Hecke Eigenforms Research Articles

Interlacing of zeros of odd period polynomials

The work of Conrey, Farmer and Imamoglu shows that all nontrivial zeros of the odd period polynomial of a Hecke eigenform of level one lie on the unit circle. We further show that nontrivial zeros of odd period polynomials of Hecke eigenforms of different weights share certain interlacing property.

A Selberg identity for the Shimura lift

AbstractWe first prove a Selberg‐type identity for the Shimura lift of the Rankin–Cohen bracket of a normalized Hecke eigenform and the theta function. We then discuss its relationship with the nonvanishing of central values of ‐functions associated to Hecke eigenforms.

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 Gross–Prasad conjecture with its refinement for (SO(5), SO(2)) and the generalized Böcherer conjecture

We investigate the Gross–Prasad conjecture and its refinement for the Bessel periods in the case of $(\mathrm {SO}(5), \mathrm {SO}(2))$. In particular, by combining several theta correspondences, we prove the Ichino–Ikeda-type formula for any tempered irreducible cuspidal automorphic representation. As a corollary of our formula, we prove an explicit formula relating certain weighted averages of Fourier coefficients of holomorphic Siegel cusp forms of degree two, which are Hecke eigenforms, to central special values of $L$-functions. The formula is regarded as a natural generalization of the Böcherer conjecture to the non-trivial toroidal character case.

Mass Equidistribution for Saito-Kurokawa Lifts

Let F be a holomorphic cuspidal Hecke eigenform for \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathrm{Sp}_{4}({\\mathbb{Z}})$\\end{document} of weight k that is a Saito–Kurokawa lift. Assuming the Generalized Riemann Hypothesis (GRH), we prove that the mass of F equidistributes on the Siegel modular variety as k⟶∞. As a corollary, we show under GRH that the zero divisors of Saito–Kurokawa lifts equidistribute as their weights tend to infinity.

Asymptotic of Fourier coefficients of Hecke eigenforms at integers represented by a binary quadratic form of a fixed discriminant

Asymptotic of Fourier coefficients of Hecke eigenforms at integers represented by a binary quadratic form of a fixed discriminant

Sub‐Weyl type range for twisted GL(2)$\mathrm{GL(2)}$ short sums

AbstractLet where is a bump function supported on the interval [1,2] and satisfying and ’s are the normalized Fourier coefficients of Hecke eigenform, and is a primitive character of conductor . In this article, we prove a sub‐Weyl type cancellation range provided that . Note that the above bound is nontrivial if .

A note on Fourier coefficients of Hecke eigenforms in short intervals

A note on Fourier coefficients of Hecke eigenforms in short intervals

Twisted Kronecker series and periods of modular forms on Γ0(N)

We introduce an infinite family of Kronecker series twisted by characters. As an application, we give a closed formula for the sum of all Hecke eigenforms on Γ0(N) multiplied by their twisted period polynomials in terms of the product of those twisted Kronecker series, when N is square free. This extends an identity of Zagier among period polynomials, Hecke eigenforms and a quotient of Jacobi theta series.

Oscillations of Fourier coefficients of product of L-functions at integers in a sparse set

Let [Formula: see text] be a normalized Hecke eigenform of weight [Formula: see text] for the full modular group [Formula: see text]. In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg [Formula: see text]-functions [Formula: see text], supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant [Formula: see text] We improve previous results in the case when the reduced form is given by [Formula: see text]

On the density hypothesis for $L$-functions associated with holomorphic cusp forms

We study the range of validity of the density hypothesis for the zeros of L -functions associated with cusp Hecke eigenforms f of even integral weight, and prove that N_{f}(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon} holds for \sigma \geq 1407/1601 . This improves upon a result of Ivić, who had previously shown the zero-density estimate in the narrower range \sigma\geq 53/60 . Our result relies on an improvement of the large value estimates for Dirichlet polynomials based on mixed moment estimates for the Riemann zeta function. The main ingredients in our proof are the Halász–Montgomery inequality, Ivić’s mixed moment bounds for the zeta function, Huxley’s subdivision argument, Bourgain’s dichotomy approach, and Heath-Brown’s bound for double zeta sums.

Zeros of modular forms and Faber polynomials

AbstractWe study the zeros of cusp forms of large weight for the modular group, which have a very large order of vanishing at infinity, so that they have a fixed number of finite zeros in the fundamental domain. We show that for large weight the zeros of these forms cluster near vertical lines, with the zeros of a weight form lying at height approximately . This is in contrast to previously known cases, such as Eisenstein series, where the zeros lie on the circular part of the boundary of the fundamental domain, or the case of cuspidal Hecke eigenforms where the zeros are uniformly distributed in the fundamental domain. Our method uses the Faber polynomials. We show that for our class of cusp forms, the associated Faber polynomials, suitably renormalized, converge to the truncated exponential polynomial of degree .

Extreme values of L$L$‐functions of newforms

AbstractIn 2008, Soundararajan showed that there exists a normalized Hecke eigenform of weight and level one such that for sufficiently large . In this note, we show that for any and for all sufficiently large , the number of normalized Hecke eigenforms of weight and level one for which is . For an odd fundamental discriminant , let be the set of all cuspidal normalized Hecke eigenforms of weight and level dividing . When the real primitive Dirichlet character satisfies , we investigate the number of for which takes extremal values.

On certain partial sums involving squares of Hecke eigenvalues

Let $a(n)$ be the $n$th Fourier coefficient of a cuspidal Hecke eigenform of even integral weight $k\ge 2$ and trivial character that is a normalized new form for some level $N$. We show that the partial sums$$H_n=\sum_{m=1}^n a(m)^2/m^k$$are not integral for $n\ge n_0$.

Weighted central limit theorems for central values of $L$-functions

We establish a central limit theorem for the central values of Dirichlet L -functions with respect to a weighted measure on the set of primitive characters modulo q as q \rightarrow \infty . Under the Generalized Riemann Hypothesis (GRH), we also prove a weighted central limit theorem for the joint distribution of the central L -values corresponding to twists of two distinct primitive Hecke eigenforms. As applications, we obtain (under GRH) positive proportions of twists for which the central L -values simultaneously grow or shrink with q as well as a positive proportion of twists for which linear combinations of the central L -values are non-zero.

Moments of ideal class counting functions

We consider the counting function of ideals in a given ideal class of a number field of degree d. This describes, at least conjecturally, the Fourier coefficients of an automorphic form on GL(d), typically not a Hecke eigenform and not cuspidal. We compute its moments, and also investigate the moments of the corresponding cuspidal projection.

On the Simultaneous Sign Changes of Coefficients of Rankin–Selberg L-Functions over a Certain Integral Binary Quadratic Form

In this paper, we consider the simultaneous sign changes of coefficients of Rankin–Selberg L-functions associated to two distinct Hecke eigenforms supported at positive integers represented by some certain primitive reduced integral binary quadratic form with negative discriminant D. We provide a quantitative result for the number of sign changes of such sequence in the interval (x, 2x] for sufficiently large x.

Harder's conjecture I

Let $f$ be a primitive form with respect to $\mathrm{SL}_2(\mathbb{Z})$. Then, we propose a conjecture on the congruence between the Klingen–Eisenstein lift of the Duke–Imamoglu–Ikeda lift of $f$ and a certain lift of a vector valued Hecke eigenform with respect to $\mathrm{Sp}_2(\mathbb{Z})$. This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.

On the Iwasawa invariants of BDP Selmer groups and BDPp-adicL-functions

AbstractLetpbe an odd prime. Letf1{f_{1}}andf2{f_{2}}be weight 2 cuspidal Hecke eigenforms with isomorphic residual Galois representations atp. Greenberg–Vatsal and Emerton–Pollack–Weston showed that ifpis a good ordinary prime for the two forms, the Iwasawa invariants of theirp-primary Selmer groups andp-adicL-functions over the cyclotomicℤp{\mathbb{Z}_{p}}-extension ofℚ{\mathbb{Q}}are closely related. The goal of this article is to generalize these results to the anticyclotomic setting. More precisely, letKbe an imaginary quadratic field wherepsplits. Suppose that the generalized Heegner hypothesis holds with respect to both(f1,K){(f_{1},K)}and(f2,K){(f_{2},K)}. We study relations between the Iwasawa invariants of the BDP Selmer groups and the BDPp-adicL-functions off1{f_{1}}andf2{f_{2}}.

On average theta functions of certain quadratic forms as sums of Eisenstein series

Abstract Let Q Q be an integral positive definite quadratic form of level N N in 2 k 2k variables. Further, we assume that ( − 1 ) k N {\left(-1)}^{k}N is a fundamental discriminant. We express the average theta function of Q Q as an explicit sum of Eisenstein series, which are Hecke eigenforms by applying Atkin-Lehner involution.