Siegel Cusp Forms Of Genus Research Articles

On sign changes of eigenvalues of Siegel cusp forms of genus 2 in prime powers

We prove that under mild assumptions there are infinitely many sign changes in the sequence of eigenvalues of a genus $2$ Siegel cuspidal eigenform supported on $p^{jn}$ for a fixed $j \geq 1$.

Rankin-Cohen Brackets on Hilbert Modular forms and Special values of certain Dirichlet series

Given a fixed Siegel cusp form of genus two, we consider a family of linear maps between the spaces of Siegel cusp forms of genus two by using the Rankin–Cohen brackets and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Siegel cusp forms of genus two constructed using this method involve special values of certain Dirichlet series of Rankin type associated to Siegel cusp forms. This is a generalisation of the work due to Kohnen (Math Z 207:657–660, 1991) and Herrero (Ramanujan J 36:529–536, 2015) in the case of elliptic modular forms to the case of Siegel cusp forms which is also considered earlier by Lee (Complex Var Theory Appl 31:97–103, 1996) for a special case.

Rankin–Cohen brackets on Siegel modular forms and special values of certain Dirichlet series

Given a fixed Siegel cusp form of genus two, we consider a family of linear maps between the spaces of Siegel cusp forms of genus two by using the Rankin–Cohen brackets and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Siegel cusp forms of genus two constructed using this method involve special values of certain Dirichlet series of Rankin type associated to Siegel cusp forms. This is a generalisation of the work due to Kohnen (Math Z 207:657–660, 1991) and Herrero (Ramanujan J 36:529–536, 2015) in the case of elliptic modular forms to the case of Siegel cusp forms which is also considered earlier by Lee (Complex Var Theory Appl 31:97–103, 1996) for a special case.

Non-vanishing and sign changes of Hecke eigenvalues for Siegel cusp forms of genus two

In this paper, we show that half of non-zero coefficients of the spinor zeta function of a Siegel cusp form of genus 2 are positive and half are negative. We also prove results concerning the non-vanishing in short intervals and strong cancellation among the coefficients evaluated at powers of a fixed prime. Our results rest on a Serre’s type density result established by Kowalski and Saha in the Appendix.

Local spectral equidistribution for Siegel modular forms and applications

AbstractWe study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson’s formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.

Classical interpretation of the Ramanujan conjecture for Siegel cusp forms of genus n

We obtain a classical interpretation of the representation theoretic statement of the Generalized Ramanujan Conjecture for Siegel cusp forms of genus n in terms of estimates on Hecke eigenvalues.

A note on the Rankin-Selberg method for Siegel cusp forms of genus 2

A note on the Rankin-Selberg method for Siegel cusp forms of genus 2

Fourier coefficients and Hecke eigenvalues

Abstract.We will give certain asymptotic relations between p-eigenvalues and certain Fourier coefficients of Siegel cusp forms of genus g. In particular, it will turn out that potential strong bounds for the Fourier coefficients will imply potential strong bounds for the eigenvalues.

On the average growth of Fourier coefficients of Siegel cusp forms of genus 2

in fact, the left-hand side is the N -th Fourier coefficients of F (( τ 0 0 τ )) (τ ∈ H = upper half-plane) which is a cusp form of weight 2k on SL2(Z). The proof of (2) is easy if one exploits the same arguments as in [3, 4] used there to derive the bound (3) a(T ) e,F (det(T ))k/2−13/36+e (e > 0) for the individual Fourier coefficients of F. Note, however, that by inserting (3) into the left-hand side of (2) and using det(T ) (tr(T )), # { T ∈ Q(2,2)|T > 0 half-integral, tr(T ) = N } N,

Fourier coefficients of Siegel cusp forms of genus n

Let F(Z) be a cusp form of integral weight k relative to the Siegel modular group Spn(Z) and let f(N) be its Fourier coefficient with index N. Making use of Rankin's convolution, one proves the estimate $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2} - \tfrac{1}{2}\delta (n)} ),$$ (1) where $$\delta (n) = \frac{{n + 1}}{{\left( {n + 1} \right)\left( {2n + \tfrac{{1 + ( - 1)^n }}{2}} \right) + 1}}.$$ Previously, for n ≥ 2 one has known Raghavan's estimate $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2}} )$$ In the case n=2, Kitaoka has obtained a result, sharper than (1), namely: $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2} - \tfrac{1}{4} + \varepsilon } ).$$ (2) At the end of the paper one investigates specially the case n=2. It is shown that in some cases the result (2) can be improved to, apparently, unimprovable estimates if one assumes some analogues of the Petersson conjecture. These results lead to a conjecture regarding the optimal estimates of f(N), n=2.

Construction of Siegel cusp forms of genus two in terms of cusp forms of genus one

One proves that the convolution of a modular form of semiintegral weight and genus one with the theta-series of an indefinite quadratic form of signature (3.2) is a Siegel modular form of genus two of an even weight. One establishes the relationship between the zeta-functions of these modular forms.

A property of the Fourier coefficients of Siegel cusp forms of genus two

Let F be a Siegel cusp modular form of genus two and of even weight k > 0. One proves that if the zeta-function ZF(s) of the form F has a pole at the point s=k, then the Fourier coefficients a(N) of this form depend only on the determinant and the divisor of the matrix N.