Forms Of Half-integral Weight Research Articles

The p-adic L-function for half-integral weight modular forms

The p-adic L-function for modular forms of integral weight is well-known. For certain weights the p-adic L-function for modular forms of half-integral weight is also known to exist, via a correspondence, established by Shimura, between them and forms of integral weight. However, we construct it here without any recourse to the Shimura correspondence, allowing us to establish its existence for all weights, including those exempt from the Shimura correspondence. We do this by employing the Rankin–Selberg method, and proving explicit p-adic congruences in the resultant Rankin–Selberg expression.

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

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

ON KOHNEN PLUS-SPACE OF JACOBI FORMS OF HALF INTEGRAL WEIGHT OF MATRIX INDEX

We introduce a plus-space of Jacobi forms, which is a certain subspace of Jacobi forms of half-integral weight of matrix index. This is an analogue to the Kohnen plus-space in the framework of Jacobi forms. We shall show a linear isomorphism between the plus-space of Jacobi forms and the space of Jacobi forms of integral weight of certain matrix index. Moreover, we shall show that this linear isomorphism is compatible with the action of Hecke operators of both spaces. This result is a kind of generalization of Eichler-Zagier-Ibukiyama correspondence, which is an isomorphism between the generalized plus-space of Siegel modular forms of general degree and Jacobi forms of index $1$ of general degree.

On newforms of half-integral weight and Jacobi forms

Let N be an odd and square-free integer. Let α be a positive integer with α=2 or α=5. Let χ modulo N be a Dirichlet character and let χ0=(4χ(−1).)χ. Let(a)χ and χ2 are primitive characters mod N, if α=2;(b)χ is the principal character if α=5. In this paper, we set up the theory of newforms for the space of cusp forms of weight k+1/2 for Γ0(2αN) with character χ0.Moreover, we prove that the space of newforms of weight k+1/2 for Γ0(32N) is trivial. Also, we set up the theory of newforms for the space of Jacobi cusp forms and skew-holomorphic Jacobi cusp forms.

Non-vanishing of products of Fourier coefficients of modular forms of half-integral weight

We prove a non-vanishing result in weight aspect for the product of two Fourier coefficients of a Hecke eigenform 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.

Modularity of generating series of winding numbers

The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.

Special values of Hecke L-functions of modular forms of half-integral weight and cohomology

We start developing a cohomology theory in the case of half-integral weight with an attempt to focus again on the connection to special values of L-functions at half-integral and integral points inside the critical strip.

Equidistribution of signs for Hilbert modular forms of half-integral weight

We prove an equidistribution of signs for the Fourier coefficients of Hilbert modular forms of half-integral weight. Our study focuses on certain subfamilies of coefficients that are accessible via the Shimura correspondence. This is a generalization of the result of Inam and Wiese to the setting of totally real number fields

Shimura lifts of weakly holomorphic modular forms

We show how to realize the Shimura lift of arbitrary level and character using the vector-valued theta lifts of Borcherds. Using the regularization of Borcherds' lift we extend the Shimura lift to take weakly holomorphic modular forms of half-integral weight to meromorphic modular forms of even integral weight having poles at CM points.

Indivisibility of relative class numbers of totally imaginary quadratic extensions and vanishing of these relative Iwasawa invariants

We study an indivisibility problem of the relative class numbers of CM fields. For prime p>3, Kohnen–Ono gave a lower bound of the number of the imaginary quadratic fields whose class numbers are prime to p by using modular forms of half-integral weight. We generalize their method to Hilbert modular forms and give a lower bound of the number of CM quadratic extensions K/F whose relative class numbers prime to p for totally real number field F which is Galois over Q and sufficiently large prime p. Combining the indivisibility result with the decomposition condition of p, we show a result on vanishing of relative Iwasawa invariants.

Zeros of L-functions attached to modular forms of half-integral weight

Zeros of L-functions attached to modular forms of half-integral weight

Generating weights for the Weil representation attached to an even order cyclic quadratic module

TextWe develop geometric methods to study the generating weights of free modules of vector-valued modular forms of half-integral weight, taking values in a complex representation of the metaplectic group. We then compute the generating weights for modular forms taking values in the Weil representation attached to cyclic quadratic modules of order 2pr, where p≥5 is a prime. We also show that the generating weights approach a simple limiting distribution as p grows, or as r grows and p remains fixed. VideoFor a video summary of this paper, please visit https://youtu.be/QNbPSXXKot4.

On special L-values attached to metaplectic modular forms

In this paper we establish some algebraic properties of special L-values attached to Siegel modular forms of half-integral weight, often called metaplectic modular forms. These results are motivated by some “exercises” left by Shimura to the reader in his marvellous book “Arithmeticity in the Theory of Automorphic Forms”.

Non-vanishing of L-functions associated to cusp forms of half-integral weight in the plus space

In this paper, we show a non-vanishing result for L-functions associated to cuspidal Hecke eigenforms of half integral weight in plus space.

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.

Supnorm of modular forms of half-integral weight in the weight aspect

We bound the supnorm of half-integral weight Hecke eigenforms in the Kohnen plus space of level $4$ in the weight aspect, by combining bounds obtained from the Fourier expansion with the amplification method using a Bergman kernel.

Zeros of modular forms of half integral weight

We study canonical bases for spaces of weakly holomorphic modular forms of level 4 and weights in $$\mathbb {Z}+\frac{1}{2}$$ and show that almost all modular forms in these bases have the property that many of their zeros in a fundamental domain for $$\Gamma _0(4)$$ lie on a lower boundary arc of the fundamental domain. Additionally, we show that at many places on this arc, the generating function for Hurwitz class numbers is equal to a particular mock modular Poincare series, and show that for positive weights, a particular set of Fourier coefficients of cusp forms in this canonical basis cannot simultaneously vanish.

Local Behavior of Arithmetical Functions with Applications to Automorphic $L$-Functions

We derive a Voronoi-type series approximation for the local weighted mean of an arithmetical function that is associated to Dirichlet series satisfying a functional equation with gamma factors. The series is exploited to study the oscillation frequency with a method of Heath-Brown and Tsang [7]. A by-product is another proof for the well-known result of no element in the Selberg class of degree 0 < d < 1. Our major applications include the sign-change problem of the coefficients of automorphic L-functions for GL m , which improves significantly some results of Liu and Wu [14]. The cases of modular forms of half-integral weight and Siegel eigenforms are also considered.

Eisenstein series in the Kohnen plus space for Hilbert modular forms

In 1975, Cohen constructed a kind of one-variable modular forms of half-integral weight, say [Formula: see text], whose [Formula: see text]th Fourier coefficient only occurs when [Formula: see text] is congruent to 0 or 1 modulo 4. The space of modular forms whose Fourier coefficients have the above property is called Kohnen plus space, initially introduced by Kohnen in 1980. Recently, Hiraga and Ikeda generalized the plus space to the spaces for half-integral weight Hilbert modular forms with respect to general totally real number fields. The [Formula: see text]th Fourier coefficients [Formula: see text] of a Hilbert modular form of parallel weight [Formula: see text] lying in the generalized Kohnen plus space does not vanish only if [Formula: see text] is congruent to a square modulo 4. In this paper, we use an adelic way to construct Eisenstein series of parallel half-integral weight belonging to the generalized Kohnen plus spaces and give an explicit form for their Fourier coefficients. These series give a generalization of the one introduced by Cohen. Moreover, we show that the Kohnen plus space is generated by the cusp forms and the Eisenstein series we constructed as a vector space over [Formula: see text].