Cusp Forms Of Half-integral Weight Research Articles

On products of Fourier coefficients of half-integral weight cusp forms

On products of Fourier coefficients of half-integral weight cusp forms

On the non-vanishing of Fourier coefficients of half-integral weight cuspforms

On the non-vanishing of Fourier coefficients of half-integral weight cuspforms

On calculations of the Fourier coefficients of cusp forms of half-integral weight given by the Shintani lift

On calculations of the Fourier coefficients of cusp forms of half-integral weight given by the Shintani lift

Estimates for the shifted convolution sum involving Fourier coefficients of cusp forms of half-integral weight

Estimates for the shifted convolution sum involving Fourier coefficients of cusp forms of half-integral weight

A short note on sign changes and non-vanishing of Fourier coefficients of half-integral weight cusp forms

We study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight.

Congruences for level $1$ cusp forms of half-integral weight

Suppose that $\ell \geq 5$ is prime. For a positive integer $N$ with $4 \mid N$, previous works studied properties of half-integral weight modular forms on $\Gamma_0(N)$ which are supported on finitely many square classes modulo $\ell$, in some cases proving that these forms are congruent to the image of a single variable theta series under some number of iterations of the Ramanujan $\Theta$-operator. Here, we study the analogous problem for modular forms of half-integral weight on $\operatorname{SL}_{2}(\mathbb{Z})$. Let $\eta$ be the Dedekind eta function. For a wide range of weights, we prove that every half-integral weight modular form on $\operatorname{SL}_{2}(\mathbb{Z})$ which is supported on finitely many square classes modulo $\ell$ can be written modulo $\ell$ in terms of $\eta^{\ell}$ and an iterated derivative of $\eta$.

Sign Changes of Fourier Coefficients of Cusp Forms of Half-Integral Weight Over Split and Inert Primes in Quadratic Number Fields

In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class $$t\mathbb {Z}^2$$ , we investigate the sign changes in the $$tp^2$$ -th coefficient as p runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. We show that infinitely many sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discriminant also divides the level.

Signs of Fourier coefficients of half-integral weight modular forms

Let g be a Hecke cusp form of half-integral weight, level 4 and belonging to Kohnen’s plus subspace. Let c(n) denote the nth Fourier coefficient of g, normalized so that c(n) is real for all n ge 1. A theorem of Waldspurger determines the magnitude of c(n) at fundamental discriminants n by establishing that the square of c(n) is proportional to the central value of a certain L-function. The signs of the sequence c(n) however remain mysterious. Conditionally on the Generalized Riemann Hypothesis, we show that c(n) < 0 and respectively c(n) > 0 holds for a positive proportion of fundamental discriminants n. Moreover we show that the sequence {c(n)} where n ranges over fundamental discriminants changes sign a positive proportion of the time. Unconditionally, it is not known that a positive proportion of these coefficients are non-zero and we prove results about the sign of c(n) which are of the same quality as the best known non-vanishing results. Finally we discuss extensions of our result to general half-integral weight forms g of level 4N with N odd, square-free.

Sign of Fourier coefficients of half-integral weight modular forms in arithmetic progressions

Let f be a half-integral weight cusp form of level 4N for odd and squarefree N and let a(n) denote its nth normalized Fourier coefficient. Assuming that all the coefficients a(n) are real, we study the sign of a(n) when n runs through an arithmetic progression. As a consequence, we establish a lower bound for the number of integers $$n\leqslant x$$ such that $$a(n)>n^{-\alpha }$$ where x and $$\alpha $$ are positive and f is not necessarily a Hecke eigenform.

Some remarks on the Fourier coefficients of cusp forms

In this paper, we consider the angular changes of Fourier coefficients of half integral weight cusp forms and sign changes of [Formula: see text]-exponents of generalized modular functions.

Determining Siegel modular forms of half-integral weight by their fundamental Fourier coefficients

We prove that a non-zero Siegel cusp form of half-integral weight and degree 2 on $\Gamma _0^{(2)}(4)$ has infinitely many non-zero Fourier coefficients indexed by semi-integral matrices having fundamental discriminant. This is the half-integral weight ve

On the Ramanujan–Petersson conjecture for modular forms of half-integral weight

Abstract We investigate the (still unknown) Ramanujan–Petersson conjecture about the growth of the Fourier coefficients of cusp forms of half-integral weight and prove that it is optimal, at least for newforms in the plus space.

A CERTAIN DIRICHLET SERIES OF RANKIN–SELBERG TYPE ASSOCIATED WITH THE IKEDA LIFT OF HALF‐INTEGRAL WEIGHT

In this article we obtain an explicit formula for certain Rankin-Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler-Zagier-Ibukiyama correspondence. The integral weight version of the main theorem had been obtained by Katsurada and Kawamura. The result of the integral weight case is a product of $L$-function and Riemann zeta functions, while half-integral weight case is a infinite summation over negative fundamental discriminants with certain infinite products. To calculate explicit formula of such Rankin-Selberg type Dirichlet series, we use a generalized Maass relation and adjoint maps of index-shift maps of Jacobi forms.

ON THE STANDARD TWIST OF THE -FUNCTIONS OF HALF-INTEGRAL WEIGHT CUSP FORMS

The standard twist$F(s,\unicode[STIX]{x1D6FC})$of$L$-functions$F(s)$in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for example the functional equation. Here we deal with a special case, where$F(s)$satisfies a functional equation with the same$\unicode[STIX]{x1D6E4}$-factor of the$L$-functions associated with the cusp forms of half-integral weight; for simplicity we present our results directly for such$L$-functions. We show that the standard twist$F(s,\unicode[STIX]{x1D6FC})$satisfies a functional equation reflecting$s$to$1-s$, whose shape is not far from a Riemann-type functional equation of degree 2 and may be regarded as a degree 2 analog of the Hurwitz–Lerch functional equation. We also deduce some results on the growth on vertical strips and on the distribution of zeros of$F(s,\unicode[STIX]{x1D6FC})$.

Zeros of $L$-functions attached to cusp forms of half-integral weight

Zeros of $L$-functions attached to cusp forms of half-integral weight

Estimates of automorphic cusp forms over quaternion algebras

In this paper, using methods from geometric analysis and theory of heat kernels, we derive qualitative estimates of automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture. We then derive quantitative estimates of classical Hilbert modular cusp forms. This is a generalization of the results from [A. Aryasomayajula, Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight, Arch. Math. 106(2) (2016) 165–173; J. S. Friedman, J. Jorgenson and J. Kramer, Uniform sup-norm bounds on average for cusp forms of higher weights, in Arbeitstagung Bonn 2013, Progress in Mathematics, Vol. 319 (Birkhäuser/Springer, Cham, 2016), pp. 127–154] to higher dimensions.

Fourier coefficients of half-integral weight cusp forms and Waring’s problem

Extending the approach of Iwaniec and Duke, we present strong uniform bounds for Fourier coefficients of half-integral weight cusp forms of level N. As an application, we consider a Waring-type problem with sums of mixed powers.

Height pairings on orthogonal Shimura varieties

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and complex multiplication points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin–Selberg convolution associated with a cusp form of half-integral weight $n/2+1$, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shimura curve, and our result is a variant of the Gross–Zagier theorem on heights of Heegner points.

Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let \({\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}\) be a cocompact Fuchsian subgroup of first kind. For \({k \in \frac{1}{2} \mathbb{Z}}\) (or \({k \in 2\mathbb{Z}}\)), let \({S^{k}_{\nu}(\Gamma)}\) denote the complex vector space of cusp forms of weight-k and nebentypus \({\nu^{2k}}\) (\({\nu^{k\slash 2}}\), if \({k \in 2\mathbb{Z}}\)) with respect to \({\Gamma}\), where \({\nu}\) is a unitary character. Let \({\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}\) denote an orthonormal basis of \({S^{k}_{\nu}(\Gamma)}\). In this article, we show that as \({k \rightarrow \infty,}\) the sup-norm for \({\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}}\) is bounded by O(k), where the implied constant is independent of \({\Gamma}\). Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when \({\Gamma}\) is cofinite.

Non-vanishing and sign changes of Hecke eigenvalues for half-integral weight cusp forms

In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form f with half-integral weight: –The first negative coefficient of the sequence {af(tn2)}n∈N,–The number of coefficients af(tn2) of same signs,–Non-vanishing of coefficients af(tn2) in short intervals and in arithmetic progressions, where af(n) is the nth Fourier coefficient of f and t is a square-free integer such that af(t)≠0.