Forms Of Half-integral Weight Research Articles

The twisted second moment of modular half-integral weight $L$-functions

Given a half-integral weight holomorphic Kohnen newform f on \Gamma_{0}(4) , we prove an asymptotic formula for large primes p with power saving error term for \sum_{\chi \pmod{p}}^{\qquad{}{}_{\scriptsize{*}}}| L(1/2,f,\chi) |^{2}. Our result is unconditional, it does not rely on the Ramanujan–Petersson conjecture for the form f . This gives a very sharp Lindelöf-on-average result for Dirichlet series attached to Hecke eigenforms without an Euler product. The Lindelöf hypothesis for such series was originally conjectured by Hoffstein. There are two main inputs. The first is a careful spectral analysis of a highly unbalanced shifted convolution problem involving the Fourier coefficients of half-integral weight forms. The second input is a bound for sums of products of Salié sums in the Pólya–Vinogradov range. Half-integrality is fully exploited to establish such an estimate. We use the closed form evaluation of the Salié sum to relate our problem to the sequence \alpha n^{2} \pmod{1} . Our treatment of this sequence is inspired by work of Rudnick–Sarnak and the second author on the local spacings of \alpha n^{2} modulo 1.

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.

L-functions associated to Jacobi forms of half-integral weight and a converse theorem

We associate certain L-functions to a Jacobi form of half-integral weight and study their analytic properties. We also prove a converse theorem for Jacobi form of half-integral weight.

A Voronoi summation formula for non-holomorphic Maass forms of half-integral weight

A Voronoi summation formula for non-holomorphic Maass forms of half-integral weight

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.

Scarcity of congruences for the partition function

abstract: The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0$ $({\rm mod}\;\ell)$ for the primes $\ell=5,7,11$, and it is known that there are no others of this form. On the other hand, for every prime $\ell\geq 5$ there are infinitely many examples of congruences of the form $p(\ell Q^m n+\beta)\equiv 0$ $({\rm mod}\;\ell)$ where $Q\geq 5$ is prime and $m\geq 3$. This leaves open the question of the existence of such congruences when $m=1$ or $m=2$ (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve $X(\ell Q)$, Galois representations and the arithmetic large sieve.

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

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.

A classification of harmonic weak Maaß forms of half-integral weight

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 an initial understanding of a theta correspondence for extensions of Harish-Chandra modules.

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

$p$-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.

The arithmetic of modular grids

A modular grid is a pair of sequences ( f m ) m $ (f_m ) _m$ and ( g n ) n $ (g_n ) _n$ of weakly holomorphic modular forms such that for almost all m and n, the coefficient of q n $q^n$ in f m $f_m$ is the negative of the coefficient of q m $q^m$ in g n $g_n$ . Zagier proved this coefficient duality in weights 1/2 and 3/2 in the Kohnen plus space, and such grids have appeared for Poincaré series, for modular forms of integral weight, and in many other situations. We give a general proof of coefficient duality for canonical row-reduced bases of spaces of weakly holomorphic modular forms of integral or half-integral weight for every group Γ ⊆ SL 2 ( R ) $\Gamma \subseteq {\text{\rm SL}}_2(\mathbb {R})$ commensurable with SL 2 ( Z ) ${\text{\rm SL}}_2(\mathbb {Z})$ . We construct bivariate generating functions that encode these modular forms, and study linear operations on the resulting modular grids.

On Jacobi forms of half-integral weight and matrix level

TextIn this paper we will consider the Kohnen plus space for Hilbert-Siegel-Jacobi forms of half-integral weight and certain type of matrix index. As in the case of classical modular forms, the Jacobi forms in the Kohnen plus space are characterized by some restrictions on their Fourier coefficients. We will show that a Jacobi form of half-integral weight is in the Kohnen plus space if and only if the representation, which is generated by the form, of the adelic metaplectic double covering of the Jacobi group is equivalent to the Weil representation and use this equivalence condition to give an isomorphism of the Kohnen plus space with the space of Jacobi forms of certain corresponding integral weight and matrix index. Finally, we will see that the given isomorphism is a Hecke isomorphism with respect to the odd places of the underlying totally real number field. VideoFor a video summary of this paper, please visit https://youtu.be/J88ahOlr0GI.

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 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.

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.

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.