonstrate that the alternative form, derived by us in an earlier paper, of the n-level densities for eigenvalues of matrices from the classical compact group U Sp(2N ) is far better suited for comparison with derivations of the n-level densities of zeros in the family of Dirichlet L-functions associated with real quadratic characters than the traditional determinantal random matrix formula. Previous authors have found ingenious proofs that the leading order term of the n-level density of the zeros agrees with the determinantal random matrix result under certain conditions, but here we show that comparison is more straightforward if the more suitable form of the random matrix result is used. For the support of the test function in [1, −1] and in [−2, 2] we compare with existing number theoretical results. For support in [−3, 3] no rigorous number theoretical result is known for the n-level densities, but we derive the densities here using random matrix theory in the hope that this may make the path to a rigorous number theoretical result clearer.

show abstract

Maass Waveforms and Low-Lying Zeros

Alpoge

Amersi

Iyer³

et al. 2015

Analytic Number Theory

View full text Add to dashboard Cite

The Katz-Sarnak Density Conjecture states that the behavior of zeros of a family of L-functions near the central point (as the conductors tend to zero) agrees with the behavior of eigenvalues near 1 of a classical compact group (as the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo and Sarnak proved that the behavior of zeros near the central point of holomorphic cusp forms agrees with the behavior of eigenvalues of orthogonal matrices for suitably restricted test functions φ . We prove similar results for families of cuspidal Maass forms, the other natural family of GL 2 /Q L-functions. For suitable weight functions on the space of Maass forms, the limiting behavior agrees with the expected orthogonal group. We prove this for supp( φ ) ⊆ (−3/2, 3/2) when the level N tends to infinity through the square-free numbers; if the level is fixed the support decreases to being contained in (−1, 1), though we still uniquely specify the symmetry type by computing the 2-level density.

show abstract

n-Level Density of the Low-lying Zeros of Quadratic Dirichlet L-Functions

Cited by 35 publications

References 17 publications

Families of L-Functions and Their Symmetry

Families of L-Functions and Their Symmetry

Symplectic n-level densities with restricted support

Maass Waveforms and Low-Lying Zeros

Contact Info

Product

Resources

About