Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the distribution of eigenvalues of Maass forms on certain moonshine groups
150
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $Gamma_0(N)^+$, where $N>1$ is a square-free integer. After we prove that $Gamma_0(N)^+$ has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an average Weyls law for the distribution of eigenvalues of Maass forms, from which we prove the classical Weyls law as a special case. The groups corresponding to $N=5$ and $N=6$ have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for $Gamma_0(5)^+$ than for $Gamma_0(6)^+$. We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyls laws. In addition, we employ Hejhals algorithm, together with recently developed refinements from [31], and numerically determine the first $3557$ of $Gamma_0(5)^+$ and the first $12474$ eigenvalues of $Gamma_0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.
rate research
Read More
We explore an algorithm which systematically finds all discrete eigenvalues of an analytic eigenvalue problem. The algorithm is more simple and elementary as could be expected before. It consists of Hejhals identity, linearisation, and Turing bounds. Using the algorithm, we compute more than one hundredsixty thousand consecutive eigenvalues of the Laplacian on the modular surface, and investigate the asymptotic and statistic properties of the fluctuations in the Weyl remainder. We summarize the findings in two conjectures. One is on the maximum size of the Weyl remainder, and the other is on the distribution of a suitably scaled version of the Weyl remainder.
Let $q:=e^{2 pi iz}$, where $z in mathbb{H}$. For an even integer $k$, let $f(z):=q^hprod_{m=1}^{infty}(1-q^m)^{c(m)}$ be a meromorphic modular form of weight $k$ on $Gamma_0(N)$. For a positive integer $m$, let $T_m$ be the $m$th Hecke operator and $D$ be a divisor of a modular curve with level $N$. Both subjects, the exponents $c(m)$ of a modular form and the distribution of the points in the support of $T_m. D$, have been widely investigated. When the level $N$ is one, Bruinier, Kohnen, and Ono obtained, in terms of the values of $j$-invariant function, identities between the exponents $c(m)$ of a modular form and the points in the support of $T_m.D$. In this paper, we extend this result to general $Gamma_0(N)$ in terms of values of harmonic weak Maass forms of weight $0$. By the distribution of Hecke points, this applies to obtain an asymptotic behaviour of convolutions of sums of divisors of an integer and sums of exponents of a modular form.
In this paper, we explicitly construct harmonic Maass forms that map to the weight one theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic information contained in the Fourier coefficients of the holomorphic part of the harmonic Maass form, establishing the main part of a conjecture of the second author.
Let $pi$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${rm U}_{mathfrak n}$ defined over a number field $F$. Under the assumption that $pi$ has a generic global Arthur parameter, we establish the non-vanishing of the central value of $L$-functions, $L(frac{1}{2},pitimeschi)$, with a certain automorphic character $chi$ of ${rm U}_1$, for the case of ${mathfrak n}=2,3,4$, and for the general ${mathfrak n}geq 5$ by assuming a conjecture on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results for the central $L$-values by means of the theory of endoscopy.
Let $lambda_{pi}(1,n)$ be the Fourier coefficients of the Hecke-Maass cusp form $pi$ for $SL(3,mathbb{Z})$. The aim of this article is to get a non trivial bound on the sum which is non-linear additive twist of the coefficients $lambda_{pi}(1,n)$. More precisely, for any $0 < beta < 1$ we have $$sum_{n=1}^{infty} lambda_{pi}(1,n) , eleft(alpha n^{beta}right) Vleft(frac{n}{X}right) ll_{pi, alpha,epsilon} X^{frac{3 }{4}+frac{3 beta}{10} + epsilon}$$ for any $epsilon>0$. Here $V(x)$ is a smooth function supported in $[1,2]$ and satisfies $V^{(j)}(x) ll_{j} 1$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
