Modular Forms Of Half-integral Weight Research Articles

Modular forms of half-integral weight on exceptional groups

We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by ${\pm }1$ . We analyze the minimal modular form $\Theta _{F_4}$ on the double cover of $F_4$ , following Loke–Savin and Ginzburg. Using $\Theta _{F_4}$ , we define a modular form of weight $\tfrac {1}{2}$ on (the double cover of) $G_2$ . We prove that the Fourier coefficients of this modular form on $G_2$ see the $2$ -torsion in the narrow class groups of totally real cubic fields.

Analogues of the Bol operator for half-integral weight weakly holomorphic modular forms

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.

Toric Periods and 𝑝-adic Families of Modular Forms of Half-Integral Weight

The primary goal of this work is to construct p p -adic families of modular forms of half-integral weight, by using Waldspurger’s automorphic framework to make the results as comprehensive and precise as possible. A secondary goal is to clarify the role of test vectors as defined by Gross-Prasad in the elucidation of general formulae for the Fourier coefficients of modular forms of half-integral weight in terms of toric periods of the corresponding modular forms of integral weight. As a consequence of our work, we develop a generalization of a classical formula due to Shintani, and make precise the conditions under which Shintani’s lift vanishes. We also give a number of results on test vectors for ramified representations which are of independent interest.

$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $\Gamma_1(4)$

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 $\theta_3$ eventually vanish modulo $p^m$. This question can be extended to a class of modular forms of half-integral weight on $\Gamma_1(4)$ 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 $\mathrm{SL}_2(\mathbb{Z})$.

Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ

Abstract Let k k be a nonnegative integer. Let K K be a number field and O K {{\mathcal{O}}}_{K} be the ring of integers of K K . Let ℓ ≥ 5 \ell \ge 5 be a prime and v v be a prime ideal of O K {{\mathcal{O}}}_{K} over ℓ \ell . Let f f be a modular form of weight k + 1 2 k+\frac{1}{2} on Γ 0 {\Gamma }_{0} (4) such that its Fourier coefficients are in O K {{\mathcal{O}}}_{K} . In this article, we study sufficient conditions that if f f has the form f ( z ) ≡ ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 ( mod v ) f\left(z)\equiv \mathop{\sum }\limits_{n=1}^{\infty }\mathop{\sum }\limits_{i=1}^{t}{a}_{f}\left({s}_{i}{n}^{2}){q}^{{s}_{i}{n}^{2}}\hspace{0.5em}\left({\rm{mod}}\hspace{0.33em}v) with square-free integers s i {s}_{i} , then f f is congruent to a linear combination of iterated derivatives of a single theta function modulo v v .

On multiplier systems and theta functions of half-integral weight for the Hilbert modular group [formula omitted

Let F be a totally real number field and 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 SL2(o). We obtain an equivalent condition that there exists a multiplier system of half-integral weight for SL2(o). We determine the condition of F that there exists a theta function which is a Hilbert modular form of half-integral weight for SL2(o). The theta function is defined by a sum on a fractional ideal of F.

Effective construction of Hilbert modular forms of half-integral weight

Given a Hilbert cuspidal newform $g$ we construct a family of modular forms of half-integral weight whose Fourier coefficients give the central values of the twisted $L$-series of $g$ by fundamental discriminants. The family is parametrized by quadratic conditions on the primes dividing the level of $g$, where each form has coefficients supported on the discriminants satisfying the conditions. These modular forms are given as generalized theta series and thus their coefficients can be effectively computed. Our construction works over arbitrary totally real number fields, except that in the case of odd degree the square levels are excluded. It includes all discriminants except those divisible by primes whose square divides the level.

Fast computation of half-integral weight modular forms

To study statistical properties of modular forms, including for instance Sato–Tate like problems, it is essential to be able to compute a large number of Fourier coefficients. We show that this can be achieved in level 4 for a large range of half-integral weights by making use of one of three explicit bases, the elements of which can be calculated via fast power series operations.

On the Shimura lift modulo 2

We study the kernel of the Shimura lifting from modular forms of half-integral weight to integral weight.

A supplement to “On the Fourier coefficients of Hilbert modular forms of half-integral weight over arbitrary algebraic number fields, Tsukuba J. Math. 37(2013), 1-11”

The purpose of this note is to prove that the Shintani lift of Hilbert modular forms over algebraic number fields commutes with the action of Hecke operators. We show our assertion using the commutativity of the Shimura lift with Hecke operators and some properties of adjoint mappings. This commutativity of the Shintani lift plays an essential role for the proof of the Waldspurger-type theorem concerning the Fourier coefficients of modular forms of half-integral weight over arbitrary algebraic number fields.

On the distribution of coefficients of half-integral weight modular forms and the Bruinier–Kohnen conjecture

This work represents a systematic computational study of the distribution of the Fourier coefficients of cuspidal Hecke eigenforms of level $\Gamma_0(4)$ and half-integral weights. Based on substantial calculations, the question is raised whether the distribution of normalised Fourier coefficients with bounded indices can be approximated by a generalised Gaussian distribution. Moreover, it is argued that the apparent symmetry around zero of the data lends strong evidence to the Bruinier-Kohnen Conjecture on the equidistribution of signs and even suggests the strengthening that signs and absolute values are distributed independently.

Voronoï Summation for Half-Integral Weight Automorphic Forms

Abstract A general Voronoï summation formula for the (metaplectic) double cover of $\operatorname{GL}_2$ 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.

Proofs of Ibukiyama’s conjectures on Siegel modular forms of half-integral weight and of degree 2

We prove Ibukiyama’s conjectures on Siegel modular forms of half-integral weight and of degree 2 by using Arthur’s multiplicity formula on the split odd special orthogonal group $${\text {SO}}_5$$ and Gan–Ichino’s multiplicity formula on the metaplectic group $${\text {Mp}}_4$$ . In the proof, the representation theory of the Jacobi groups also plays an important role.

English

Suppose that f is a primitive Hecke eigenform or a Mass cusp form for Γ0(N) with normalized eigenvalues λf (n) and let X > 1 be a real number. We consider the sum $${{\cal S}_k}(X): = \sum\limits_{n < X} {\sum\limits_{n = {n_1},{n_2}, \ldots ,{n_k}} {{\lambda _f}({n_1}){\lambda _f}({n_2}) \ldots {\lambda _f}({n_k})}}$$ and show that $${{\cal S}_k}(X){\ll _{f,\varepsilon }}{X^{1 - 3/(2(k + 3)) + \varepsilon}}$$ for every k ⩾ 1 and e > 0. The same problem was considered for the case N = 1, that is for the full modular group in Lu (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for k ⩾ 5. Since the result is valid for arbitrary level, we obtain, as a corollary, estimates on sums of the form $${{\cal S}_k}(X)$$ , where the sum involves restricted coefficients of some suitable half integral weight modular forms.

Mod p modular forms and simple congruences

In this article, we first give a complete description of the algebra of integer weight modular forms on the congruence subgroup $$\Gamma _0(2)$$ modulo a prime $$p\ge 3$$ . This result parallels results of Swinnerton-Dyer in the $$SL_2(\mathbb {Z})$$ case, Katz on the subgroup $$\Gamma (N)$$ for $$N\ge 3$$ , Gross on the subgroup $$\Gamma _1(N)$$ for $$N\ge 4$$ and Tupan on modular forms of half-integral weight on $$\Gamma _1(4)$$ . Next, we use the theory of mod p modular forms on $$\Gamma _0(2)$$ to prove the non-existence of simple congruences for Fourier coefficients of quotients of certain integer weight Eisenstein series on $$\Gamma _0(2)$$ . The non-existence of simple congruences for coefficients of quotients of Eisenstein series on $$SL_2(\mathbb {Z})$$ has been shown by Dewar.

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 $\eta$.

Shimura lifts of certain classes of modular forms of half-integral weight

Shimura defined a family of maps from the space of modular forms of half-integral weight to the space of modular forms of integral weight. Selberg in his unpublished work found explicitly this correspondence (the first Shimura map [Formula: see text]) for the class of forms which are products of a Hecke eigenform of level one and a Jacobi theta function. Later, Cipra generalized the work of Selberg to the case where Jacobi theta functions are replaced by the theta functions associated to Dirichlet character of prime power moduli, and the level one Hecke eigenforms are replaced by newforms of arbitrary level. Hansen and Naqvi generalized Cipra’s work (on the image of a class of modular forms under the first Shimura map [Formula: see text]) to cover theta functions associated to Dirichlet characters of arbitrary moduli. In this paper, we show that the earlier results can be modified to get similar results for the [Formula: see text]th Shimura lifts [Formula: see text], for any positive square-free integer [Formula: see text].

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.

Half-integral weight modular forms and application to neutrino mass models

We generalize the modular invariance approach to include the half-integral weight modular forms. Accordingly the modular group should be extended to its metaplectic covering group for consistency. We introduce the well-defined half-integral weight modular forms for congruence subgroup $\Gamma(4N)$ and show that they can be decomposed into the irreducible multiplets of finite metaplectic group $\widetilde{\Gamma}_{4N}$. We construct concrete expressions of the half-integral/integral modular forms for $\Gamma(4)$ up to weight 6 and arrange them into the irreducible representations of $\widetilde{\Gamma}_4$. We present three typical models with $\widetilde{\Gamma}_4$ modular symmetry for neutrino masses and mixing, and the phenomenological predictions of each model are analyzed numerically.

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.