Low-lying Zeros Research Articles

AN EQUIDISTRIBUTION THEOREM FOR HOLOMORPHIC SIEGEL MODULAR FORMS FOR AND ITS APPLICATIONS

We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for$\mathit{GSp}_{4}/\mathbb{Q}$in various aspects. A main tool is Arthur’s invariant trace formula. While Shin [Automorphic Plancherel density theorem,Israel J. Math.192(1) (2012), 83–120] and Shin–Templier [Sato–Tate theorem for families and low-lying zeros of automorphic$L$-functions,Invent. Math.203(1) (2016) 1–177] used Euler–Poincaré functions at infinity in the formula, we use a pseudo-coefficient of a holomorphic discrete series to extract holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms$A,B_{1}$in Theorem 1.1 which have not been studied and a mysterious second term$B_{2}$also appears in the second main term coming from the semisimple elements. Furthermore our explicit study enables us to treat more general aspects in the weight. We also give several applications including the vertical Sato–Tate theorem, the unboundedness of Hecke fields and low-lying zeros for degree 4 spinor$L$-functions and degree 5 standard$L$-functions of holomorphic Siegel cusp forms.

Low-lying zeros of quadratic Dirichlet -functions: lower order terms for extended support

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the generalized Riemann hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$, which is valid when the support of the Fourier transform of the corresponding even test function $\unicode[STIX]{x1D719}$ is contained in $(-2,2)$. We uncover a phase transition when the supremum $\unicode[STIX]{x1D70E}$ of the support of $\widehat{\unicode[STIX]{x1D719}}$ reaches $1$, both in the main term and in the lower order terms. A new lower order term appearing at $\unicode[STIX]{x1D70E}=1$ involves the quantity $\widehat{\unicode[STIX]{x1D719}}(1)$, and is analogous to a lower order term which was isolated by Rudnick in the function field case.

Low-lying zeros for $L$-functions associated to Hilbert modular forms of large level

We determine the 1-level density of families of Hilbert modular forms, and show the answer agrees only with orthogonal random matrix ensembles.

Low-lying zeros of elliptic curve L-functions: Beyond the Ratios Conjecture

AbstractWe study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb{Q}$. We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise expression for the corresponding 1-level density. In particular, for test functions whose Fourier transforms have sufficiently restricted support, we are able to compute the 1-level density up to an error term that is significantly sharper than the square-root error term predicted by the L-functions Ratios Conjecture.

A NOTE ON SIMPLE ZEROS OF PRIMITIVE DIRICHLET -FUNCTIONS

In this paper, by using the theory of reproducing kernel Hilbert spaces and the pair correlation formula constructed by Chandee et al. [‘Simple zeros of primitive Dirichlet $L$-functions and the asymptotic large sieve’, Q. J. Math.65(1) (2014), 63–87], we prove that at least 93.22% of low-lying zeros of primitive Dirichlet $L$-functions are simple in a proper sense, under the assumption of the generalised Riemann hypothesis.

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

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 $$^{L}G=\widehat{G} \rtimes \mathrm{Gal}(\bar{F}/F)$$ be the associated $$L$$ -group and $$r:{}^L G\rightarrow \mathrm{GL}(d,\mathbb {C})$$ a continuous homomorphism which is irreducible and does not factor through $$\mathrm{Gal}(\bar{F}/F)$$ . The families under consideration consist of discrete automorphic representations of $$G(\mathbb {A}_F)$$ 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 low-lying zeros of the associated family of $$L$$ -functions $$L(s,\pi ,r)$$ , 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 $$r^\vee $$ then the symmetry type is unitary. Otherwise there is a bilinear form on $$\mathbb {C}^d$$ which realizes the isomorphism between $$r$$ and $$r^\vee $$ . If the bilinear form is symmetric (resp. alternating) then $$r$$ is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).

Local spectral equidistribution for degree two Siegel modular forms in level and weight aspects

We prove an equidistribution result for the Satake parameters of the local representations attached to Siegel cusp forms of degree 2 of increasing level and weight, counted with a certain arithmetic weight. We then apply this to compute the symmetry type of a similarly weighted distribution of the low-lying zeros of L-functions attached to these cusp forms.

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

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions in the family of all characters modulo $q$, with $Q/2 < q\leq Q$. 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 $Q^{-\frac 12 +\epsilon}$ in the Ratios Conjecture is best possible. We also give more precise results when the support of the Fourier Transform of the test function is restricted to the interval $[-1,1]$. Finally we show how natural conjectures on the distribution of primes in arithmetic progressions allow one to extend the support. The most powerful conjecture is Montgomery's, which implies that the Ratios Conjecture's prediction holds for any finite support up to an error $Q^{-\frac 12 +\epsilon}$.

Highly biased prime number races

Chebyshev observed in a letter to Fuss that there tends to be more primes of the form $4n+3$ than of the form $4n+1$. The general phenomenon, which is referred to as Chebyshev's bias, is that primes tend to be biased in their distribution among the different residue classes $\bmod q$. It is known that this phenomenon has a strong relation with the low-lying zeros of the associated $L$-functions, that is if these $L$-functions have zeros close to the real line, then it will result in a lower bias. According to this principle one might believe that the most biased prime number race we will ever find is the Li$(x)$ versus $\pi(x)$ race, since the Riemann zeta function is the $L$-function of rank one having the highest first zero. This race has density 0.99999973..., and we study the question of whether this is the highest possible density. We will show that it is not the case, in fact there exists prime number races whose density can be arbitrarily close to 1. An example of race whose density exceeds the above number is the race between quadratic residues and non-residues modulo 4849845, for which the density is 0.999999928... We also give fairly general criteria to decide whether a prime number race is highly biased or not. Our main result depends on the General Riemann Hypothesis and on a hypothesis on the multiplicity of the zeros of a certain Dedekind zeta function. We also derive more precise results under a linear independence hypothesis.

Small first zeros of $$L$$ L -functions

From a family of \(L\)-functions with unitary symmetry, Hughes and Rudnick obtained results on the height of its lowest zero. We extend their results to other families of \(L\)-functions according to the type of symmetry coming from statistics for low-lying zeros.

Quantum Graphs Whose Spectra Mimic the Zeros of the Riemann Zeta Function

One of the most famous problems in mathematics is the Riemann hypothesis: that the nontrivial zeros of the Riemann zeta function lie on a line in the complex plane. One way to prove the hypothesis would be to identify the zeros as eigenvalues of a Hermitian operator, many of whose properties can be derived through the analogy to quantum chaos. Using this, we construct a set of quantum graphs that have the same oscillating part of the density of states as the Riemann zeros, offering an explanation of the overall minus sign. The smooth part is completely different, and hence also the spectrum, but the graphs pick out the low-lying zeros.

Low-Lying Zeros of Maass Form L-Functions

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 $kN \to\infty$ 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 ${\rm GL}_2$ automorphic forms over $\mathbb{Q}$: Maass forms. To facilitate the analysis, we use smooth weight functions in the Kuznetsov formula which, among other restrictions, vanish to order $M$ at the origin. For test functions with Fourier transform supported inside $\left(-2 + \frac{2}{2M+1}, 2 - \frac{2}{2M+1}\right)$, 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.

Low-lying Zeros of Quadratic Dirichlet L-Functions, Hyper-elliptic Curves and Random Matrix Theory

The statistics of low-lying zeros of quadratic Dirichlet L-functions were conjectured by Katz and Sarnak to be given by the scaling limit of eigenvalues from the unitary symplectic ensemble. The n-level densities were found to be in agreement with this in a certain neighborhood of the origin in the Fourier domain by Rubinstein in his Ph.D. thesis in 1998. An attempt to extend the neighborhood was made in the Ph.D. thesis of Peng Gao (n-level density of the low-lying zeros of quadratic Dirichlet L-functions, 2005), who under GRH gave the density as a complicated combinatorial factor, but it remained open whether it coincides with the Random Matrix Theory factor. For n ≤ 7 this was recently confirmed by Levinson and Miller. We resolve this problem for all n, not by directly doing the combinatorics, but by passing to a function field analogue, of L-functions associated to hyper-elliptic curves of given genus g over a field of q elements. We show that the answer in this case coincides with Gao’s combinatorial factor up to a controlled error. We then take the limit of large finite field size q → ∞ and use the Katz–Sarnak equidistribution theorem, which identifies the monodromy of the Frobenius conjugacy classes for the hyperelliptic ensemble with the group USp(2g). Further taking the limit of large genus g→ ∞ allows us to identify Gao’s combinatorial factor with the RMT answer.

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

The Density Conjecture of Katz and Sarnak associates a classical compact group to each reasonable family of $L$-functions. Under the assumption of the Generalized Riemann Hypothesis, Rubinstein computed the $n$-level density of low-lying zeros for the family of quadratic Dirichlet $L$-functions in the case that the Fourier transform $\hat{f}(u)$ of any test function $f$ is supported in the region $\sum^n_{j=1}u_j < 1$ and showed that the result agrees with the Density Conjecture. In this paper, we improve Rubinstein's result on computing the $n$-level density for the Fourier transform $\hat{f}(u)$ being supported in the region $\sum^n_{j=1}u_j < 2$.

The n-level densities of low-lying zeros of quadratic Dirichlet L-functions

ABSTRACT. Previous work by Rubinstein [Rub] and Gao [Gao] computed the n-level densities for families of quadratic Dirichlet L-functions for test functions φ1, . . . , φn supported in ∑ n i=1 |ui| < 2, and showed agreement with random matrix theory predictions in this range for n ≤ 3 but only in a restricted range for larger n. We extend these results and show agreement for n ≤ 7, and reduce higher n to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form, which facilitates the comparison of the two sides.

Low-lying zeros of number field L -functions

One of the most important statistics in studying the zeros of L-functions is the 1-level density, which measures the concentration of zeros near the central point. Fouvry and Iwaniec (2003) [FI] proved that the 1-level density for L-functions attached to imaginary quadratic fields agrees with results predicted by random matrix theory. In this paper, we show a similar agreement with random matrix theory occurring in more general sequences of number fields. We first show that the main term agrees with random matrix theory, and similar to all other families studied to date, is independent of the arithmetic of the fields. We then derive the first lower order term of the 1-level density, and see the arithmetic enter. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=zpb-gu3G8i0.

Local spectral equidistribution for Siegel modular forms and applications

AbstractWe study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson’s formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.

Moments of the Rank of Elliptic Curves

Abstract Fix an elliptic curve E/Qand assume the Riemann Hypothesis for the L-function L(ED, s) for every quadratic twist ED of E by D ϵ Z. We combine Weil's explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of ED. We derive from this an upper bound for the density of low-lying zeros of L(ED, s) that is compatible with the randommatrixmodels of Katz and Sarnak. We also show that for any unbounded increasing function f on R, the analytic rank and (assuming in addition the Birch and Swinnerton-Dyer conjecture) the number of integral points of ED are less than f (D) for almost all D.

One level density of low-lying zeros of families of L-functions

AbstractIn this paper, we prove some one level density results for the low-lying zeros of families of L-functions. More specifically, the families under consideration are that of L-functions of holomorphic Hecke eigenforms of level 1 and weight k twisted with quadratic Dirichlet characters and that of cubic and quartic Dirichlet L-functions.

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

We study one-level and two-level densities for low-lying zeros of symmetric power L-functions in the level aspect. This allows us to completely determine the symmetry types of some families of symmetric power L-functions with prescribed sign of functional equation. We also compute the moments of one-level density and exhibit mock-Gaussian behavior discovered by Hughes & Rudnick.