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In this article, we first give a complete description of the algebra of integer weight modular forms on the congruence subgroup $$\Gamma _0(2)$$ modulo a prime $$p\ge 3$$ . This result parallels results of Swinnerton-Dyer in the $$SL_2(\mathbb {Z})$$ case, Katz on the subgroup $$\Gamma (N)$$ for $$N\ge 3$$ , Gross on the subgroup $$\Gamma _1(N)$$ for $$N\ge 4$$ and Tupan on modular forms of half-integral weight on $$\Gamma _1(4)$$ . Next, we use the theory of mod p modular forms on $$\Gamma _0(2)$$ to prove the non-existence of simple congruences for Fourier coefficients of quotients of certain integer weight Eisenstein series on $$\Gamma _0(2)$$ . The non-existence of simple congruences for coefficients of quotients of Eisenstein series on $$SL_2(\mathbb {Z})$$ has been shown by Dewar.

We prove a Voronoi–Oppenheim summation formula for divisor functions associated with Gaussian integers. This formula is a direct generalization of Oppenheim’s summation formula for classical divisor functions. To prove the formula we construct an Eisenstein series and study its properties. Our method of proof is similar to Beineke and Bump’s proof of the classical Oppenheim summation formula.

Based on the values of the Weierstrass elliptic function $$\wp (z|\tau )$$ at $$z=\pi \tau /2$$ , $$(\pi +\pi \tau )/{2}, (\pi +\pi \tau )/{4},(\pi +2\pi \tau )/{4}$$ and the theory of modular forms on the arithmetic group $$\Gamma _0(2)$$ , we decompose $$\theta _2^{2n}(\tau )$$ as sum of Eisenstein series and cusp forms. Using the recurrence relation of $$\wp ^{(2n)}(z|\tau )$$ , we provide an algorithm to determine the exact form of these cusp forms.

We prove that for a graded algebra $$\mathcal {A}$$ with a derivation $$D_{\mathcal {A}}$$ satisfying certain conditions, and a bi-graded algebra $$\mathcal {A}[q]$$ with an extended derivation D of $$D_{\mathcal {A}}$$ , there are only finitely many $$D_{\mathcal {A}}$$ - and D-invariant (or differential with respect to $$D_{\mathcal {A}}$$ and D) principal prime ideals of $$\mathcal {A}$$ and of $$\mathcal {A}[q]$$ , respectively. As its application, we prove the algebraic independence of the values of certain Eisenstein series for the arithmetic Hecke groups H(m) for $$m=4,6, \infty $$ , which is an extension of Nesterenko’s result for the Eisenstein series for SL $$_2(\mathbb {Z})$$ .

In this paper, we investigate the non-modular solutions to the Schwarz differential equation $\{f,\tau \}=sE_4(\tau)$ where $E_4(\tau)$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we provide explicit solutions for each $s=2\pi^2(n/6)^2$ with $n\equiv 1\mod 12$. These solutions are obtained as integrals of meromorphic weight 2 modular forms. As a consequence, we find explicit solutions to the differential equation $\displaystyle y''+\frac{\pi^2n^2}{36}\,E_4\,y=0$ for each $n\equiv 1\mod 12$ generalizing the work of Hurwitz and Klein on the case $n=1$. Our investigation relies on the theory of equivariant functions on the complex upper half-plane. This paper supplements a previous work where we determine all the parameters $s$ for which the above Schwarzian equation has a modular solution., Comment: 20 pages

In 1914, Srinivasa Ramanujan published several hypergeometric series for $$1/\pi $$ . One of these series was used by Bill Gosper in 1985 in a world-record-computation of $$\pi $$ . Shortly after this, the Chudnovskys found a faster series for $$1/\pi $$ based on the largest Heegner number and the Borweins proved Ramanujan’s series. Lately, the Chudnovskys’ series has often been used in practice to calculate digits of $$\pi $$ , it reads: $$\begin{aligned} \frac{\sqrt{640320^3}}{12 \pi } = \sum _{n=0}^\infty \frac{\left( 6n\right) !}{\left( 3n\right) !\left( n!\right) ^3}\,\frac{13591409+ 545140134 n}{\left( -640320^3\right) ^n}. \end{aligned}$$ In this paper, we calculate the coefficients in two of Ramanujan’s series for $$1/\pi $$ and those in the Chudnovskys’ series. For our calculation, we don’t require special software packages, but only the Fourier expansions of the Eisenstein series with a precision of $$\approx 20$$ decimals. We also prove the exactness of our calculations by proving that the values of certain non-holomorphic modular functions are algebraic integers. Our proof uses the division values of the Weierstras $$\wp $$ function.

The purpose of this paper is to consider the computation of the Fourier coefficients of real-analytic Eisenstein series associated to general cusps of Hecke congruence subgroups of arbitrary level. Explicit formulas for some coefficients in the Fourier expansion of the Eisenstein series of weight zero for general cusps are given.

We calculate the Jacobi Eisenstein series of weight $k \ge 3$ for a certain representation of the Jacobi group, and evaluate these at $z = 0$ to give coefficient formulas for a family of modular forms $Q_{k,m,\beta}$ of weight $k \ge 5/2$ for the (dual) Weil representation on an even lattice. The forms we construct always contain all cusp forms within their span. We explain how to compute the representation numbers in the coefficient formulas for $Q_{k,m,\beta}$ and the Eisenstein series of Bruinier and Kuss $p$-adically to get an efficient algorithm. The main application is in constructing automorphic products., Comment: 30 pages. Corrected typos and other minor edits

In a well known 1914 paper, Ramanujan gave a number of rapidly converging series for $$1/\pi $$ which are derived using modular functions of higher level. Chudnovsky and Chudnovsky in their 1988 paper derived an analogous series representing $$1/\pi $$ using the modular function J of level 1, which results in highly convergent series for $$1/\pi $$ , often used in practice. In this paper, we explain the Chudnovsky method in the context of elliptic curves, modular curves, and the Picard–Fuchs differential equation. In doing so, we also generalize their method to produce formulae which are valid around any singular point of the Picard–Fuchs differential equation. Applying the method to the family of elliptic curves parameterized by the absolute Klein invariant J of level 1, we determine all Chudnovsky–Ramanujan type formulae which are valid around one of the three singular points: $$0, 1, \infty $$ .

The main object of study in this paper is the double holomorphic Eisenstein series $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ having two complex variables $$\mathbf{s}=(s_1,s_2)$$ and two parameters $$\mathbf{z}= (z_1,z_2)$$ which satisfies either $$\mathbf{z}\in (\mathfrak {H}^+)^2$$ or $$\mathbf{z}\in (\mathfrak {H}^-)^2$$ , where $$\mathfrak {H}^{\pm }$$ denotes the complex upper and lower half-planes, respectively. For $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ , its transformation properties and asymptotic aspects are studied when the distance $$|z_2-z_1|$$ becomes both small and large under certain natural settings on the movement of $$\mathbf{z}\in (\mathfrak {H}^{\pm })^2$$ . Prior to the proofs our main results, a new parameter $$\eta $$ , which plays a pivotal role in describing our results, is introduced in connection with the difference $$z_2-z_1$$ . We then establish complete asymptotic expansions for $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ when $$\mathbf{z}$$ moves within the poly-sector either $$(\mathfrak {H}^+)^2$$ or $$(\mathfrak {H}^-)^2$$ , so as to $$\eta \rightarrow 0$$ through $$|\arg \eta

If \(\rho \) denotes a finite-dimensional complex representation of \(\mathbf {SL}_{2}(\mathbf {Z})\), then it is known that the module \(M(\rho )\) of vector-valued modular forms for \(\rho \) is free and of finite rank over the ring M of scalar modular forms of level one. This paper initiates a general study of the structure of \(M(\rho )\). Among our results are absolute upper and lower bounds, depending only on the dimension of \(\rho \), on the weights of generators for \(M(\rho )\), as well as upper bounds on the multiplicities of weights of generators of \(M(\rho )\). We provide evidence, both computational and theoretical, that a stronger three-term multiplicity bound might hold. An important step in establishing the multiplicity bounds is to show that there exists a free basis for \(M(\rho )\) in which the matrix of the modular derivative operator does not contain any copies of the Eisenstein series \(E_6\) of weight six.

We consider two problems concerning oscillations of the q-exponents of generalized modular functions. First, we prove a quantitative result for the number of sign changes for the sequence of q-exponents at primes in short intervals. Second, we investigate the growth of the q-exponents.

We discuss polyharmonic Maass forms of even integer weight on PSL(2, Z)\H, which are a generalization of classical Maass forms. We explain the role of real-analytic Eisenstein series E_k(z, s) and the differential operator $\partial/\partial s$ in this theory.

Let m be a positive integer \(\ge \)3 and \(\lambda =2\cos \frac{\pi }{m}\). The Hecke group \(\mathfrak {G}(\lambda )\) is generated by the fractional linear transformations \(\tau + \lambda \) and \(-\frac{1}{\tau }\) for \(\tau \) in the upper half plane \(\mathbb H\) of the complex plane \(\mathbb C\). We consider a set of functions \(\mathfrak {f}_0, \mathfrak {f}_i\) and \(\mathfrak {f}_{\infty }\) automorphic with respect to \(\mathfrak {G}(\lambda )\), constructed from the conformal mapping of the fundamental domain of \(\mathfrak {G}(\lambda )\) to the upper half plane \(\mathbb H\), and establish their connection with the Legendre functions and a class of hyper-elliptic functions. Many well-known classical identities associated with the cases of \(\lambda =1\) and 2 are preserved. As an application, we will establish a set of identities expressing the reciprocal of \(\pi \) in terms of the hypergeometric series.

We show that there is the Shimura lifting map of \(\mathbf {M}_{3/2}(N,\chi _{0})\) to \(\mathbf {M}_{2}(N/2,\chi _{0}^{2})\) for any natural number N divisible by 4 and for any odd Dirichlet character \(\chi _{0}\), which was already proved in the case of higher weight (Tsuyumine, Tsukuba J Math 23:465–483, 1999). As the application, we show that the lift of the weighted average of theta series of quadratic forms of an odd number of variables in a class, is an Eisenstein series of integral weight.

The aim of this paper is to study certain multiple series which can be regarded as multiple analogues of Eisenstein series. As a prior research, the second-named author considered double analogues of Eisenstein series and expressed them as polynomials in terms of ordinary Eisenstein series. This fact was derived from the analytic observation of infinite series involving hyperbolic functions which were based on the study of Cauchy, and also Ramanujan. In this paper, we prove an explicit relation formula among these series. This gives an alternative proof of this fact by using the technique of partial fraction decompositions of multiple series which was introduced by Gangl, Kaneko and Zagier. By the same method, we further show a certain multiple analogue of this fact and give some examples of explicit formulas. Finally we give several remarks about the relation between our present result and the previous work for infinite series involving hyperbolic functions., 8 pages

We study the algebra $${{\mathrm{{\mathcal {MD}}}}}$$ of generating functions for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in $${\mathbb {Q}}$$ arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra $${{\mathrm{{\mathcal {MD}}}}}$$ is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in $${{\mathrm{{\mathcal {MD}}}}}$$ . The (quasi-)modular forms for the full modular group $${{\mathrm{SL}}}_2({\mathbb {Z}})$$ constitute a subalgebra of $${{\mathrm{{\mathcal {MD}}}}}$$ , and this also yields linear relations in $${{\mathrm{{\mathcal {MD}}}}}$$ . Generating functions of multiple divisor sums can be seen as a q-analogue of multiple zeta values. Studying a certain map from this algebra into the real numbers we will derive a new explanation for relations between multiple zeta values, including those of length 2, coming from modular forms.

We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals $K$ and $E$. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.

We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in dimensions 2 and 3 leads naturally to close connections between modular forms and hypergeometric series.

From the theory of modular forms, there are exactly $[(k-2)/6]$ linear relations among the Eisenstein series $E_k$ and its products $E_{2i}E_{k-2i}\ (2\le i \le [k/4])$. We present explicit formulas among these modular forms based on the partial fraction decomposition, and use them to determining a basis of the space of modular forms of weight $k$ on ${\rm SL}_2({\mathbb Z})$., 8 pages

We revisit the classical theorem of Euler regarding special values of the Riemann zeta function as well as Hecke’s generalization of this to Dirichlet’s \(L\)-functions and derive an elliptic analogue. We also discuss transcendence questions that arise from this analogue.

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.

A systematic procedure for generating cubic multisections of Eisenstein series is given. The relevant series are determined from Fourier expansions for Eisenstein series by restricting the congruence class of the summation index modulo three. We prove that the resulting series are rational functions of \eta(\tau) and \eta(3\tau), where \eta is the Dedekind eta function. A more general treatment of cubic dissection formulas is given by describing the dissection operators in terms of linear transformations. These operators exhibit properties that mirror those of similarly defined quintic operators.

Using the properties of conformal mappings and differential equations, we develop a class of elliptic functions associated with the hypergeometric function ${_{2}}F_{1}(\frac{1}{4},\frac{3}{4};1;z)$ . A detailed comparison is made with the classical Jacobi elliptic functions. Within the frame work of this theory, we provide a proof and new insight into a set of identities of Ramanujan associated with the above hypergeometric function.

We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence of a prime dividing the $k-1$-th generalized Bernoulli number and the existence of non-trivial Hermitian cusp forms of weight $k$. We will conclude by giving numerical examples for each case., Comment: 12 pages

In this paper, computing a double integral of convolution type in two ways, we give certain formulas for general multiple series. The method is based on that of Kanemitsu–Tanigawa–Yoshimoto in their previous work. As concrete examples, considering multiple zeta-functions of Barnes type and Euler–Zagier type, and Epstein zeta-functions, we give new formulas for multiple series involving these zeta-functions.

Kurokawa and Wakayama (Ramanujan J. 10:23–41, 2005) studied a family of elliptic functions defined by certain q-series. This family, in particular, contains the Weierstrass ℘-function. In this paper, we prove that elliptic functions in this family satisfy certain non-linear algebraic differential equations whose coefficients are essentially given by rational functions of the first few Eisenstein series of the modular group.

We prove an explicit formula for local densities of inhomogeneous quadratic forms. The formula is derived by calculating explicitly the Fourier coefficients of various types of Eisenstein series.

We study a class of elliptic functions associated with the hypergeometric function \({_{2}}F_{1}(\frac{1}{6},\frac{5}{6};1;z)\). From the perspective of the properties of conformal mappings and differential equations, we provide new insight into a set of identities of Ramanujan associated with the above hypergeometric function.

We give proofs for new Ramanujan-type modular equations discovered by M. Somos and establish applications of some of them.

Let P,Q, and R denote the Ramanujan Eisenstein series. We compute algebraic relations in terms of P(qi) (i=1,2,3,4), Q(qi) (i=1,2,3), and R(qi) (i=1,2,3). For complex algebraic numbers q with 0

In this paper, we consider the space of second order cusp forms. We determine that this space is precisely the same as a certain subspace of mixed mock modular forms. Based upon Poincare series of Diamantis and O’Sullivan (Trans. Am. Math. Soc. 360:5629–5666, 2008) which span the space of second order cusp forms, we construct Poincare series which span a natural (more general) subspace of mixed mock modular forms.

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).

We discuss the problem of the vanishing of Poincar\'e 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 Poincar\'e series and the connection to Ramanujan's mock theta functions.

We establish (Theorem 3.6) polynomial-growth estimates for the Fourier coefficients of holomorphic logarithmic vector-valued modular forms.

We employ a new constructive approach to study modular forms of level five by evaluating the Weierstrass elliptic functions at points of order five on the period parallelogram. A significant tool in our analysis is a nonlinear system of coupled differential equations analogous to Ramanujan’s differential system for the Eisenstein series on SL(2,ℤ). The resulting relations of level five may be written as a coupled system of differential equations for quintic Eisenstein series. Some interesting combinatorial and analytic consequences result, including an alternative proof of a famous identity of Ramanujan involving the Rogers–Ramanujan continued fraction.

In this paper, we extend previous results to prove that generalized modular forms with rational Fourier expansions whose divisors are supported only at the cusps and certain other points in the upper half plane are actually classical modular forms. We discuss possible limitations to this extension and pose questions about possible zeroes for modular forms of prime level., Comment: 12 pages

In this paper we obtain a weighted average formula for special values of $L$-functions attached to normalized elliptic modular forms of weight $k$ and full level. These results are obtained by studying the pullback of a Siegel Eisenstein series and working out an explicit spectral decomposition., Comment: 12 pages

In this paper, we determine the q-expansions of vector-valued modular forms (Knopp and Mason in Ill. J. Math. 48:1345–1366, 2004; Acta Arith. 110(2): 117–124, 2003) of large negative weight on the full modular group where we allow poles in the upper half plane and at infinity.

We prove explicit formulas decomposing cusp forms of even weight for the modular group, in terms of generators having rational periods, and in terms of generators having rational Fourier coefficients. Using the Shimura correspondence, we also give a decomposition of Hecke cusp forms of half integral weight k+1/2 with k even in terms of forms with rational Fourier coefficients, given by Rankin–Cohen brackets of theta series with Eisenstein series.

Ramanujan’s differential equations for the classical Eisenstein series are of great importance to many areas in number theory and special functions. H.H. Chan recently demonstrated that these differential equations can be derived from the triple product identity and the quintuple product identity in an elementary manner. In this article, we extend this method in a uniform manner to derive corresponding differential equations for the Eisenstein series of level 2. Several applications of these differential equations are also given.

In this paper, we shall reveal the hidden structure in recent results of Katsurada as the Meijer G-function hierarchy. In Sect. 1, we consider the holomorphic Eisenstein series and show that Katsurada’s two new expressions are variants of the classical Chowla–Selberg integral formula (Fourier expansion) with or without the beta-transform of Katsurada being incorporated. In Sect. 2, we treat the Taylor series expansion of the Lipschitz–Lerch transcendent in the perturbation variable. In the proofs, we make an extensive use of the beta-transform (used to be called the Mellin–Barnes formula).

Kurokawa introduced q-multiple gamma functions and q-multiple sine functions. We show that the Appell’s O-function is expressed via the q-multiple gamma function. We also give some applications of this result. For example, we obtain a formula for the “Stirling modular form” and calculate special values of the q-multiple sine function. Moreover, we give some formulas of Eisenstein series and double cotangent functions and its generalization. Then the former gives an infinite product expression of the double sine function explicitly and a result of Kurokawa.

We discuss some properties of exponents of q-product expansions of specific generalized modular functions on the congruence subgroup Γ0(N).

Kaneko and Koike gave the “extremal” quasimodular forms of depth 1 for PSL2(ℤ) and modular differential equations they satisfy. In this paper, we study modular solutions of their modular differential equations.

The Rankin–Cohen product of two modular forms is known to be a modular form. The same formula can be used to define the Rankin–Cohen product of two holomorphic functions f and g on the upper half-plane. Assuming that this product is a modular form, we prove that both f and g are modular forms if one of them is. We interpret this result in terms of solutions of linear ordinary differential equations.

Using certain representations for Eisenstein series, we uniformly derive several Ramanujan-type series for 1/π.

B.C. Berndt (J. Reine Angew. Math. 272:182–193, 1975; 304:332–365, 1978) has derived a number of new transformation formulas, in particular, the transformation formulae of the logarithms of the classical theta functions, by using a transformation formula for a more general class of Eisenstein series. In this paper, we continue his study. By using a transformation formula for a class of twisted generalized Eisenstein series, we generalize a transformation formula given by J. Lehner (Duke Math. J. 8:631–655, 1941) and give a new proof for transformation formulas proved by Y. Yang (Bull. Lond. Math. Soc. 36:671–682, 2004).

Using results that were well-known to Ramanujan, we give proofs of some results for Eisenstein series in the lost notebook. Our proofs have the additional advantage that it is not necessary to know the results in advance; that is, the proofs are derivations as opposed to verifications.

In this paper, we define the normalized Eisenstein series ℘, e, and $\mathcal{Q}$ associated with Γ0(2), and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on Γ0(2) and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a formula involving the triangular numbers.