Random Matrix Theory and the Fourier Coefficients of Half-Integral-Weight Forms (original) (raw)
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TURKISH JOURNAL OF MATHEMATICS, 2021
This work represents a systematic computational study of the distribution of the Fourier coefficients of cuspidal Hecke eigenforms of level Γ 0 (4) and half-integral weights. Based on substantial calculations, the question is raised whether the distribution of normalised Fourier coefficients with bounded indices can be approximated by a generalised Gaussian distribution. Moreover, it is argued that the apparent symmetry around zero of the data lends strong evidence to the Bruinier-Kohnen Conjecture on the equidistribution of signs and even suggests the strengthening that signs and absolute values are distributed independently.
On the Fourier coefficients of modular forms of half-integral weight
Forum Mathematicum, 2010
We prove a formula relating the Fourier coefficients of a modular form of half-integral weight to the special values of L-functions. The form in question is an explicit theta lift from the multiplicative group of an indefinite quaternion algebra over Q. This formula has applications to proving the nonvanishing of this lift and to an explicit version of the Rallis inner product formula.
On the mean values of L-functions in orthogonal and symplectic families
Proceedings of The London Mathematical Society, 2007
Hybrid Euler-Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L-functions, in the context of the calculation of moments and connections with Random Matrix Theory. According to the Katz-Sarnak classification, these are believed to represent families of L-function with unitary symmetry. We here extend the formalism to families with orthogonal & symplectic symmetry. Specifically, we establish formulae for real quadratic Dirichlet L-functions and for the L-functions associated with primitive Hecke eigenforms of weight 2 in terms of partial Euler and Hadamard products. We then prove asymptotic formulae for some moments of these partial products and make general conjectures based on results for the moments of characteristic polynomials of random matrices.
Correlations of zeros of families of L-functions with orthogonal or symplectic symmetry
The aim of my work is to gain new insights into the properties of L-functions by using their connections with Random Matrix Theory. Random Matrix Theory was developed by physicists trying to model the quantum states of complicated atomic nuclei, while L-functions arise from almost every area of number theory. The surprising link between these two areas of mathematics (usually considered entirely separate) has yet to be fully proven, but the numerical evidence indicates a definite connection. The simplest L-function to consider is the Riemann zeta-function, a complex function with its nontrivial zeros contained in a vertical strip, called the critical strip, on the complex plane. The Riemann hypothesis says that these zeros all lie on a single vertical line, called the critical line, so rather than worrying about where the zeros lie horizontally in the critical strip we can look at how they are distributed vertically. In the 1970s Montgomery proved – with certain conditions – a striking similarity between the pair correlation of the zeros of the Riemann zeta-function (as the height on the critical line tends to infinity) and the pair correlation of the eigenvalues of random unitary matrices (as the size of the matrices tends to infinity). Rudnick and Sarnak showed this result can be extended to n-correlation of the Riemann zeta-function and also generalised these results to apply to all L-functions. Katz and Sarnak conjectured that if we arrange L-functions into families, we can also model statistics across these families using groups of appropriate random matrices. Unfortunately random matrices do not directly give us all the lower-order terms for the n-correlation of zeros of the Riemann zeta-function. Luckily Conrey and Snaith showed how we can use calculations on random unitary matrices as a guide for calculating the n-correlation of the Riemann zeta-function explicitly. In this thesis, I have explicitly calculated all lower order terms for the n-correlation of zeros of certain families of L-functions. These calculations follow from Conrey and Snaith’s similar work for the Riemann zeta function. Katz and Sarnak have argued that the zero statistics of families of L-functions have an underlying symmetry relating to certain ensembles of random matrices. With this in mind, we have looked at a family with orthogonal symmetry (even twists of the Hasse-Weil L-function of a given elliptic curve) and a family with symplectic symmetry (Dirichlet Lfunctions). Assuming the ratios conjectures of Conrey, Farmer, and Zirnbauer, we prove a formula which explicitly gives all of the lower order terms in the n-correlation. For the families relating to elliptic curves, this formula agrees with the known results of Huynh, Keating and Snaith for n = 1 and as the conductor tends to infinity the 2-correlation matches that of eigenangles of random orthogonal matrices under Haar measure. The method used in this thesis works by first calculating n-correlation of eigenangles of SO(2N) and USp(2N) via ratios of characteristic polynomials. In a similar manner to Conrey and Snaith’s work on U(N), we can identify which terms remain in the n-correlation of eigenangles of random orthogonal or symplectic matrices when restrictions are placed on the support of the test function. It is hoped that this will allow for an easier way of checking results with L-functions match those predicted by random matrix theory.
A note on the Fourier coefficients of half-integral weight modular forms
Archiv der Mathematik, 2014
In this note, we show that the algebraicity of the Fourier coefficients of half-integral weight modular forms can be determined by checking the algebraicity of the first few of them. We also give a necessary and sufficient condition for a half-integral weight modular form to be in Kohnen's +-subspace by considering only finitely many terms.
Fourier coefficients of half-integral weight modular forms modulo ell
1996
For each prime ℓ, let |·|_ℓ be an extension to of the usual ℓ-adic absolute value on . Suppose g(z) = ∑_n=0^∞ c(n)q^n ∈ M_k+(N) is an eigenform whose Fourier coefficients are algebraic integers. Under a mild condition, for all but finitely many primes ℓ there are infinitely many square-free integers m for which |c(m)|_ℓ = 1. Consequently we obtain indivisibility results for "algebraic parts" of central critical values of modular L-functions and class numbers of imaginary quadratic fields. These results partially answer a conjecture of Kolyvagin regarding Tate-Shafarevich groups of modular elliptic curves. Similar results were obtained earlier by Jochnowitz by a completely different method. Our method uses standard facts about Galois representations attached to modular forms, and pleasantly uncovers surprising Kronecker-style congruences for L-function values. For example if Δ(z) is Ramanujan's cusp form and g(z)=∑_n=1^∞c(n)q^n is the cusp form for which L(Δ_D,6)=()πD^6...
Artin–Schreier L-functions and random unitary matrices
Journal of Number Theory
We give a new derivation of an identity due to Z. Rudnick and P. Sarnak about the n-level correlations of eigenvalues of random unitary matrices as well as a new proof of a formula due to M. Diaconis and P. Shahshahani expressing averages of trace products over the unitary matrix ensemble. Our method uses the zero statistics of Artin-Schreier L-functions and a deep equidistribution result due to N. Katz.
Low-lying Zeros of Quadratic Dirichlet L-Functions, Hyper-elliptic Curves and Random Matrix Theory
Geometric and Functional Analysis, 2013
The statistics of low-lying zeros of quadratic Dirichlet Lfunctions 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 (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.