Half-integral Weight Research Articles

On effective multiplicity one for modular forms of half-integral weight

On effective multiplicity one for modular forms of half-integral weight

Correction to: Period-like polynomials for L-series associated with half-integral weight cusp forms

AbstractWe correct an error affecting Propositions 5.2 and Theorem 5.4 of [1]

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.

On the Shintani lifting of integral weight modular forms to half-integral weight modular forms

On the Shintani lifting of integral weight modular forms to half-integral weight modular forms

The Weil bound for generalized Kloosterman sums of half-integral weight

Abstract Let L be an even lattice of odd rank with discriminant group L ′ / L {L^{\prime}/L} , and let α , β ∈ L ′ / L {\alpha,\beta\in L^{\prime}/L} . We prove the Weil bound for the Kloosterman sums S α , β ⁢ ( m , n , c ) {S_{\alpha,\beta}(m,n,c)} of half-integral weight for the Weil Representation attached to L. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen’s identity for plus space Kloosterman sums with the theta multiplier system.

Period-like polynomials for L-series associated with half-integral weight cusp forms

Given the L-series of a half-integral weight cusp form, we construct polynomials behaving similarly to the classical period polynomial of an integral weight cusp form. We also define a lift of half-integral weight cusp forms to integral weight cusp forms that are compatible with the L-series of the respective forms.

On products of Fourier coefficients of half-integral weight cusp forms

On products of Fourier coefficients of half-integral weight cusp forms

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.

Uniform bounds for Kloosterman sums of half-integral weight with applications

Abstract Sums of Kloosterman sums have deep connections with the theory of modular forms, and their estimation has many important consequences. Kuznetsov used his famous trace formula and got a power-saving estimate with respect to x with implied constants depending on m and n. Recently, in 2009, Sarnak and Tsimerman obtained a bound uniformly in x, m and n. The generalized Kloosterman sums are defined with multiplier systems and on congruence subgroups. Goldfeld and Sarnak bounded sums of them with main terms corresponding to exceptional eigenvalues of the hyperbolic Laplacian. Their error term is a power of x with implied constants depending on all the other factors. In this paper, for a wide class of half-integral weight multiplier systems, we get the same bound with the error term uniformly in x, m and n. Such uniform bounds have great applications. For the eta-multiplier, Ahlgren and Andersen obtained a uniform and power-saving bound with respect to m and n, which resulted in a convergent error estimate on the Rademacher exact formula of the partition function p ⁢ ( n ) {p(n)} . We also establish a Rademacher-type exact formula for the difference of partitions of rank modulo 3, which allows us to apply our power-saving estimate to the tail of the formula for a convergent error bound.

On the non-vanishing of Fourier coefficients of half-integral weight cuspforms

On the non-vanishing of Fourier coefficients of half-integral weight cuspforms

A SURVEY ON RELATION BETWEEN THE RAMANUJAN’S CONGRUENCE FORM AND THE BACK BACKGROUND OF THE MODULUS FORM

In this paper, we will be covering the background of modular forms and explain how they are used to prove many important results about partition function congruence. We first go over the background of molecular forms mainly focusing on modular forms over SL 2(z). In addition to this, we will also introduce more general modular forms and half-integral weight molecular forms by Ramanujan's congruence proof. Apart from this, an additional family of Congruences was proven using the theory of modular forms over SL₂(z) and also using some computer-checkable computation, we will also modify the Method of ono to give a more general framework of proof. It allows us to straightforwardly extend the congruence on proving modules 13.17, 19, and 23 to give similar congruence to modules 29 and 31.

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.

Sign changes of coefficients of half-integral weight Hecke eigenforms

Let f \mathfrak {f} be a cusp form of half-weight k + 1 / 2 k+1/2 and at most quadratic nebentype character whose Fourier coefficients are denoted as a f ( n ) \mathfrak {a}_\mathfrak {f}(n) . We study sign changes of the family { a f ( t p 2 ) } p ∈ P \{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}} where t t is a square-free number and p p runs through the prime numbers. By taking use of the relationship between the cusp form f \mathfrak {f} of half-integral weight and the Shimura lift f t f_t , we establish that under certain conditions, the Hecke eigenforms of half-integral weight could be determined by sign changes of the sequence { a f ( t p 2 ) } p ∈ P \{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}} .

Infinitely many zeros of additively twisted L-functions on the critical line

For f a cuspidal modular form for the group Γ0(N) of integral or half-integral weight, N a multiple of 4 in case the weight is half-integral, we study the zeros of the L-function attached to f twisted by an additive character e2πinpq with pq∈Q. We prove that for certain f and pq∈Q, the additively twisted L-function has infinitely many zeros on the critical line. We develop a variant of the Hardy-Littlewood method which uses automorphic distributions to prove the result.

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.