Square-free Level Research Articles

An explicit trace formula of Jacquet–Zagier type for Hilbert modular forms

We give an exact formula of the average of adjoint L-functions of holomorphic Hilbert cusp forms with a fixed weight and a square-free level, which is a generalization of Zagier's formula known for the case of elliptic cusp forms on SL2(Z). As an application, we prove that the Satake parameters of Hilbert cusp forms with a fixed weight and with growing square-free levels are equidistributed in an ensemble constructed by values of the adjoint L-functions.

A generalized cubic moment and the Petersson formula for newforms

Using a cubic moment, we prove a Weyl-type subconvexity bound for the quadratic twists of a holomorphic newform of square-free level, trivial nebentypus, and arbitrary even weight. This generalizes work of Conrey and Iwaniec in that the newform that is being twisted may have arbitrary square-free level, and also that the quadratic character may have even conductor. One of the new tools developed in this paper is a more general Petersson formula for newforms of square-free level.

Refined dimensions of cusp forms, and equidistribution and bias of signs

We refine known dimension formulas for spaces of cusp forms of squarefree level, determining the dimension of subspaces generated by newforms both with prescribed global root numbers and with prescribed local signs of Atkin–Lehner operators. This yields precise results on the distribution of signs of global functional equations and sign patterns of Atkin–Lehner eigenvalues, refining and generalizing earlier results of Iwaniec, Luo and Sarnak. In particular, we exhibit a strict bias towards root number +1 and a phenomenon that sign patterns are biased in the weight but perfectly equidistributed in the level. Another consequence is lower bounds on the number of Galois orbits.

On sign changes of Fourier coefficients of modular forms

It is known that if a nonzero cusp form has real Fourier coefficients, then its Fourier coefficients change signs infinitely often. In this paper, we prove that there is a codimension one subspace in the space of holomorphic modular forms of square-free level such that all of its non-zero forms have similar sign change property.

The Fourier coefficients of the McKay\u2013Thompson series and the traces of CM values

The elliptic modular function j(tau ) enjoys many beautiful properties. Its Fourier coefficients are related to the Monster group, and its CM values generate abelian extensions over imaginary quadratic fields. Kaneko gave an arithmetic formula for the Fourier coefficients expressed in terms of the traces of the CM values. In this article, we are concerned with analogues of Kaneko’s result for the McKay–Thompson series of square-free level.

One-level density for holomorphic cusp forms of arbitrary level

In 2000 Iwaniec, Luo, and Sarnak proved for certain families of L-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Milićević, which is of use for other problems as well.

Siegel paramodular forms of weight 2 and squarefree level

We compute the space [Formula: see text] of weight 2 Siegel paramodular cusp forms of squarefree level [Formula: see text]. In conformance with the paramodular conjecture of Brumer and Kramer, the space is only the additive (Gritsenko) lift space of the Jacobi cusp form space [Formula: see text] except for [Formula: see text], when it further contains one nonlift newform. For these two values of [Formula: see text], the Hasse–Weil [Formula: see text]-Euler factors of a relevant abelian surface match the spin [Formula: see text]-Euler factors of the nonlift newform for the first two primes [Formula: see text].

Newforms in the Kohnen plus space

In this paper we want to define the Kohnen plus space for Hilbert modular forms with a odd square-free level and a quadratic character by a representation-theoretic way. We will show that in the classical case the one we defined is the same with the one given by Kohnen. Also, we will interpret the actions of the Hecke operators on the new forms in the plus space and give a characterization for the new forms using Hecke operators. All the results with respect to Hecke operators can are comparable with the integral weight case.

Non-vanishing of fundamental Fourier coefficients of paramodular forms

We prove that paramodular newforms of odd square-free level have infinitely many non-zero fundamental Fourier coefficients.

Trace formula for Jacobi forms of odd squarefree level

We study the traces of Hecke operators acting on the space of Jacobi forms and give the trace formula in the case of general odd squarefree level N by using results of Skoruppa and Zagier. As an application, we obtain some trace relations of them, and construct an isomorphism map from the space of all Jacobi cusp new forms of level N and index 1 having the eigenvalue +1 with respect to any Atkin–Lehner operator on the level to the space of all Jacobi cusp new forms of level 1 and index N whose eigenvalue with respect to any Atkin–Lehner operator on the index is equal to +1.

Dimension formulas of paramodular forms of squarefree level and comparison with inner twist

In this paper, we give an explicit dimension formula for the spaces of Siegel paramodular cusp forms of degree two of squarefree level. As an application, we propose a conjecture on symplectic group version of Eichler–Jacquet–Langlands type correspondence. It is a generalization of the previous conjecture of the first named author for prime levels published in 1985, where inner twists corresponding to binary quaternion hermitian forms over definite quaternion algebras were treated. Our present study contains also the case of indefinite quaternion algebras. Additionally, we give numerical examples of $L$ functions which support the conjecture. These comparisons of dimensions and examples give also evidence for conjecture on a certain precise lifting theory. This is related to the lifting theory from pairs of elliptic cusp forms initiated by Y. Ihara in 1964 in the case of compact twist, but no such construction is known in the case of non-split symplectic groups corresponding to quaternion hermitian groups over indefinite quaternion algebras and this is new in that sense.

Hecke eigenvalues and relations for Siegel Eisenstein series of arbitrary degree, level, and character

We evaluate the action of Hecke operators on Siegel Eisenstein series of arbitrary degree, level and character. For square-free level, we simultaneously diagonalize the space with respect to all the Hecke operators, computing the eigenvalues explicitly, and obtain a multiplicity-one result. For arbitrary level, we simultaneously diagonalize the space with respect to the Hecke operators attached to primes not dividing the level, again computing the eigenvalues explicitly.

The explicit Sato-Tate Conjecture and densities pertaining to Lehmer-type questions

Let $f(z)=\sum_{n=1}^\infty a(n)q^n\in S^{\text{new}}_ k (\Gamma_0(N))$ be a newform with squarefree level $N$ that does not have complex multiplication. For a prime $p$, define $\theta_p\in[0,\pi]$ to be the angle for which $a(p)=2p^{( k -1)/2}\cos \theta_p $. Let $I\subset[0,\pi]$ be a closed subinterval, and let $d\mu_{ST}=\frac{2}{\pi}\sin^2\theta d\theta$ be the Sato-Tate measure of $I$. Assuming that the symmetric power $L$-functions of $f$ satisfy certain analytic properties (all of which follow from Langlands functoriality and the Generalized Riemann Hypothesis), we prove that if $x$ is sufficiently large, then \[ \left|\#\{p\leq x:\theta_p\in I\} -\mu_{ST}(I)\int_2^x\frac{dt}{\log t}\right|\ll\frac{x^{3/4}\log(N k x)}{\log x} \] with an implied constant of $3.34$. By letting $I$ be a short interval centered at $\frac{\pi}{2}$ and counting the primes using a smooth cutoff, we compute a lower bound for the density of positive integers $n$ for which $a(n)\neq0$. In particular, if $\tau$ is the Ramanujan tau function, then under the aforementioned hypotheses, we prove that \[ \lim_{x\to\infty}\frac{\#\{n\leq x:\tau(n)\neq0\}}{x}>1-1.54\times10^{-13}. \] We also discuss the connection between the density of positive integers $n$ for which $a(n)\neq0$ and the number of representations of $n$ by certain positive-definite, integer-valued quadratic forms.

A note on the Fourier coefficients of a Cohen–Eisenstein series

We prove a formula for the coefficients of a weight [Formula: see text] Cohen–Eisenstein series of square-free level [Formula: see text]. This formula generalizes a result of Gross, and in particular, it proves a conjecture of Quattrini. Let [Formula: see text] be an odd prime number. For any elliptic curve [Formula: see text] defined over [Formula: see text] of rank zero and square-free conductor [Formula: see text], if [Formula: see text], under certain conditions on the Shafarevich–Tate group [Formula: see text], we show that [Formula: see text] divides [Formula: see text] if and only if [Formula: see text] divides the class number [Formula: see text] of [Formula: see text]

The special L-value of the winding quotient of square-free level

We give an explicit formula that expresses the algebraic part of the special L-value of the winding quotient of square-free level as a rational number, and interpret it in terms of the Birch–Swinnerton-Dyer conjecture.

Bounds for eigenforms on arithmetic hyperbolic 3-manifolds

On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we prove an upper bound for the sup-norm of Hecke-Maass cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the volume. By a novel combination of diophantine and geometric arguments in a noncommutative setting, we obtain bounds as strong as the best corresponding results on arithmetic surfaces.

Obstruction criteria for modular deformation problems

For a cuspidal newform [Formula: see text] of weight [Formula: see text] and a prime [Formula: see text] of [Formula: see text], the deformation problem for its associated mod [Formula: see text] Galois representation is unobstructed for all primes outside some finite set. Previous results gave an explicit bound on this finite set for [Formula: see text] of squarefree level; we modify this bound and remove the squarefree hypothesis. We also show that if the [Formula: see text]-adic deformation problem for [Formula: see text] is unobstructed, then [Formula: see text] is not congruent mod [Formula: see text] to a newform of lower level.

Certain aspects of holomorphic function theory on some genus-zero arithmetic groups

There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group $\operatorname{PSL}(2,\mathbb{Z})$, including the following statements: the ring of holomorphic modular forms is generated by the holomorphic Eisenstein series of weights four and six, denoted by $E_{4}$ and $E_{6}$; the smallest-weight cusp form $\unicode[STIX]{x1D6E5}$ has weight twelve and can be written as a polynomial in $E_{4}$ and $E_{6}$; and the Hauptmodul $j$ can be written as a multiple of $E_{4}^{3}$ divided by $\unicode[STIX]{x1D6E5}$. The goal of the present article is to seek generalizations of these results to some other genus-zero arithmetic groups $\unicode[STIX]{x1D6E4}_{0}(N)^{+}$ with square-free level $N$, which are related to ‘Monstrous moonshine conjectures’. Certain aspects of our results are generated from extensive computer analysis; as a result, many of the space-consuming results are made available on a publicly accessible web site. However, we do present in this article specific results for certain low-level groups.

The index of an Eisenstein ideal and multiplicity one

Mazur’s fundamental work on the Eisenstein ideal of prime level has a variety of arithmetic applications. In this article, we generalize some of his work to square-free level. More specifically, we attempt to compute the index of an Eisenstein ideal and the dimension of the \({\mathfrak {m}}\)-torsion of the modular Jacobian variety, where \({\mathfrak {m}}\) is an Eisenstein maximal ideal. In many cases, the dimension of the \({\mathfrak {m}}\)-torsion is 2, in other words, a multiplicity one theorem holds.

Saito–Kurokawa lifts of square-free level

Let f∈S2κ−2(Γ0(M)) be a Hecke eigenform with κ≥2 even and M≥1 and odd and square-free. In this paper we survey the construction of the Saito–Kurokawa lifting from the classical point of view. We also provide some arithmetic results on the Fourier coefficients of Saito–Kurokawa liftings. We then calculate the norm of the Saito–Kurokawa lift.