Forms Of Half-integral Weight Research Articles

Integral traces of weak Maass forms of genus zero odd prime level

Duke and the second author defined a family of linear maps from spaces of weakly holomorphic modular forms of negative integral weight and level 1 into spaces of weakly holomorphic modular forms of half-integral weight and level 4 and showed that these lifts preserve the integrality of Fourier coefficients. We show that the generalization of these lifts to modular forms of genus 0 odd prime level also preserves the integrality of Fourier coefficients.

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.

The Kohnen plus space for Hilbert–Siegel modular forms

The Kohnen plus space, roughly speaking, is a space consisting of modular forms of half integral weight with some property in Fourier coefficients. For example, the n-th coefficient of a normal modular form of weight k+1/2 in the plus space is 0 unless (−1)kn is congruent to some square modulo 4. The concept of plus space was initially introduced by Kohnen in 1980. Eichler and Zagier showed that the plus space is isomorphic to the space of Jacobi forms in the one variable case. Later, Ibukiyama generalized these results to the cases for Siegel modular forms in 1992. Also, Hiraga and Ikeda generalized these results to the cases for Hilbert modular forms in 2013. In this paper, we continue to consider the case of Hilbert–Siegel modular forms. An analogue of the previous results will be given.

Lifting from two elliptic modular forms to Siegel modular forms of half-integral weight of even degree

The aim of this paper is to construct lifts from two elliptic modular forms to Siegel modular forms of half-integral weight of even degree under the assumption that the constructed Siegel modular form is not identically zero. The key ingredient of the proof is a new Maass relation for Siegel modular forms of half-integral weight and any degree.

ON SIGN CHANGES FOR ALMOST PRIME COEFFICIENTS OF HALF‐INTEGRAL WEIGHT MODULAR FORMS

For a half-integral weight modular form of weight on such that () are real, we prove for a fixed suitable natural number that changes sign infinitely often as varies over numbers having at most prime factors, assuming the analog of the Ramanujan conjecture for Fourier coefficients of half-integral weight forms.

Structures and Dimensions of Vector Valued Jacobi Forms of Degree Two

We give a complete characterization of vector valued holomorphic Jacobi forms of degree two of index one in the sense of Ziegler by the Taylor expansion and vector valued Siegel modular forms of various weights. By this characterization, we also give explicit dimension formulas for spaces of vector valued holomorphic Jacobi forms of index one of degree two, using those for vector valued Siegel modular forms and a certain surjectivity theorem on the Witt operator (the restriction operator to the diagonals). Our characterization also gives a concrete way to give the plus subspace of the space of Siegel modular forms of half-integral weight.

Metaplectic theta functions and global integrals

We convolve a theta function on an n-fold cover of GL3 with an automorphic form on an n′-fold cover of GL2 for suitable n, n′. To do so, we induce the theta function to the n-fold cover of GL4 and use a Shalika integral. We show that when n=n′=3 this construction gives a new Eulerian integral for an automorphic form on the 3-fold cover of GL2, and when n=4, n′=2, it gives a Dirichlet series with analytic continuation and functional equation that involves both the Fourier coefficients of an automorphic form of half-integral weight and quartic Gauss sums. The analysis of these cases is based on the uniqueness of the Whittaker model for the local exceptional representation. The constructions studied here may be put in the context of a larger family of global integrals constructed using automorphic representations on covering groups. We sketch this wider context and some related conjectures.

Evaluating Igusa functions

The moduli space of principally polarized abelian surfaces is parametrized by three Igusa functions. In this article we investigate a new way to evaluate these functions by using the Siegel Eisenstein series. We explain how to compute the Fourier coefficients of certain Siegel modular forms using classical modular forms of half-integral weight. One of the results in this paper is an explicit algorithm to evaluate the Igusa functions to a prescribed precision.

On indivisibility of relative class numbers of totally imaginary quadratic extensions and these relative Iwasawa invariants

In this paper, we announce some results on indivisibility of relative class numbers of CM quadratic extensions $K/F$ of a fixed totally real number field $F$ which is Galois over $\mathbf{Q}$ and on vanishing of these relative Iwasawa $\lambda_{p}$-, $\mu_{p}$-invariants. In particular, we give a lower bound of the number of such CM extensions $K/F$ with bounded (norm of) relative discriminants. To prove them, we use Hilbert modular forms of half-integral weight.

Hecke operators in half-integral weight

In [6], Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm’s bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators $T_{p^\ell }$ with $p$ prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators $T_{p^{\ell }}$ commute with the Shimura correspondence.

On the Kohnen plus space for Hilbert modular forms of half-integral weight I

AbstractIn this paper, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half-integral weight. The Kohnen plus space can be characterized by the eigenspace of a certain Hecke operator. It can be also characterized by the behavior of the Fourier coefficients. For example, in the parallel weight case, a modular form of weight $\kappa + (1/ 2)$ with $\xi \mathrm{th} $ Fourier coefficient $c(\xi )$ belongs to the Kohnen plus space if and only if $c(\xi )= 0$ unless $\mathop{(- 1)}\nolimits ^{\kappa } \xi $ is congruent to a square modulo $4$. The Kohnen subspace is isomorphic to a certain space of Jacobi forms. We also prove a generalization of the Kohnen–Zagier formula.

ON SHIMURA'S DECOMPOSITION

Let k be an odd integer ≥ 3 and N be a positive integer such that 4|N. Let χ be an even Dirichlet character modulo N. Shimura decomposes the space of half-integral weight cusp forms Sk/2(N,χ) as a direct sum [Formula: see text] where F runs through all newforms of weight k - 1, level dividing N/2 and character χ2, the space Sk/2(N,χ,F) is the subspace of forms that are "Shimura equivalent" to F, and the space S0(N,χ) is the subspace spanned by single-variable theta-series. The explicit computation of this decomposition is important for practical applications of a theorem of Waldspurger relating the critical values of L-functions of quadratic twists of newforms of even integral weight to coefficients of modular forms of half-integral weight. In this paper, we give a more precise definition of the summands Sk/2(N,χ,F) whilst proving that it is equivalent to Shimura's definition. We use our definition to give a practical algorithm for computing Shimura's decomposition, and illustrate this with some examples.

On the Fourier coefficients of Hilbert modular forms of half-integral weight over arbitrary algebraic number fields

In Theorem 2.5 in previous paper [4], we determined the Fourier coefficients of the image of Shimura correspondence of modular forms f of half integral weight over arbitrary algebraic number fields in terms of those of f. It seems that there is a gap in the proof. We give a correct proof of Theorem 2.5 in [4]. Moreover, we deduce useful formulas between the product of Fourier coefficients of f and the central value of quadratic twisted L-series associated with the image of Shimura correspondence of f .

On harmonic weak Maass forms of half integral weight

We show that certain space of vector valued harmonic weak Maass forms of half integral weight is isomorphic to a space of scalar valued ones whose Fourier coefficients are supported on suitable progressions. This kind of result for holomorphic modular forms was proved by Eichler and Zagier.

A short remark on sign changes of Fourier coefficients of half-integral weight modular forms

We give an upper bound on the first sign change of the Fourier coefficients of a cusp form of half-integral weight.

Metaplectic Ramanujan Conjecture Over Function Fields with Applications to Quadratic Forms

We formulate and prove the analogue of the Ramanujan Conjectures for modular forms of half-integral weight subject to some ramification restriction in the setting of a polynomial ring over a finite field. This is applied to give an effective solution to the problem of representations of elements of the ring by ternary quadratic forms. Our proof develops the theory of half-integral weight forms and Siegel's theta functions in this context as well as the analogue of an explicit Waldspurger formula. As in the case over the rationals, the half-integral weight Ramanujan Conjecture is in this way converted into a question of estimating special values of members of a special family of L-functions. These polynomial functions have a growing number of roots (all on the unit circle thanks to Drinfeld and Deligne's work) which are shown to become equidistributed. This eventually leads to the key estimate for Fourier coefficients of half-integral weight cusp forms.

Explicit application of Waldspurger’s theorem

AbstractFor a given cusp form $\phi $ of even integral weight satisfying certain hypotheses, Waldspurger’s theorem relates the critical value of the $\mathrm{L} $-function of the $n\mathrm{th} $ quadratic twist of $\phi $ to the $n\mathrm{th} $ coefficient of a certain modular form of half-integral weight. Waldspurger’s recipes for these modular forms of half-integral weight are far from being explicit. In particular, they are expressed in the language of automorphic representations and Hecke characters. We translate these recipes into congruence conditions involving easily computable values of Dirichlet characters. We illustrate the practicality of our ‘simplified Waldspurger’ by giving several examples.

Eichler integrals for Maass cusp forms of half-integral weight

In this paper, we define and discuss Eichler integrals for Maass cusp forms of half-integral weight on the full modular group. We discuss nearly periodic functions associated to the Eichler integrals, introduce period functions for such Maass cusp forms, and show that the nearly periodic functions and the period functions are closely related. Those functions are extensions of the periodic functions and period functions for Maass cusp forms of weight $0$ on the full modular group introduced by Lewis and Zagier.

Jacobi forms over complex quadratic fields via the cubic Casimir operators

We prove that the center of the algebra of differential operators invariant under the action of the Jacobi group over a complex quadratic field is generated by two cubic Casimir operators, which we compute explicitly. In the spirit of Borel, we consider Jacobi forms over complex quadratic fields that are also eigenfunctions of these Casimir operators, a new approach in the complex case. Theta functions and Eisenstein series provide standard examples. In addition, we introduce an analog of Kohnen's plus space for modular forms of half-integral weight over K=\mathbb{Q}(i) , and provide a lift from it to the space of Jacobi forms over K=\mathbb{Q}(i) .

Non-split sums of coefficients of GL(2)-automorphic forms

Given a cuspidal automorphic form π on GL2, we study smoothed sums of the form $$\sum\nolimits_n {{a_\pi }({n^2} + d)V({n \over x})} $$ . The error term we get is sharp in that it is uniform in both d and Y and depends directly on bounds towards Ramanujan for forms of half-integral weight and Selberg eigenvalue conjecture. Moreover, we identify (at least in the case where the level is square-free) the main term as a simple factor times the residue as s = 1 of the symmetric square L-function L(s, sym2 π). In particular there is no main term unless d > 0 and π is a dihedral form.