A study on twisted Koecher–Maass series of Siegel cusp forms via an integral kernel

The Ramanujan Journal - The authors wish to draw the attention to a mistake which appears in the proof of Proposition 3 of the above quoted paper [4].     

Nonvanishing of Siegel Poincaré series

We prove that, under suitable conditions, certain Siegel Poincaré series of exponential type of even integer weight and degree 2 do not vanish identically. We also find estimates for twisted Kloosterman sums and Salié sums in all generality.     

Shifted convolution sums related to Hecke–Maass forms

The Ramanujan Journal - Let $$\phi (z)$$ be a primitive Hecke–Maass cusp forms with Laplace eigenvalue $$\tfrac{1}{4}+t^2$$ . Denote by $$L(s, \mathrm{sym}^m\phi )$$ the m-th symmetric power...     

Nonvanishing of Rankin–Selberg L-functions for Hilbert modular forms

Let F be a totally real number field of degree n over $\mathbb{Q}$ with ring of integers $\mathcal{O}$ and narrow class number one. Let S _2k (Γ) be the vector space of cuspidal Hilbert modular forms of parallel weight 2k for $\varGamma=\mathrm{SL}_{2}(\mathcal{O})$ , and let B _2k be an orthogonal Hecke eigenbasis for this space. For any fixed Hecke eigenform f∈S _2k (Γ) and any ε>0, we prove that $$\# \biggl\{ g \in B_{2k}: L \biggl(f \times g, \frac{1}{2} \biggr) \ne0 \biggr\} \gg k^{n- \varepsilon}, $$ where L(f×g,s) is the Rankin–Selberg L–function of f and g.     

p-adic Siegel–Eisenstein series of degree two

We prove an explicit formula for Fourier coefficients of Siegel–Eisenstein series of degree two with a primitive character of any conductor. Moreover, we prove that there exists the p-adic analytic family which consists of Siegel–Eisenstein series of degree two and a certain p-adic limit of Siegel–Eisenstein series of degree two is actually a Siegel–Eisenstein series of degree two.     

Eisenstein series attached to lattices¶and modular forms on orthogonal groups

We study certain vector valued Eisenstein series on the metaplectic cover of SL₂(ℝ), which transform with the Weil representation associated with the discriminant group of an even lattice L. We find a closed formula for the Fourier coefficients in terms of Dirichlet L-series and representation numbers of L modulo “bad” primes. Such Eisenstein series naturally occur in the context of Borcherds' theory of automorphic products. We indicate some applications to modular forms on the orthogonal group of L with zeros on Heegner divisors. Received: 27 September 2001     

Projections of modular forms on Eisenstein series and its application to Siegel’s formula

The Ramanujan Journal - Let $$k ge 2$$ and N be positive integers and let $$chi $$ be a Dirichlet character modulo N. Let f(z) be a modular form in $$M_k(Gamma _0(N),chi )$$ . Then we have a...     

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.     

Linear relations among Poincaré series via harmonic weak Maass forms

We discuss the problem of the vanishing of Poincaré series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan??s mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincaré series.     

The first simultaneous sign change for Fourier coefficients of Hecke–Maass forms

The Ramanujan Journal - Let f and g be two Hecke–Maass cusp forms of weight zero for $$SL_2({mathbb {Z}})$$ with Laplacian eigenvalues $$frac{1}{4}+u^2$$ and $$frac{1}{4}+v^2$$ ,...     

Maass–Jacobi Poincaré series and Mathieu Moonshine

Mathieu moonshine attaches a weak Jacobi form of weight zero and index one to each conjugacy class of the largest sporadic simple group of Mathieu. We introduce a modification of this assignment, whereby weak Jacobi forms are replaced by semi-holomorphic Maass–Jacobi forms of weight one and index two. We prove the convergence of some Maass–Jacobi Poincaré series of weight one, and then use these to characterize the semi-holomorphic Maass–Jacobi forms arising from the largest Mathieu group.     

On the Fourier coefficients of Siegel–Eisenstein series of degree 2 for odd level

The Ramanujan Journal - We calculate the Fourier coefficients of the Siegel–Eisenstein series of degree 2, level $$p^n$$ with trivial character. The main result is to compute the Siegel...     

Derivative of power series attached to Γ-transform of p-adic measures

We extend the result of Anglès (2007) [1], namely for the Iwasawa power series . For the derivative , a numerical polynomial Q on Zp, and a prime π in over p, we show that if and only if i.e. for all x∈Zp. This result comes from a similar assertion for the power series attached to the Γ-transform of a p-adic measure which is related to a certain rational function in .     

Oscillations of coefficients of primitive cusp forms over some special sequences

Let λ _f (n) be the Fourier coefficient of a Hecke primitive eigencuspform f(z) of even integral weight k for the full modular group. In this paper we study the cancelation of the function λ _f (n) over some special sequences. This work is supported by the National Natural Science Foundation of China (Grant No. 10701048).     

Explicit formulas for the twisted Koecher–Maaß series of the Duke–Imamoglu–Ikeda lift and their applications

We give an explicit formula for the twisted Koecher–Maaß series of the Duke–Imamoglu–Ikeda lift. As an application we prove a certain algebraicity result for the values of twisted Rankin–Selberg series at integers of half-integral weight modular forms, which was not treated by Shimura (J Math Soc Japan 33:649–672, 1981).     

Generators of Jacobi forms are Poincaré series

The ring of Jacobi forms of even weights is generated by the weak Jacobi forms \(\phi _{-2,1}\) and \(\phi _{0,1}\). Bringmann and the first author expressed \(\phi _{-2,1}\) as a specialization of a Maass–Jacobi–Poincaré series. In this paper, we extend the domain of absolute convergence of Maass–Jacobi–Poincaré series which allows us to show that \(\phi _{0,1}\) is also a Poincaré series.     

Fourier coefficients of half-integral weight cusp forms and Waring’s problem

Extending the approach of Iwaniec and Duke, we present strong uniform bounds for Fourier coefficients of half-integral weight cusp forms of level N. As an application, we consider a Waring-type problem with sums of mixed powers.     

On the first simultaneous sign change for Fourier coefficients associated with Hecke–Maass forms

The Ramanujan Journal - Let f and g be two distinct Hecke–Maass cusp forms of weight zero for $$SL(2,\mathbb {Z})$$ with Laplacian eigenvalues $$\frac{1}{4}+u^{2}$$ and $$\frac{1}{4}+v^{2}$$...