Modular Forms Of Half-integral Weight Research Articles

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.

Algebraicity of metaplectic L-functions

We give a precise determination of the algebraicity of the critical values of L-functions associated to Siegel modular forms of half-integral weight and arbitrary degree. We generalise and improve on similar results for the integral-weight case by adapting the Rankin-Selberg method to this setting, with the aid of Shimura's theory of half-integral weight modular forms and recent work on precise algebraicity results by Bouganis. An essential ingredient of this work is a proof of a new analogue of Garrett's conjecture on the algebraicity of Klingen Eisenstein series, a result which is also of independent interest.

Periodicities for Taylor coefficients of half-integral weight modular forms

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 $\tau=i$ 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.

Half-integral weight modular forms and real quadratic $p$-rational fields

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.

$$\Lambda $$-adic families of Jacobi forms

We show that Hida’s families of p-adic elliptic modular forms generalize to p-adic families of Jacobi forms. We also construct p-adic versions of theta lifts from elliptic modular forms to Jacobi forms. Our results extend to Jacobi forms previous works by Hida and Stevens on the related case of half-integral weight modular forms.

Romik's conjecture for the Jacobi theta function

Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point i. He then conjectured that the Taylor coefficients d(n) either vanish or are periodic modulo any prime p; this was proved by the combined efforts of Scherer and Guerzhoy-Mertens-Rolen, with the latter trio considering arbitrary half integral weight modular forms. We refine previous work for p≡1(mod4) by displaying a concise algebraic relation between d(n+p−12) and d(n) related to the p-adic factorial, from which we can deduce periodicity with an effective period.

Determining Siegel modular forms of half-integral weight by their fundamental Fourier coefficients

We prove that a non-zero Siegel cusp form of half-integral weight and degree 2 on $\Gamma _0^{(2)}(4)$ has infinitely many non-zero Fourier coefficients indexed by semi-integral matrices having fundamental discriminant. This is the half-integral weight ve

Bounds for twisted symmetric square L-functions via half-integral weight periods

Abstract We establish the first moment bound $$\begin{align*}\sum_{\varphi} L(\varphi \otimes \varphi \otimes \Psi, \tfrac{1}{2}) \ll_\varepsilon p^{5/4+\varepsilon} \end{align*}$$ for triple product L-functions, where $\Psi $ is a fixed Hecke–Maass form on $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ and $\varphi $ runs over the Hecke–Maass newforms on $\Gamma _0(p)$ of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent $5/4$ is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases. Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime p, all but very few Hecke–Maass newforms on $\Gamma _0(p) \backslash \mathbb {H}$ of bounded eigenvalue have very uniformly distributed mass after pushforward to $\operatorname {\mathrm {SL}}_2(\mathbb {Z}) \backslash \mathbb {H}$ . Our main result turns out to be closely related to estimates such as $$\begin{align*}\sum_{|n| < p} L(\Psi \otimes \chi_{n p},\tfrac{1}{2}) \ll p, \end{align*}$$ where the sum is over those n for which $n p$ is a fundamental discriminant and $\chi _{n p}$ denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke–Iwaniec.

Half-integral weight modular forms and modular forms for Weil representations

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 $$\Gamma _0(4)$$ with Kohnen’s plus condition and modular forms for the Weil representation associated to the lattice with Gram matrix (2). With such an isomorphism, we prove the Zagier duality and express the Borcherds lifts in the case of $$\text {O}(2,1)$$ explicitly.

On the Ramanujan–Petersson conjecture for modular forms of half-integral weight

Abstract We investigate the (still unknown) Ramanujan–Petersson conjecture about the growth of the Fourier coefficients of cusp forms of half-integral weight and prove that it is optimal, at least for newforms in the plus space.

A variant of multiplicity one theorems for half-integral weight modular forms

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.

The p-adic L-function for half-integral weight modular forms

The p-adic L-function for modular forms of integral weight is well-known. For certain weights the p-adic L-function for modular forms of half-integral weight is also known to exist, via a correspondence, established by Shimura, between them and forms of integral weight. However, we construct it here without any recourse to the Shimura correspondence, allowing us to establish its existence for all weights, including those exempt from the Shimura correspondence. We do this by employing the Rankin–Selberg method, and proving explicit p-adic congruences in the resultant Rankin–Selberg expression.

Sums of quadratic functions with two discriminants

In a 1999 paper, Zagier discusses a construction of a function Fk,D(x) defined for an even integer k≥2, and a positive discriminant D. This construction is intimately related to half-integral weight modular forms. In particular, the average value of this function is a constant multiple of the D-th Fourier coefficient of weight k+1/2 Eisenstein series constructed by H. Cohen.In this note we consider a construction which works both for even and odd positive integers k. Our function Fk,D,d(x) depends on two discriminants d and D with signs sgn(d)=sgn(D)=(−1)k, degenerates to Zagier's function when d=1, namely,Fk,D,1(x)=Fk,D(x), and has very similar properties. In particular, we prove that the average value of Fk,D,d(x) is again a Fourier coefficient of H. Cohen's Eisenstein series of weight k+1/2, while now the integer k≥2 is allowed to be both even and odd.

Non-vanishing of products of Fourier coefficients of modular forms of half-integral weight

We prove a non-vanishing result in weight aspect for the product of two Fourier coefficients of a Hecke eigenform of half-integral weight.

Modularity of generating series of winding numbers

The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.

Special values of Hecke L-functions of modular forms of half-integral weight and cohomology

We start developing a cohomology theory in the case of half-integral weight with an attempt to focus again on the connection to special values of L-functions at half-integral and integral points inside the critical strip.

Equidistribution of signs for Hilbert modular forms of half-integral weight

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

Nonzero coefficients of half-integral weight modular forms mod $$\ell $$ ℓ

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 $$\ell $$ l . Among the applications of this we show that there are $$\gg \sqrt{X}/\log \log X$$ ≫ X / log log X integers $$n \le X$$ n ≤ X for which the partition function p(n) is not divisible by $$\ell $$ l , and that there are $$\gg \sqrt{X}/\log \log X$$ ≫ X / log log X values of $$n \le X$$ n ≤ X for which c(n), the nth Fourier coefficient of the j-invariant, is odd.

On Hecke L-functions attached to half-integral weight modular forms

We investigate non-vanishing properties of L(f,s) on the real line, when f is a Hecke eigenform of half-integral weight k+12 on Γ0(4).

Shimura lifts of weakly holomorphic modular forms

We show how to realize the Shimura lift of arbitrary level and character using the vector-valued theta lifts of Borcherds. Using the regularization of Borcherds' lift we extend the Shimura lift to take weakly holomorphic modular forms of half-integral weight to meromorphic modular forms of even integral weight having poles at CM points.