Holomorphic Cusp Forms Research Articles

On special Bessel periods and the Gross–Prasad conjecture for $$\mathrm {SO}(2n+1) \times \mathrm {SO}(2)$$

In this paper we investigate one instance of the global Gross–Prasad conjecture. Our main result proves that if an irreducible cuspidal automorphic representation \(\pi \) of an odd dimensional special orthogonal group, whose local component \(\pi _{w}\) at some finite place w is generic, admits the special Bessel model corresponding to a quadratic extension E over a base field F, then the central L-value \(L(1/2,\pi )L(1/2,\pi \times \chi _E)\) does not vanish. Here \(\chi _E\) denotes the quadratic character of \(\mathbb A_F^\times \) corresponding to E. As an application, we obtain the equivalence between the non-vanishing of the special Bessel period and that of the corresponding central L-value when \(\pi \) is associated to a full modular holomorphic Siegel cusp form of degree two, which is a Hecke eigenform, and E is an imaginary quadratic extension of \(\mathbb Q\).

$L$-series for Vector-valued Modular Forms

Motivated by the recent works of Bringmann, Guerzhoy, Kent, and Ono [4] and Bringmann, Fricke, and Kent [3], we introduce $L$-series for vector-valued weakly holomorphic cusp forms, and mock modular period polynomials for vector-valued harmonic weak Maass forms. In particular, we will discuss an integral representation of this new $L$-series and the limiting behavior of special values. Moreover, we also give relations between mock modular periods and $L$-series for vector-valued harmonic weak Maass forms.

On the sparsity of positive-definite automorphic forms within a family

Baker and Montgomery have proved that almost all Fekete polynomials with respect to a certain ordering have at least one zero on the interval (0, 1). It is also known that a Fekete polynomial has no zeros on the interval (0, 1) if and only if the corresponding automorphic form is positive-definite. Generalizing their result, we formulate an axiomatic result about sets of automorphic forms π satisfying certain averages when suitably ordered to ensure that almost all p’s are not positive-definite within such sets. We then apply the result to various families, including the family of holomorphic cusp forms, the family of Hilbert class characters of imaginary quadratic fields, and the family of elliptic curves. In the appendix, we apply the result to general families of automorphic forms defined by Sarnak, Shin, and Templer.

Lower bound for the higher moment of symmetric square $L$-functions

Let $\mathcal {S}_k(N)$ be the space of holomorphic cusp forms of weight $k$, level $N$ and let $\mathcal {B}_k(N)$ be an orthogonal basis of $\mathcal {S}_k(N)$ consisting of newforms. Let $L(s, \textup {sym}^2 f)$ be the symmetric square $L$-function of $f\in \mathcal {B}_k(N)$. In this paper, the lower bound of the higher moment of $L(1/2,\textup {sym}^2 f)$ is established, i.e., for any even positive number $r$, \[ \sum _{f\in \mathcal {B}_k(N)}\omega _f^{-1}L\bigg (\frac 12, \textup {sym}^2 f\bigg )^r \gg (\log N)^{{r(r+1)}/{2}} \] holds for $N\rightarrow \infty $.

Zeros of a family of approximations of Hecke L-functions associated with cusp forms

We consider a family of approximations of a Hecke L-function \(L_f(s)\) attached to a holomorphic cusp form f of positive integral weight k with respect to the full modular group. These families are of the form $$\begin{aligned} L_f(X;s):=\sum _{n\le X}\frac{a(n)}{n^s}+(-1)^{k/2}(2\pi )^{-(1-2s)}\frac{\Gamma \left( \tfrac{k+1}{2}-s\right) }{\Gamma \left( \tfrac{k-1}{2}+s\right) }\sum _{n\le X}\frac{a(n)}{n^{1-s}}, \end{aligned}$$ where \(s=\sigma +it\) is a complex variable and a(n) is a normalized Fourier coefficient of f. From an approximate functional equation, one sees that \(L_f(X;s)\) is a good approximation to \(L_f(s)\) when \(X=t/2\pi \). We obtain vertical strips where most of the zeros of \( L_f(X;s) \) lie. We study the distribution of zeros of \(L_f(X;s)\) when X is independent of t. For \(X=1\) and 2, we prove that all the complex zeros of \(L_f(X;s)\) lie on the critical line \(\sigma =1/2\). We also show that as \(T\rightarrow \infty \) and \( X=T^{o(1)} \), \(100\,\%\) of the complex zeros of \( L_f(X;s) \) up to height T lie on the critical line. Here by \(100\,\%\) we mean that the ratio between the number of simple zeros on the critical line and the total number of zeros up to height T approaches 1 as \(T\rightarrow \infty \).

THE SHIFTED CONVOLUTION OF DIVISOR FUNCTIONS

We prove an asymptotic formula for the shifted convolution of the divisor functions $d_3(n)$ and $d(n)$, which is uniform in the shift parameter and which has a power-saving error term. The method is also applied to give analogous estimates for the shifted convolution of $d_3(n)$ and Fourier coefficents of holomorphic cusp forms. These asymptotics improve previous results obtained by several different authors.

Cusp forms on the exceptional group of type

Let $\mathbf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let ${\rm\Gamma}=\mathbf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\geqslant 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to ${\rm\Gamma}$ from a Hecke cusp form in $S_{2k-8}(\text{SL}_{2}(\mathbb{Z}))$. We follow Ikeda’s idea of using Siegel’s Eisenstein series, their Fourier–Jacobi expansions, and the compatible family of Eisenstein series.

Approximate functional equation and mean value formula for the derivatives of $L$-functions attached to cusp forms

Let $f$ be a holomorphic cusp form of weight $k$ with respect to the full modular group $SL_2(\mathbb{Z})$. We suppose that $f$ is a normalized Hecke eigenform. Let $L_f(s)$ be the $L$-function attached to the form $f$. Good gave the approximate functional equation and mean square formula of $L_f(s)$. In this paper, we shall generalize these formulas for the derivatives of $L_f(s)$.

Higher Moments of Fourier Coefficients of Cusp Forms

AbstractLet Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(z, ℤ). Let be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms , and h(z) ∊ Sk3 (Γ), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as twisted by the arithmetic function λf(n).

English

Let $S_k$ be the space of holomorphic cusp forms of weight $k$ with respect to $SL_2(\mathbb{Z})$. Let $f \in S_k$ be a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper we consider the distribution of zeros of $L_f(s)$ in the strip $\sigma \leq \Re s \leq 1$ for fixed $\sigma>1/2$ with respect to the imaginary part. We study estimates of \[ N_f(\sigma,T) = #\{\rho\in\mathbb{C} \mid L_f(\rho)=0, leq \Re\rho \leq 1, 0 \leq \Im\rho \leq T} \] for $1/2 \leq \sigma \leq1$ and large $T>0$. Using the methods of Karatsuba and Voronin we shall give another proof for Ivic's method.

Gaps between zeros of [formula omitted]L-functions

Let L(s,f) be an L-function associated to a primitive (holomorphic or Maass) cusp form f on GL(2) over Q. Combining mean-value estimates of Montgomery and Vaughan with a method of Ramachandra, we prove a formula for the mixed second moments of derivatives of L(1/2+it,f) and, via a method of Hall, use it to show that there are infinitely many gaps between consecutive zeros of L(s,f) along the critical line that are at least 3=1.732… times the average spacing. Using general pair correlation results due to Murty and Perelli in conjunction with a technique of Montgomery, we also prove the existence of small gaps between zeros of any primitive L-function of the Selberg class. In particular, when f is a primitive holomorphic cusp form on GL(2) over Q, we prove that there are infinitely many gaps between consecutive zeros of L(s,f) along the critical line that are at most 0.823 times the average spacing.

Oscillations of coefficients of symmetric square L-functions over primes

Let L(s, sym2f) be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL(2,ℤ), with tf(n, 1) denoting the nth coefficient of the Dirichlet series for it. It is proved that, for N ⩾ 2 and any α ∈ ℝ, there exists an effective positive constant c such that Σn⩽NΛ(n)tf(n, 1)e(nα) ≪ N\(\exp ( - c\sqrt {\log N} )\), where Λ(n) is the von Mangoldt function, and the implied constant only depends on f. We also study the analogue of Vinogradov’s three primes theorem associated to the coefficients of Rankin-Selberg L-functions.

Arithmeticity of holomorphic cuspforms on Hermitian symmetric domains

We prove Galois equivariance of ratios of Petersson inner products of holomorphic cuspforms on symplectic, unitary, or Hermitian orthogonal groups. As a consequence, we show that the ratios of Petersson norms of such cuspforms with the same Hecke eigenvalues are algebraic. We also show that spaces of such cuspforms of sufficiently high fixed weight and level are spanned by theta series.

Moments of L-functions attached to the twist of modular form by Dirichlet characters

Let f(z) be a holomorphic cusp form of weight κ with respect to the full modular group SL2(ℤ). Let L(s, f) be the automorphic L-function associated with f(z) and χ be a Dirichlet character modulo q. In this paper, the authors prove that unconditionally for \(k = \tfrac{1} {n}\) with n ∈ ℕ, $$M_k \left( {q,f} \right) = \sum\limits_{\begin{array}{*{20}c} {\chi (\bmod q)} \\ {\chi \ne \chi _0 } \\ \end{array} } {\left| {L\left( {\frac{1} {2},f \otimes \chi } \right)} \right|^{2k} < < _k \varphi \left( q \right)(\log q)^{k^2 } ,}$$ and the result also holds for any real number 0 < k < 1 under the GRH for L(s, f ⊗ χ). The authors also prove that under the GRH for L(s, f ⊗ χ), $$M_k \left( {q,f} \right) > > _k \varphi (q)(log q)^{k^2 }$$ for any real number k > 0 and any large prime q.

Meromorphic analogues of modular forms generating the kernel of Shimura’s lift

We study the meromorphic modular forms defined as sums of -k (k>1) powers of integral quadratic polynomials with negative discriminant. These functions can be viewed as meromorphic analogues of the holomorphic modular forms defined in the same way with positive discriminant, first investigated by Zagier in connection with the Doi-Naganuma map and then by Kohnen and Zagier in connection with the Shimura-Shintani lifts. We compute the Fourier coefficients of these meromorphic modular forms and we show that they split into the sum of a meromorphic modular form with computable algebraic Fourier coefficients and a holomorphic cusp form.

Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions

We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments, and depend crucially on results on the independence of monodromy groups related to products of Kloosterman sums.

Exponential sums twisted by Fourier coefficients of automorphic cusp forms for SL(2, ℤ)

Let f be a holomorphic cusp form of weight k for SL(2, ℤ) with Fourier coefficients λf(n). We study the sum ∑n&gt;0λf(n)ϕ(n/X)e(αn), where [Formula: see text]. It is proved that the sum is rapidly decaying for α close to a rational number a/q where q2 &lt; X1-ε. The main techniques used in this paper include Dirichlet's rational approximation of real numbers, a Voronoi summation formula for SL(2, ℤ) and the asymptotic expansion for Bessel functions.

On effective determination of symmetric-square lifts, level aspect

Let F be the symmetric-square lift with Laplace eigenvalue λF(Δ) = 1 + 4μ2. Suppose that |μ| ≤ Λ. It is proved that F is uniquely determined by the central values of Rankin–Selberg L-functions L(s, F ⊗ h), where h runs over the set of holomorphic cusp forms of weight 10 and level q ≈ Λϱ+ϵ with [Formula: see text] for any ϵ &gt; 0. Here θ is the exponent towards the Ramanujan conjecture for GL2 Maass forms. We also prove an unconditional result in weight aspect.

On effective determination of cusp forms by L-values, level aspect

Let f be a normalized holomorphic Hecke newform of weight k≤K and level q≤Q with trivial nebentypus. We give the approximate formulas for the first moments of L(1/2,f⊗g) and L′(1/2,f⊗g), where g runs over Hl(N,χN), the normalized Hecke eigen-basis of holomorphic cusp forms of weight l and level N with nebentypus χN=(N⋅). As an application, we obtain some quantitative results that f is uniquely determined by the central values of L(s,f⊗g) and L′(s,f⊗g), where g runs over Hl(N,χN).

On effective determination of symmetric-square lifts

Abstract Let F be the symmetric-square lift with Laplace eigenvalue λ F (Δ) = 1+4µ2. Suppose that |µ| ≤ Λ. We show that F is uniquely determined by the central values of Rankin-Selberg L-functions L(s, F ⋇ h), where h runs over the set of holomorphic Hecke eigen cusp forms of weight κ ≡ 0 (mod 4) with κ≍ϱ+ɛ, t9 = max {4(1+4θ)/(1−18θ), 8(2−9θ)/3(1−18θ)} for any 0 ≤ θ &lt; 1/18 and any ∈ &gt; 0. Here θ is the exponent towards the Ramanujan conjecture for GL2 Maass forms.