In this paper we define Banach spaces of overconvergent half-integral weight $p$-adic modular forms and Banach modules of families of overconvergent half-integral weight $p$-adic modular forms over admissible open subsets of weight space. Both spaces are equipped with a continuous Hecke action for which $U_{p^2}$ is moreover compact. The modules of families of forms are used to construct an eigencurve parameterizing all finite-slope systems of eigenvalues of Hecke operators acting on these spaces. We also prove an analog of Coleman's theorem stating that overconvergent eigenforms of suitably low slope are classical.
We construct a rigid-analytic map from the the author's half-integral weight cuspidal eigencurve to its integral weight counterpart that interpolates the classical Shimura lifting.
In this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of k-th invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid-analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the U operator, th…
We give a necessary and sufficient criterion for the existence of Borcherds products of half-integral weight associated to even lattices that split two hyperbolic planes. In particular we prove that half-integral weight Borcherds products exist for lattices of arbitrary rank.
We show that signs of Fourier coefficients, on certain sub-families, determine the half-integral weight cuspidal eigenform uniquely, up to a positive constant. We also study sign change results for the product of the Fourier coefficients of two distinct half-integral weight eigenforms.
We investigate non-vanishing properties of $L(f,s)$ on the real line, when $f$ is a Hecke eigenform of half-integral weight $k+{1\over 2}$ on $Γ_0(4).$
Systematic choice of the Hecke eigenforms of half-integral weight is an interesting problem in the theory of modular forms. In this paper, we find all Dedekind-eta products of half-integral weight which are Hecke eigenforms up to weight 15/2 with varying levels. Proof is based on the Shimura lift.
In this article, we prove non-vanishing results for $L$-functions associated to holomorphic cusp forms of half-integral weight on average (over an orthogonal basis of Hecke eigenforms). This extends a result of W. Kohnen to forms of half-integral weight.
We define an analogue of the Bol operator on spaces of weakly holomorphic modular forms of half-integral weight. We establish its main properties and relation with other objects.
We prove a strengthening of Muić's integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups and use it to prove a result on non-vanishing of L-functions associated to cusp forms of half-integral weight.
Using half-integral weight modular forms we give a criterion for the existence of real quadratic $p$-rational fields. For $p=5$ we prove the existence of infinitely many real quadratic $p$-rational fields.
In this paper, we investigate the sign changes of Fourier coefficients of half-integral weight Hecke eigenforms and give two quantitative results on the number of sign changes.
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.
In this article we show analytic properties of certain Rankin-Selberg type Dirichlet series for holomorphic Jacobi cusp forms of integral weight and of half-integral weight. The numerators of these Dirichlet series are the inner products of Fourier-Jacobi coefficients of two Jacobi cusp forms. The denominators and the range of summation of these Dirichlet series are like the ones of the Koecher-Maass series. The meromorphic continuations and functional equations of these Dirichlet series are obtained. Moreover, an identity between the Petersson norms of Jacobi forms with respect to linear isom…
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.
In \cite{Shimura}, 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.
We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half-plane $\re s\textgreater{}\tfrac{1}{2}$. This exhibits a high fluctuation of the coefficients at squarefree integers.
We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.
A general Voronoï summation formula for the (metaplectic) double cover of $\text{GL}_2$ is derived via the representation theoretic framework à la Ichino--Templier. The identity is also formulated classically and used to establish Voronoï summation formulae for half-integral weight modular forms and Maaß forms.
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.
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.
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.
The aim of this paper is to show lifts from pairs of 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 of the proof is to introduce a certain generalization of the Maass relation for Siegel modular forms of half-integral weight of general degree.
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 w…
Let $F$ be a totally real number field and $\mathfrak{o}$ the ring of integers of $F$. We study theta functions which are Hilbert modular forms of half-integral weight for the Hilbert modular group $\mathrm{SL}_2(\mathfrak{o})$. We obtain an equivalent condition that there exists a multiplier system of half-integral weight for $\mathrm{SL}_2(\mathfrak{o})$. We determine the condition of $F$ that there exists a theta function which is a Hilbert modular form of half-integral weight for $\mathrm{SL}_2(\mathfrak{o})$. The theta function is defined by a sum on a fractional ideal $\mathfrak{a}$ of $…
Bounding sup-norms of modular forms in terms of the level has been the focus of much recent study. In this work the sup norm of a half integral weight cusp form is bounded in terms of the level.
For a half-integral weight modular form $f = \sum_{n=1}^{\infty} a_f(n)n^{\frac{k-1}{2}} q^n$ of weight $k = l +\frac{1}{2}$ on $Γ_0(4)$ such that $a_f(n)$ ($n$ $\in$ $\mathbb{N}$) are real, we prove for a fixed suitable natural number $r$ that $a_f(n)$ changes sign infinitely often as $n$ varies over numbers having at most $r$ prime factors, assuming the analog of the Ramanujan conjecture for half-integral weight forms.
In this paper we study special bases of certain spaces of half-integral weight weakly holomorphic modular forms. We establish a criterion for the integrality of Fourier coefficients of such bases. By using recursive relations between Hecke operators, we derive relations of Fourier coefficients of each basis element and obtain congruences of the Fourier coefficients, which extend known congruences for traces of singular moduli.
We introduce a geometric formalism for studying modular forms of half-integral weight and explore some of its basic properties. Geometric Hecke operators are constructed and some basic spaces of $p$-adic forms are introduced. The $p$-adic theory is greatly expanded in subsequent papers, making that part of this paper largely obsolete.
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
In analogy with the classical theory of Eichler integrals for integral weight modular forms, Lawrence and Zagier considered examples of Eichler integrals of certain half-integral weight modular forms. These served as early prototypes of a new type of object, which Zagier later called a quantum modular form. Since then, a number of others have studied similar examples. Here we develop the theory in a general context, giving rise to a well-defined class of quantum modular forms. Since elements of this class show up frequently in examples of combinatorial and number theoretical interest, we propo…
We establish an isomorphism between certain complex-valued and vector-valued modular form spaces of half-integral weight, generalizing the well-known isomorphism between modular forms for $Γ_0(4)$ with Kohnen's plus condition and modular forms for the Weil representation associated to the discriminant form for the lattice with Gram matrix $(2)$. With such an isomorphism, we prove the Zagier duality and write down the Borcherds lifts explicitly.
We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry-Ganguly-Kowalski-Michel and Kowalski-Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution which comes from the asymptotic mixed distribution of Salié sums.
Weights of modular forms on $\mathrm{SO}^{+}(2,l)$ and congruences between Eisenstein series and cusp forms of half-integral weight on $\mathrm{SL}_{2}$
To study statistical properties of modular forms, including for instance Sato-Tate like problems, it is essential to have a large number of Fourier coefficients. In this article, we exhibit three bases for the space of modular forms of any half-integral weight and level 4, which have the property that many coefficients can be computed (relatively) quickly on a computer.
In this paper, we explicitly construct mock modular forms whose shadows are Eisenstein series of arbitrary integral and half-integral weight, level and character at the cusps $\infty$ and $0$. As an application, we give explicit construction of harmonic weak Maass forms which are Hecke eigenforms and are the preimages of $Θ^k, k \in \{ 3, 5, 7\}$ under the shadow operator, where $Θ$ is the classical Jacobi theta function.
We classify Harish-Chandra modules generated by the pullback to the metaplectic group of harmonic weak Maaß forms with exponential growth allowed at the cusps. This extends work by Schulze-Pillot and parallels recent work by Bringmann-Kudla, who investigated the case of integral weights. We realize each of our cases via a regularized theta lift of an integral weight harmonic weak Maaß form. Harish-Chandra modules in both integral and half-integral weight that occur need not be irreducible. Therefore, our display of the role that the theta lifting takes in this picture, we hope, contributes to …
Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has been much less studied. Recently, Romik made a conjecture about the periodicity of coefficients around $τ=i$ of the classical Jacobi theta function. Here, we prove this conjecture and generalize the phenomenon observed by Romik to a general class of modular 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 $Γ_0(N)$ 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 $Θ$-operator. Here, we study the analogous problem for modular forms of half-integral weight on $\operatorname{SL}_{2}(\mathbb{Z})$. Let $η$ be the Dedekind eta function. For a wide range of weights, we prove that every…
In this thesis, we construct a half-integral weight multiplier system on the group SU(2,1). In order to do so, we first find a formula for a 2-cocycle representing the double cover of SU(2,1)(k), where k is a local field. For each non-archimedean local field k, we describe how the cocycle splits on a compact open subgroup. The multiplier system is then expressed in terms of the product of the local splittings at each prime.
We prove a subconvexity bound in the conductor aspect for $L(s,f,χ)$ where $f$ is a half integer weight modular form. This $L$-function has analytic continuation and functional equation, but no Euler product. Due to the lack of an Euler product, one does not expect a Riemann hypothesis for half integer weight modular forms. Nevertheless one may speculate a Lindelof-type hypothesis, and this current subconvexity result is an indication towards its truth.
Let $f$ be a half-integral weight cusp form of level $4N$ for odd and squarefree $N$ and let $a(n)$ denote its $n^{\rm th}$ 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 $n\le x$ such that $a(n)>n^{-α}$ where $x$ and $α$ are positive and $f$ is not necessarily a Hecke eigenform.
In this paper, we study the parallel cases of Zagier's and Folsom-Ono's grids of weakly holomorphic (resp. weakly holomorphic and mock modular) forms of weights 3/2 and 1/2, investigating their $p$-adic properties under the action of Hecke operators.
In this article, using the heat kernel approach from \cite{bouche}, we derive sup-norm bounds for cusp forms of integral and half integral weight. Let $Γ\subset \mathrm{PSL}_{2}(\mathbb{R})$ be a cocompact Fuchsian subgroup of first kind. For $k\in\frac{1}{2}\mathbb{Z}$ (or $k\in 2\mathbb{Z}$), let $S^{k}(Γ)$ denote the complex vector space of weight-$k$ cusp forms. Let $\lbrace f_{1},\ldots,f_{j_{k}} \rbrace$ denote an orthonormal basis of $S^{k}(Γ)$. 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 th…
For a prime $p\equiv 3$ $(\text{mod }4)$ and $m\ge 2$, Romik raised a question about whether the Taylor coefficients around $\sqrt{-1}$ of the classical Jacobi theta function $θ_3$ eventually vanish modulo $p^m$. This question can be extended to a class of modular forms of half-integral weight on $Γ_1(4)$ and CM points; in this paper, we prove an affirmative answer to it for primes $p\ge5$. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on $\mathrm{SL}_2(\mathbb{Z})$.
We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime $\ell$. Among the applications of this we show that there are $\gg \sqrt{X}/\log \log X$ integers $n \leq X$ for which the partition function $p(n)$ is not divisible by $\ell$, and that there are $\gg \sqrt{X}/\log \log X$ values of $n \leq X$ for which $c(n)$, the $n$th Fourier coefficient of the $j$-invariant, is odd.
We construct a basis for the space of half-integral weight Siegel Eisenstein series of level 4N where N is odd and square-free. Then we restrict our attention to those Eisenstein series generated from elements of $Γ_0(4)$, commenting on why this restriction is necessary for our methods. We directly apply to these forms all Hecke operators attached to odd primes, and we realize the images explicitly as linear combinations of Siegel Eisenstein series. Using this information, we diagonalize the subspace of Eisenstein series generated from elements of $Γ_0(4)$, obtaining a multiplicity-one result.
We introduce an alternate set of generators for the Hecka algebra, and give an explicit formula for the action of these operators on Fourier coefficients. With this, we compute the eigenvalues of Hecke operators acting on average Siegel theta series with half-integral weight (provided the prime associated to the operators does not divide the level of the theta series). Next, we bound the eigenvalues of these operators in terms of bounds on Fourier coefficients. Then we show that the half-integral weight Kitaoka subspace is stable under all Hecke operators. Finally, we observe that an obvious i…
We locate all but $O(\sqrt{k\log{k}})$ zeroes of the half integral weight Eisenstein series $E_\infty(z,k)$ of $Γ_0(4)$ for $k$ sufficiently large. To do this, we relate $E_\infty(z,k)$ to $Γ_0(4)$'s other Eisenstein series, $E_0(z,k)$ and $E_\frac{1}{2}(z,k)$, which are easier to study in the regions of which zeroes occur. We will use trigonometric approximations of $E_0(z,k)$ and $E_\frac{1}{2}(z,k)$ in order to locate the zeroes.
