An additive problem in the Fourier coefficients of cusp forms
Gergely Harcos
备注：
16 pages, LaTeX2e; v2: lots of changes, Theorem 2 is new, notation changed to standard one, abstract and further references added; v3: minor changes, some restriction imposed in Theorem 2, additional references; v4: introduction revised, references added, typos corrected; v5: final, revised version incorporating suggestions by the referee (e.g. Section 5 was added)
We establish an estimate on sums of shifted products of Fourier coefficients coming from holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums are analogous to the binary additive divisor sum which has been studied extensively. As an application we derive, extending work of Duke, Friedlander and Iwaniec, a subconvex estimate on the critical line for L-functions associated to character twists of these cusp forms.
The aim of this article is to study the fields generated by the Fourier coefficients of Hilbert newforms at arbitrary cusps. Precisely, given a cuspidal Hilbert newform $f$ and a matrix $σ$ in (a suitable conjugate of) the Hilbert modular group, we give a cyclotomic extension of the field generated by the Fourier coefficients at infinity which contains all the Fourier coefficients of $f||_kσ$.
Kohnen and Sengupta proved that two cusp forms of different integral weights with real algebraic Fourier coefficients have infinitely many Fourier coefficients of the same as well as of opposite sign, up to the action of a Galois automorphism. Recently Gun, Kohnen and Rath strengthen their result by comparing the simultaneous sign changes of Fourier coefficients of two cusp forms with arbitrary real Fourier coefficients. The simultaneous sign changes of Fourier coefficients of two same integral weight cusp forms follow from an earlier work of Ram Murty. In this note we compare the signs of the…
Let $ν_{f}(n)$ be the $n$-th nomalized Fourier coefficient of a Hecke--Maass cusp form $f$ for ${\rm SL}(2,\Z)$ and let $α$ be a real number. We prove strong oscillations of the argument of $ν_{f}(n)μ(n) \exp (2πi n α)$ as $n$ takes consecutive integral values.
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 establish quantitative results for sign changes in certain subsequences of primitive Fourier coefficients of a non-zero Siegel cusp form of arbitrary degree over congruence subgroups. As a corollary of our result for degree two Siegel cusp forms, we get sign changes of its diagonal Fourier coefficients. In the course of our proofs, we prove the non-vanishing of certain type of Fourier-Jacobi coefficients of a Siegel cusp form and all theta components of certain Jacobi cusp forms of arbitrary degree over congruence subgroups, which are also of independent interest.
In this article, we study the simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different integral weights. We also study the simultaneous non-vanishing of Fourier coefficients, of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights, at powers of a fixed prime ideal.
This paper studies the Fourier expansion of Hecke-Maass eigenforms for $GL(2, \mathbb Q)$ of arbitrary weight, level, and character at various cusps. Translating well known results in the theory of adelic automorphic representations into classical language, a multiplicative expression for the Fourier coefficients at any cusp is derived. In general, this expression involves Fourier coefficients at several different cusps. A sufficient condition for the existence of multiplicative relations among Fourier coefficients at a single cusp is given. It is shown that if the level is 4 times (or in some…
In this short note, we treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms by a rather simple argument. Our result improves previous results established by more advanced approaches.
Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler--Shimura isomorphism and contain information about automorphic $L$-functions. In this paper we prove that central values of additive twists of the $L$-function associated to a holomorphic cusp form $f$ of even weight $k$ are asymptotically normally distributed. This generalizes (to $k\geq 4$) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore we give as an application an asymptotic formula for the averages of certain 'wide' fam…
In this paper we study the Fourier coefficients of theta functions attached to Dirichlet characters at cusps other than infinity. The method is based on expressing them in terms of explicit elements of the adelic Schwartz space and studying the action of the adelic metaplectic group on these elements. We derive explicit formulae for the Fourier coefficients at all cusps. For the sake of simplicity, some restrictions are placed on the Dirichlet characters considered.
We study sums of additively twisted Fourier coefficients of a holomorphic cusp form, a Maass cusp form, and the symmetric-square lift of a holomorphic cusp form. We obtain bounds that are uniform with respect to both the form and the terms of the additive twist. We also improve a bound on a sum of additively twisted Fourier coefficients of a holomorphic cusp form that have also been twisted by a nonlinear exponential term. This last bound is uniform with respect to the terms of the additive twist but it is not uniform with respect to the form.
The purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form, $a(n)a(n + r)$ $(n \geq 1)$ for a fixed positive integer $r$, concerning both non-vanishing and non-negativity.
In this article, the authors give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. They also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus were dealt in an earlier work of Choie, the first author and Kohnen.
We estimate the mean square of a short exponential sum involving Fourier coefficients of a cusp form with a linear twist, a smooth weight function, and a relatively short averaging interval.
Given a finite index subgroup of $SL_2(\mathbb Z)$ with modular curve defined over $\mathbb Q$, under the assumption that the space of weight $k$ ($ \ge 2$) cusp forms is $1$-dimensional, we show that a form in this space with Fourier coefficients in $\mathbb Q$ has bounded denominators if and only if it is a congruence modular form.
In this paper, we investigate the "angular changes" behavior of some subfamilies of Fourier coefficients of both integral and half-integral weight holomorphic cusp forms, thus one gets information about signs of the real an imaginary parts of these subfamilies. These give an extension of some recent results of Kohnen and his collaborators.
We prove that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square-free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which are also treated in this paper.
We treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms. As a consequence, we obtain an upper bound for correlation of three Hecke eigenvalues of holomorphic cusp forms $\sum_{H\leq h\leq 2H}W\big(\frac{h}{H}\big)\sum_{X\leq n\leq 2X}λ_{1}(n-h)λ_{2}(n)λ_{3}(n+h)$, which is nontrivial provided that $H\geq X^{2/3+\varepsilon}$. The result can be viewed as a cuspidal analogue of a recent result of Blomer on triple correlations of divisor functions.
In this paper, we discuss questions related to the oscillatory behavior and the equidistribution of signs for certain subfamilies of Fourier coefficients of integral weight newforms with a non-trivial nebentypus as well as Fourier coefficients of eigenforms of half-integral weight reachable by the Shimura correspondence.
We prove that a Siegel cusp form of degree 2 for the full modular group is determined by its set of Fourier coefficients a(S) with 4 det(S) ranging over odd squarefree integers. As a key step to our result, we also prove that a classical cusp form of half-integral weight and level 4N, with N odd and squarefree, is determined by its set of Fourier coefficients a(d) with d ranging over odd squarefree integers, a result that was previously known only for Hecke eigenforms.
The summatory function of $t_j(n^2)$ is estimated, where $H_j(s) = \sum_{n=1}^\infty t_j(n)n^{-s}$ is the Hecke series of a non-holomorphic cusp form. The analogous problem of holomorphic cusp forms is also treated.
We study the simultaneous sign change of Fourier coefficients of a pair of distinct normalized newforms of integral weight supported on primes power indices, we also prove some equidistribution results. Finally, we consider an analogous question for Fourier coefficients of a pair of half-integral weight Hecke eigenforms.
We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level. In contrast to previous work on special cases of this problem we use elementary methods for the computation of Petersson products and avoid using the Rankin $L$-function.
In this work we prove a prime number type theorem involving the normalised Fourier coefficients of holomorphic and Maass cusp forms, using the classical circle method. A key point is in a recent paper of Fouvry and Ganguly, based on Hoffstein-Ramakrishnan's result about the non-existence of the Siegel zeros for $GL(2)$ $L$-functions, which allows us to improve preceding estimates.
We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group $\mathrm{SL}_3(\mathbb Z)$. When the underlying form is self-dual, we show that there are $\gg_\varepsilon X^{5/6-\varepsilon}$ sign changes among the coefficients $\{A(m,1)\}_{m\leq X}$ and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients $A(m,m)$ for generic $\mathrm{GL}_3$ cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of $\…
Let $\bf f$ be a primitive Hilbert cusp form of weight $k$ and level $\mathfrak{n}$ with Fourier coefficients $c_{\bf f}(\mathfrak{m})$. We prove a non-trivial upper bound for almost all Fourier coefficients $c_{\bf f}(\mathfrak{m})$ of $\bf f$. This generalizes the bounds obtained by Luca, Radziwiłł and Shparlinski. We also prove the existence of infinitely many integral ideals $\mathfrak{m}$ for which the Fourier coefficients $c_{\bf f}(\mathfrak{m})$ have the improved upper bound and further we obtain a refinement of these integral ideals in terms of prime powers. In particular, this enable…
This paper develops a general theory of the Fourier-Jacobi expansion of cusp forms on the real symplectic group of degree two including generic cusp forms. An explicit description of such expansion is available for cusp forms generating discrete series representations, generalized principal series representations induced from the Jacobi parabolic subgroup and principal series representations, where we note that the latter two cases include non-spherical representations. As the archimedean local ingredients we need Fourier-Jacobi type spherical functions and Whittaker functions, whose explicit …
This paper treats the problem of determining conditions for the Fourier coefficients of a Maass-Hecke newform at cusps other than infinity to be multiplicative. To be precise, the Fourier coefficients are defined using a choice of matrix in SL(2, Z) which maps infinity to the cusp in question. Let c and d be the entries in the bottom row of this matrix, and let N be the level. In earlier work with Dorian Goldfeld and Min Lee, we proved that the coefficients will be multiplicative whenever N divides 2cd. This paper proves that they will not be multiplicative unless N divides 576cd. Further, if …
We show that if a modular cuspidal eigenform $f$ of weight $2k$ is $2$-adically close to an elliptic curve $E/\mathbb{Q}$, which has a cyclic rational $4$-isogeny, then $n$-th Fourier coefficient of $f$ is non-zero in the short interval $(X, X + cX^{\frac{1}{4}})$ for all $X \gg 0$ and for some $c > 0$. We use this fact to produce non-CM cuspidal eigenforms $f$ of level $N>1$ and weight $k > 2$ such that $i_f(n) \ll n^{\frac{1}{4}}$ for all $n \gg 0$.
We prove a non-vanishing result of modular L-values with quadratic twists, where the quadratic discriminants are in a short interval. Using this fact and Waldspurger's theorem, we improve the results of Balog-Ono[The chebotarev density theorem in short intervals and some questions of Serre, Journal of number theory. 91(2):356-371(2001)] on the non-vanishing of Fourier coefficients of half-integral weight eigenform.
Let $a(n)$ be the Fourier coefficients of a holomorphic cusp form of weight $κ=2n\geqslant12$ for the full modular group and $A(x)=\sum\limits_{n\leqslant x}a(n)$. In this paper, we establish an asymptotic formula of the fourth power moment of $A(x)$ and prove that \begin{equation*} \int_1^TA^4(x)\mathrm{d}x=\frac{3}{64κπ^4}s_{4;2}(\tilde{a}) T^{2κ}+O\big(T^{2κ-δ_4+\varepsilon}\big) \end{equation*} with $δ_4=1/8$, which improves the previous result.
In a previous paper we formulated an analogue of the Ichino-Ikeda conjectures for Whittaker-Fourier coefficients of cusp forms on quasi-split groups, as well as the metaplectic group of arbitrary rank. In this paper we reduce the conjecture for the metaplectic group to a local conjectural identity. We motivate this conjecture by giving a heuristic argument for the case $\widetilde{SL}_2$. In a subsequent paper we will prove the local identity in the $p$-adic case.
In this paper, we investigate Fourier expansions of meromorphic modular forms. Over the years, a number of special cases of meromorphic modular forms were shown to have Fourier expansions closely resembling the expansion of the reciprocal of the weight $6$ Eisenstein series which was computed by Hardy and Ramanujan. By investigating meromorphic modular forms within a larger space of so-called polar harmonic Maass forms, we prove in this paper that all negative-weight meromorphic modular forms (and furthermore all quasi-meromorpic modular forms) have Fourier expansions of this type, granted tha…
In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class $t\mathbb{Z}^2$, we investigate the sign changes in the $tp^2$-th coefficient as $p$ runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. We show that sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discrim…
This was originally an appendix to our paper `Fourier expansions at cusps' [arXiv:1807.00391]. The purpose of this note is to give a proof of a theorem of Shimura on the action of $\mathrm{Aut}(\mathbb{C})$ on modular forms for $Γ(N)$ from the perspective of algebraic modular forms. As the theorem is well-known, we do not intend to publish this note but want to keep it available as a preprint.
We prove that the central values of additive twists of a cuspidal $L$-function define a quantum modular form in the sense of Zagier, generalizing recent results of Bettin and Drappeau. From this we deduce a reciprocity law for the twisted first moment of multiplicative twists of cuspidal $L$-functions, similar to reciprocity laws discovered by Conrey for the twisted second moment of Dirichlet $L$-functions. Furthermore we give an interpretation of quantum modularity at infinity for additive twists of $L$-functions of weight 2 cusp forms in terms of the corresponding functional equations.
From a result of Waldspurger, it is known that the normalized Fourier coefficients $a(m)$ of a half-integral weight holomorphic cusp eigenform $\f$ are, up to a finite set of factors, one of $\pm \sqrt{L(1/2, f, χ_m)}$ when $m$ is square-free and $f$ is the integral weight cusp form related to $\f$ by the Shimura correspondence. In this paper we address a question posed by Kohnen: which square root is $a(m)$? In particular, if we look at the set of $a(m)$ with $m$ square-free, do these Fourier coefficients change sign infinitely often? By partially analytically continuing a related Dirichlet s…
Associated to an (adelic) Hilbert modular form is a sequence of `Fourier coefficients' which uniquely determine the form. In this paper we characterize Hilbert modular cusp forms by the size of their Fourier coefficients. This answers in the affirmative a question posed by Winfried Kohnen.
We consider a special theta lift $θ(f)$ from cuspidal Siegel modular forms $f$ on $\mathrm{Sp}_4$ to "modular forms" $θ(f)$ on $\mathrm{SO}(4,4)$. This lift can be considered an analogue of the Saito-Kurokawa lift, where now the image of the lift is representations of $\mathrm{SO}(4,4)$ that are quaternionic at infinity. We relate the Fourier coefficients of $θ(f)$ to those of $f$, and in particular prove that $θ(f)$ is nonzero and has algebraic Fourier coefficients if $f$ does. Restricting the $θ(f)$ to $G_2 \subseteq \mathrm{SO}(4,4)$, we obtain cuspidal modular forms on $G_2$ of arbitrarily…
Let $f$ be a Hecke-Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $λ_f(Δ)=1/4+μ^2$ and let $λ_f(n)$ be its $n$-th normalized Fourier coefficient. It is proved that, uniformly in $α, β\in \mathbb{R}$, $$ \sum_{n \leq X}λ_f(n)e\left(αn^2+βn\right) \ll X^{7/8+\varepsilon}λ_f(Δ)^{1/2+\varepsilon}, $$ where the implied constant depends only on $\varepsilon$. We also consider the summation function of $λ_f(n)$ and under the Ramanujan conjecture we are able to prove $$ \sum_{n \leq X}λ_f(n)\ll X^{1/3+\varepsilon}λ_f(Δ)^{4/9+\varepsilon} $$ with the implied constant depending only …
In this article, we establish an average behaviour of the normalised Fourier coefficients of the Hecke eigenforms supported at the integers represented by any primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when the class number $h(D) = 1$. We also obtain a quantitative result for the number of sign changes of the sequence of the normalised Fourier coefficients $λ_{f}(n)$ of the Hecke eigenforms $f$ where $n$ is represented by any primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when the class number $h(D) = 1$ in…
We show, for levels of the form $N = p^a q^b N'$ with $N'$ squarefree, that in weights $k \geq 4$ every cusp form $f \in \mathcal{S}_k(N)$ is a linear combination of products of certain Eisenstein series of lower weight. In weight $k=2$ we show that the forms $f$ which can be obtained in this way are precisely those in the subspace generated by eigenforms $g$ with $L(g, 1) \neq 0$. As an application of such representations of modular forms we can calculate Fourier expansions of modular forms at arbitrary cusps and we give several examples of such expansions in the last section.
Let $\G\subset \mathrm{SL}_{2}(\R)$ be a cofinite Fuchsian subgroup, and let $i\infty$ be a cusp of $\G$. For $k\in\Z_{\geq 0}$, let $\Sk$ denote the complex vector space of cusp forms of weight-$k$, with respect to the Fuchsian subgroup $\G$. Let $f\in\Sk$ be a cusp form of weight-$k$, which is normalized, with respect to the Petersson inner-product on $\Sk$. For any $n\in\Z_{\geq 1}$, let $a_{n}$ denote the $n$-th Fourier coefficient of $f$ at $i\infty$. Then, for any $k\in\Z_{\geq 5}$, we show that \begin{align*} \big|a_{n}\big|=O_{f,\G}\big(n^{\frac{k-1}{2}+\frac{2}{k}}\big), \end{align*} …
We consider the Fourier expansion of a Hecke (resp.\ Hecke--Maaß) cusp form of general level $N$ at the various cusps of $Γ_{0}(N)\bs\Hb$. We explain how to compute these coefficients via the local theory of $p$-adic Whittaker functions and establish a classical Voronoï summation formula allowing an arbitrary additive twist. Our discussion has applications to bounding sums of Fourier coefficients and understanding the (generalised) Atkin--Lehner relations.
We prove that given any $ε> 0$ and a primitive adelic Hilbert cusp form $f$ of weight $k=(k_1,k_2,...,k_n) \in (2 \mathbb{Z})^n$ and full level, there exists an integral ideal $\mathfrak{m}$ with $N(\mathfrak{m}) \ll_ε Q_{f}^{9/20+ ε} $ such that the $\mathfrak{m}$-th Fourier coefficient of $C_{f} (\mathfrak{m})$ of $f$ is negative. Here $n$ is the degree of the associated number field, $N(\mathfrak{m})$ is the norm of integral ideal $\mathfrak{m}$ and $Q_{f}$ is the analytic conductor of $f$. In the case of arbitrary weights, we show that there is an integral ideal $\mathfrak{m}$ with $N(\mat…
We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of Saito-Kurokawa type, we prove an improved bound in the $k$-aspect for the smallest prime at which its Hecke eigenvalue is negative. Secondly, we show that there are infinitely many sign changes among the Hecke eigenvalues of $F$ at primes lying in an arithmetic progression. Third, we show that there are infinitely many positive as well as infinitely many negat…
We derive explicit formulas for some Kloosterman sums on $Γ_0(N)$, and for the Fourier coefficients of Eisenstein series attached to arbitrary cusps, around a general Atkin-Lehner cusp.
We obtain upper bounds for the fourth and higher moments of short exponential sums involving Fourier coefficients of holomorphic cusp forms twisted by rational additive twists with small denominators.
We formulate an analogue of the Ichino-Ikeda conjectures for the Whittaker-Fourier coefficients of automorphic forms on quasi-split reductive groups. This sharpens the conjectures of Sakellaridis-Venkatesh in the case at hand.
