Cusp Forms Of Weight Research Articles

Quantitative versions of the joint distributions of Hecke eigenvalues

In 2009, Omar and Mazhouda proved that as k→∞, {λf(p2):f∈Hk} and {λf(p3):f∈Hk} are equidistributed with respect to some measures respectively, where Hk is the set of all the normalized primitive holomorphic cusp forms of weight k for SL2(Z). In this paper, we obtain a quantitative version of Omar and Mazhouda's result. Moreover, we find out that {λf(p4):f∈Hk} and {λf(pr)−λf(pr−2):f∈Hkandr≥2} follow some nice distribution laws respectively as k→∞ and get quantitative versions of these distributions. In the context of Maass cusp forms, we establish analogous results.

The Eichler–Shimura cohomology theorem for Jacobi forms

Let \(\Gamma \) be a subgroup of finite index in \(\mathrm {SL}(2,\mathbb {Z})\). Eichler defined the first cohomology group of \(\Gamma \) with coefficients in a certain module of polynomials. Eichler and Shimura established that this group is isomorphic to the direct sum of two spaces of cusp forms on \(\Gamma \) with the same integral weight. These results were generalized by Knopp to cusp forms of real weights. In this paper, we define the first parabolic cohomology groups of Jacobi groups \(\Gamma ^{(1,j)}\) and prove that these are isomorphic to the spaces of (skew-holomorphic) Jacobi cusp forms of real weights. We also show that if \(j=1\) and the weights of Jacobi cusp forms are in \(\frac{1}{2}\mathbb {Z}-\mathbb {Z}\), then these isomorphisms can be written in terms of special values of partial L-functions of Jacobi cusp forms.

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 $.

Non-vanishing and sign changes of Hecke eigenvalues for half-integral weight cusp forms

In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form f with half-integral weight: –The first negative coefficient of the sequence {af(tn2)}n∈N,–The number of coefficients af(tn2) of same signs,–Non-vanishing of coefficients af(tn2) in short intervals and in arithmetic progressions, where af(n) is the nth Fourier coefficient of f and t is a square-free integer such that af(t)≠0.

Feynman integrals and critical modular $L$-values

Broadhurst conjectured that the Feynman integral associated to the polynomial corresponding to $t=1$ in the one-parameter family $(1+x_1+x_2+x_3)(1+x_1^{-1}+x_2^{-1}+x_3^{-1})-t$ is expressible in terms of $L(f,2),$ where $f$ is a cusp form of weight $3$ and level $15$. Bloch, Kerr and Vanhove have recently proved that the conjecture holds up to a rational factor. In this paper, we prove that Broadhurst's conjecture is true. Similar identities involving Feynman integrals associated to other polynomials in the same family are also established.

On direct integration for mirror curves of genus two and an almost meromorphic Siegel modular form

This work considers aspects of almost holomorphic and meromorphic Siegel modular forms from the perspective of physics and mathematics. The first part is concerned with (refined) topological string theory and the direct integration of the holomorphic anomaly equations. Here, a central object to compute higher genus amplitudes, which serve as the generating functions of various enumerative invariants, is provided by the so-called propagator. We derive a universal expression for the propagator for geometries that have mirror curves of genus two which is given by the derivative of the logarithm of Igusa's cusp form of weight 10. In addition, we illustrate our findings by solving the refined topological string on the resolutions of the three toric orbifolds of order three, five and six. In the second part, we give explicit expressions for lowering and raising operators on Siegel modular forms, and define almost holomorphic Siegel modular forms based on them. Extending the theory of Fourier-Jacobi expansions to almost holomorphic Siegel modular forms and building up on recent work by Pitale, Saha, and Schmidt, we can show that there is no analogue of the almost holomorphic elliptic second Eisenstein series. In the case of genus 2, we provide an almost meromorphic substitute for it. This, in particular, leads us to a generalization of Ramanujan's differential equation for the second Eisenstein series. The two parts are intertwined by the observation that the meromorphic analogue of the almost holomorphic second Eisenstein series coincides with the physical propagator. In addition, the generalized Ramanujan identities match precisely the physical consistency conditions that need to be imposed on the propagator.

Modular forms, de Rham cohomology and congruences

In this paper we show that Atkin and Swinnerton- Dyer type of congruences hold for weakly modular forms (modular forms that are permitted to have poles at cusps). Unlike the case of original congruences for cusp forms, these congruences are nontrivial even for congruence subgroups. On the way we provide an explicit interpretation of the de Rham cohomology groups associated to modular forms in terms of ``differentials of the second kind''. As an example, we consider the space of cusp forms of weight 3 on a certain genus zero quotient of Fermat curve $X^N+Y^N=Z^N$. We show that the Galois representation associated to this space is given by a Gr\ossencharacter of the cyclotomic field $\Q(\zeta_N)$. Moreover, for $N=5$ the space does not admit a ``$p$-adic Hecke eigenbasis'' for (non-ordinary) primes $p\equiv 2, 3 \pmod{; ; 5}; ; $, which provides a counterexample to Atkin and Swinnerton-Dyer's original speculation.

Lattices with many Borcherds products

We prove that there are only nitely many isometry classes of even lattices L of signature (2;n) for which the space of cusp forms of weight 1 + n=2 for the Weil representation of the discriminant group of L is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of L can be realized as the divisor of a Borcherds product. We obtain similar classication results in greater generality for nite quadratic modules.

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 $L_f(s)$.

Linear relations among half-integral weight Poincaré series

We construct an infinite family of half-integral weight Poincaré series coming from vector valued harmonic weak Maass forms, and investigate linear relations among the Poincaré series by applying a pairing between weakly holomorphic modular forms and harmonic weak Maass forms together with properties of Maass Poincaré series. We also show that the Poincaré series are characterized by the Fourier coefficients of half-integral weight cusp forms.

Ikeda's conjecture on the period of the Duke–Imamoḡlu–Ikeda lift

Let k and n be positive even integers. For a cus- pidal Hecke eigenform h in the Kohnen plus subspace of weight k − n/2 + 1/2 for 0(4), let In(h) be the Duke-Imamoglu-Ikeda lift of h in the space of cusp forms of weight k for Spn(Z), and f the primitive form of weight 2k − n for SL2(Z) corresponding to h under the Shimura correspondence. We then express the ratio hIn(h),In(h)i/hh,hi of the period of In(h) to that of h in terms of special values of certain L-functions of f. This proves the conjecture proposed by Ikeda concerning the period of the Duke- Imamoglu-Ikeda lift. One of the fascinating problems in the theory of modular forms is to find the relation between the periods (or the Petersson products) of cuspidal Hecke eigenforms which are related with each other through their L-functions. In particular, there are several important results concerning the relation between the period of a cuspidal Hecke eigen- form g for an elliptic modular group Γ � SL2(Z) and that of its lift bHere, by a lift of g we mean a cuspidal Hecke eigenform for another modular group Γ 0 (e.g. the symplectic group, the orthogonal group, the unitary group, etc.) whose certain L-function can be expressed in terms of certain L-functions related with g:Thus we propose the following problem: Problem A. Express the ratio hbbi=hg;gi e in terms of arithmetic invariants of g;for example, the special values of certain L-functions related with g for some integer e: For instance, Zagier (Zag77) solved Problem A for the Doi-Naganuma

Nonvanishing of Koecher–Maass series attached to Siegel cusp forms

We prove a nonvanishing result for Koecher–Maass series attached to Siegel cusp forms of weight k and degree n in certain strips on the complex plane. When n=2, we prove such a result for forms orthogonal to the space of the Saito–Kurokawa lifts ‘up to finitely many exceptions’, in bounded regions.

Note on Igusa's cusp form of weight 35

A congruence relation satisfied by Igusa's cusp form of weight 35 is presented. As a tool to confirm the congruence relation, a Sturm-type theorem for the case of odd-weight Siegel modular forms of degree 2 is included.

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.

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.

ON SPACES OF MODULAR FORMS OF EVEN WEIGHT

In the article we study the structure of space of cusp forms of an even weight and a level N with the help of cusp forms of minimal weight of the same level. The exact cutting is studied when each cusp form is a product of fixed function and a modular form of a smaller weight. Except the levels 17 and19 the cutting function is a multiplicative eta - product. In the common case the space f(z) M k-l(Γ0(N)) is not equal to the space Sk (Γ0(N)), the structure of additional space is competely studied. The result is depended on the value of the level modulo 12. Dimensions of spaces are calculated by the Cohen - Oesterle formula, the orders in cusps are calculated by the Biagioli formula.

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.

Construction of vector-valued modular integrals and vector-valued mock modular forms

It is known that, given a vector-valued modular form of negative weight, its Fourier coefficients can be calculated based on the principal part of the form. In this paper we start with an arbitrary principal part and complete the Fourier expansion using the calculation. We show that the so-obtained function is a vector-valued modular integral of negative weight on the full modular group. Next, we construct the supplementary function associated to a vector-valued modular cusp form of positive weight. The constructions are inspired by the construction of Eichler integrals by Knopp. We conclude with a comparison of these forms and their integrals to vector-valued weak harmonic Maass forms.

Automorphic products of singular weight for simple lattices

We classify the simple even lattices of square free level and signature (2,n) for n > 3. A lattice is called simple if the space of cusp forms of weight 1+n/2 for the dual Weil representation of the lattice is trivial. For a simple lattice every formal principal part obeying obvious conditions is the principal part of a vector valued modular form. Using this, we determine all holomorphic Borcherds products of singular weight (arising from vector valued modular forms with non-negative principal part) for the simple lattices. We construct the corresponding vector valued modular forms by eta products and compute expansions of the automorphic products at different cusps.