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We develop a theory of modular forms on the groups SO ( 3 , n + 1 ) , n ≥ 3 . This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and Gan-Gross-Savin. We prove the results analogous to those of earlier papers of the author on modular forms on exceptional groups, except now in the familiar setting of classical groups. Moreover, in the setting of SO ( 3 , n + 1 ) , there is a family of absolutely convergent Eisenstein series, which are modular forms. We prove that these Eisenstein series have algebraic Fourier coefficients, like the classical holomorphic Eisenstein series on SO ( 2 , n ) . As an application, using a local result of Savin, we prove that the so-called “next-to-minimal” modular form on quaternionic E 8 has rational Fourier expansion.

For all Eichler orders with a same squarefree level in a definite quaternion algebra over the field of rational numbers, we prove that a weighted sum of Jacobi theta series associated to these orders is a Jacobi Eisenstein series. Multiply the Fourier coefficients of the latter by a power of 2 to get our modified Hurwitz class numbers. As its corollaries, the modified Hurwitz class numbers can be used to calculate the traces of Brandt matrices and the type number of the Eichler orders.

We show that the projection of a product of two or three Eisenstein series of level one onto the cuspidal subspace is not an eigenform unless the dimension of the cuspidal subspace is one.

In recent years, Rogers and Zudilin developed a method to write $L$-values attached to elliptic curves as periods. In order to apply this method to a broader collection of $L$-values, we study Eisenstein series and determine their Fourier series at cusps. Subsequently, we write the $L$-values of an elliptic curve of conductor 32 as an integral of Eisenstein series and evaluate the value at $k>1$ explicitly as a period. As a side result, we give simple integral expressions for the generating functions of $L(E,k)$ when even (or odd) $k$ runs over positive integers., 25 pages

We give a lower bound of the number of zeros of the modular forms E k + E l + − E k + l + for large enough k and l on each of the arc boundary and on the vertical boundary of the fundamental domain for the Fricke group Γ 0 + ( 2 ) , where E k + is the Eisenstein series of weight k for Γ 0 + ( 2 ) .

Notable results on the special values of $L$-functions of Siegel modular forms were obtained by J. Sturm in the case when the degree $n$ is even and the weight $k$ is an integer. In this paper we extend this method to half-integer weights $k$ and arbitrary degree $n$, determining the algebraic field in which they lie. This method hinges on the Rankin-Selberg method; our extension of this is aided by the theory of half-integral modular forms developed by G. Shimura. In the second half, an analogue of P. B. Garrett's conjecture is proved in this setting, a result that is of independent interest but that bears direct applications to our first results. It determines exactly how the decomposition of modular forms into cusp forms and Eisenstein series preserves algebraicity and, ultimately, the full range of special values.

Let E k ( z ) be the normalized Eisenstein series of weight k for the modular group SL ( 2 , Z ) . We study the zeros of E k to prove that the equation ∏ i = 1 n E k i = ∏ j = 1 m E l j has no solutions, except for those given by known relationships between E 4 , E 6 , E 8 , E 10 , and E 14 . We go on to discuss some implications of this result.

We combine Zagier's theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of SL ( 2 , R ) due to Bernstein and Reznikov to study triple products of Eisenstein series of arbitrary (in particular, non-arithmetic) non-compact finite-volume hyperbolic surfaces.

In this paper, we introduce modular polynomials for the congruence subgroup $\Gamma_0(M)$ when $ X_0(M) $ has genus zero and therefore the polynomials are defined by a Hauptmodul of $ X_0(M) $. We show that the intersection number of two curves defined by two modular polynomials can be expressed as the sum of the numbers of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of positive definite binary quadratic forms over $\mathbb{Z}$. We also show that the intersection numbers can be also combinatorially written by Fourier coefficients of the Siegel Eisenstein series of degree 2, weight 2 with respect to $\mathrm{Sp}_2(\mathbb{Z})$., Comment: 27 pages, 2 tables

The classical Kronecker limit formula describes the constant term in the Laurent expansion at the first order pole of the non-holomorphic Eisenstein series associated to the cusp at infinity of the modular group. Recently, the meromorphic continuation and Kronecker limit type formulas were investigated for non-holomorphic Eisenstein series associated to hyperbolic and elliptic elements of a Fuchsian group of the first kind by Jorgenson, Kramer and the first named author. In the present work, we realize averaged versions of all three types of Eisenstein series for Γ0(N) as regularized theta lifts of a single type of Poincaré series, due to Selberg. Using this realization and properties of the Poincaré series we derive the meromorphic continuation and Kronecker limit formulas for the above Eisenstein series. The corresponding Kronecker limit functions are then given by the logarithm of the absolute value of the Borcherds product associated to a special value of the underlying Poincaré series., Journal of Number Theory, 225, ISSN:0022-314X, ISSN:1096-1658

Poincare and Eisenstein series are building blocks for every type of automorphic forms. We define Poincare series for Jacobi forms of lattice index and show that they reproduce Fourier coefficients of cusp forms under the Petersson scalar product. We compute the Fourier expansions of Poincare and Eisenstein series and give an explicit formula for the Fourier coefficients of the trivial Eisenstein series in terms of values of Dirichlet L-functions at negative integers. For even weight and fixed index, finite linear combinations of Fourier coefficients of non-trivial Eisenstein series are equal to finite linear combinations of Fourier coefficients of the trivial one. This result is used to obtain formulas for the Fourier coefficients of Eisenstein series associated with isotropic elements of small order.

We define a generalized hyperbolic Eisenstein series for a pair of a hyperbolic manifold of finite volume and its submanifold. We prove the convergence, the differential equation and the precise spectral expansion associated to the Laplace–Beltrami operator. We also derive the analytic continuation with the location of the possible poles and their residues from the spectral expansion.

Eisenstein derived addition formulas for the Weierstrass zeta function from the addition formula for the cotangent function and the fact that the Weierstrass zeta function can be represented as an infinite sum of the cotangent functions. In this paper, we apply this idea of Eisenstein's to the addition type formula for the double cotangent function, established by the author. We show that the elliptic digamma function, defined by the logarithmic derivative of the elliptic gamma function, satisfies an addition type formula. This formula includes the addition formula for the Weierstrass zeta function, evaluation formulas for the double Eisenstein series introduced by Tsumura and the double shuffle relations for the double Eisenstein series, proved by Gangl-Kaneko-Zagier.

A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for ζ ( 2 m + 1 ) . The formula for ζ 2 ( 2 m + 1 ) is then generalized in two different directions, one, by considering the generalized divisor function σ z ( n ) , and the other, by studying a more general analogue of the aforementioned Eisenstein series, consisting of one more parameter N . A number of important special cases are derived from the first generalization. For example, we obtain a series representation for ζ ( 1 + ω ) ζ ( − 1 − ω ) , where ω is a non-trivial zero of ζ ( z ) . We also evaluate a series involving the modified Bessel function of the second kind in the form of a rational linear combination of ζ ( 4 k − 1 ) and ζ ( 4 k + 1 ) for k ∈ N .

We consider the Eisenstein series E ( z , s ; k , χ , N ) of weight k = ( n + 3 ) / 2 , level N > 1 and a Dirichlet character χ modulo N such that χ 2 = 1 . Shimura proved that E ( z , k / 2 ; k , χ , N ) is a nearly holomorphic function. We prove that E ( z , k / 2 ; k , χ , N ) generates an indecomposable reducible ( g , K ) -module of length 2. These are new examples of indecomposable reducible ( g , K ) -modules generated by nearly holomorphic modular forms.

We invert the period map defined by the second structure connection of quantum cohomology of P 2 . For small quantum cohomology the inverse is given explicitly in terms of the Eisenstein series E 4 and E 6 , while for big quantum cohomology the inverse is determined perturbatively as a Taylor series expansion whose coefficients are quasi-modular forms.

We define the unit circle for global function fields. We demonstrate that this unit circle, endearingly termed the q-unit circle (pronounced “cue-nit”), after the finite field F q of q elements, enjoys all of the properties akin to the classical unit circle: center, curvature, roots of unity in completions, integrality conditions, embedding into a finite-dimensional vector space over the real line, a partition of the ambient space into concentric circles, Mobius transformations, a Dirichlet approximation theorem, a reciprocity law, and much more. In addition, we extend the polynomial exponential action of Carlitz to an action by all points on the real line; we show that mutually tangent horoballs solve a Descartes-type relation arising from reciprocity; we define the hyperbolic plane, which we prove is uniquely determined by the q-unit circle; and we give the associated modular forms and Eisenstein series.

We study the algebra of certain q-series, called bi-brackets, whose coefficients are given by weighted sums over partitions. These series incorporate the theory of modular forms for the full modular group as well as the theory of multiple zeta values (MZV) due to their appearance in the Fourier expansion of regularized multiple Eisenstein series. Using the conjugation of partitions we obtain linear relations between bi-brackets, called the partition relations, which yield naturally two different ways of expressing the product of two bi-brackets similar to the stuffle and shuffle product of multiple zeta values. In a recent work, the author and K. Tasaka defined (shuffle) regularized multiple Eisenstein series G ⧢ , by using an explicit connection to the coproduct on formal iterated integrals introduced by Goncharov. These satisfy the shuffle product formula. Applying the same concept for the coproduct on quasi-shuffle algebras enables us to define (stuffle) regularized multiple Eisenstein series G ⁎ satisfying the stuffle product formula. We show that both G ⧢ and G ⁎ are given by linear combinations of products of MZV and bi-brackets.

We obtain a Voronoi–Oppenheim summation formula for divisor functions of totally real number fields. This generalizes a formula proved by Oppenheim in 1927. We use a similar method to the one developed by Beineke and Bump in order to prove the classical Oppenheim summation using a certain Eisenstein series and representation theory. Our formula has a simple formulation for real quadratic number fields.

We locate the zeros of the modular forms E k 2 ( τ ) + E 2 k ( τ ) , E k 3 ( τ ) + E 3 k ( τ ) , and E k ( τ ) E l ( τ ) + E k + l ( τ ) , where E k ( τ ) is the Eisenstein series for the full modular group SL 2 ( Z ) . By utilizing work of F.K.C. Rankin and Swinnerton-Dyer, we prove that for sufficiently large k , l , all zeros in the standard fundamental domain are located on the lower boundary A = { e i θ : π / 2 ≤ θ ≤ 2 π / 3 } .

In this paper, we proved an arithmetic Siegel–Weil formula and the modularity of some arithmetic theta function on the modular curve X 0 ( N ) when N is square free. In the process, we also constructed some generalized Delta function for Γ 0 ( N ) and proved some explicit Kronecker limit formula for Eisenstein series on X 0 ( N ) .

We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain "growth-killing" invariant differential operators on $\mathbf H_2$ and those on $\mathbf{H}_1 \times \mathbf{C}$ (here $\mathbf H_n$ denotes the Siegel upper half space of degree $n$). As applications, we can understand better the growth of Petersson norms of Fourier Jacobi coefficients of Klingen Eisenstein series, which in turn has applications to finer issues about representation numbers of quadratic forms, and as a by-product we also show that \textit{any} Siegel modular form of degree $2$ is determined by its `fundamental' Fourier coefficients., 41pp

In this article, we study the Q -rational torsion subgroups of the Jacobian varieties of modular curves. The main result is that, for any positive integer N, J 0 ( N ) ( Q ) tor [ q ∞ ] = 0 if q is a prime not dividing 6 ⋅ N ⋅ ∏ p | N ( p 2 − 1 ) . To prove the result, we explicitly construct a collection of Eisenstein series with rational Fourier expansions, and then determine their constant terms to control the size of the rational torsion subgroups.

We examine the zeros of newform Eisenstein series E χ 1 , χ 2 , k ( z ) of weight k on Γ 0 ( q 1 q 2 ) , where χ 1 and χ 2 are primitive characters modulo q 1 and q 2 , respectively. We determine the location of a significant fraction of the zeros of these Eisenstein series for k sufficiently large.

This paper investigates the relations between modular graph forms, which are generalizations of the modular graph functions that were introduced in earlier papers motivated by the structure of the low energy expansion of genus-one Type II superstring amplitudes. These modular graph forms are multiple sums associated with decorated Feynman graphs on the world-sheet torus. The action of standard differential operators on these modular graph forms admits an algebraic representation on the decorations. First order differential operators are used to map general non-holomorphic modular graph functions to holomorphic modular forms. This map is used to provide proofs of the identities between modular graph functions for weight less than six conjectured in earlier work, by mapping these identities to relations between holomorphic modular forms which are proven by holomorphic methods. The map is further used to exhibit the structure of identities at arbitrary weight.

Let $f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}q^{n(n-1)/2}x^n$ ($0, Comment: 26 pages

We show, for levels of the form $N = p^a q^b N'$ with $N'$ squarefree, that in weights $k \geq 4$ every cusp form $f \in \mathcal{S}_k(N)$ is a linear combination of products of certain Eisenstein series of lower weight. In weight $k=2$ we show that the forms $f$ which can be obtained in this way are precisely those in the subspace generated by eigenforms $g$ with $L(g, 1) \neq 0$. As an application of such representations of modular forms we can calculate Fourier expansions of modular forms at arbitrary cusps and we give several examples of such expansions in the last section., Comment: This is an update to a previous version of the article. In the new version we describe an algorithm (available at https://github.com/michaelneururer/products-of-eisenstein-series) that computes Fourier expansions at cusps

A new generalization of the modified Bessel function of the second kind K z ( x ) is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby anticipating a rich theory that it may possess. The motivation behind introducing this generalization is to have a function which gives a new pair of functions reciprocal in the Koshliakov kernel cos ⁡ ( π z ) M 2 z ( 4 x ) − sin ⁡ ( π z ) J 2 z ( 4 x ) and which subsumes the self-reciprocal pair involving K z ( x ) . Its application towards finding modular-type transformations of the form F ( z , w , α ) = F ( z , i w , β ) , where α β = 1 , is given. As an example, we obtain a beautiful generalization of a famous formula of Ramanujan and Guinand equivalent to the functional equation of a non-holomorphic Eisenstein series on S L 2 ( Z ) . This generalization can be considered as a higher level analogue of the general theta transformation formula. We then use it to evaluate an integral involving the Riemann Ξ-function and consisting of a sum of products of two confluent hypergeometric functions.

The modular function h ( τ ) = q ∏ n = 1 ∞ ( 1 − q 16 n ) 2 ( 1 − q 2 n ) ( 1 − q n ) 2 ( 1 − q 8 n ) is called a level 16 analogue of Ramanujan's series for 1 / π . We prove that h ( τ ) generates the field of modular functions on Γ 0 ( 16 ) and find its modular equation of level n for any positive integer n. Furthermore, we construct the ray class field K ( h ( τ ) ) modulo 4 over an imaginary quadratic field K for τ ∈ K ∩ H such that Z [ 4 τ ] is the integral closure of Z in K, where H is the complex upper half plane. For any τ ∈ K ∩ H , it turns out that the value 1 / h ( τ ) is integral, and we can also explicitly evaluate the values of h ( τ ) if the discriminant of K is divisible by 4.

We propose a third order generalization of the Kaneko–Zagier modular differential equation, which has two parameters. We describe modular and quasimodular solutions of integral weight in the case where one of the exponents at infinity is a multiple root of the indicial equation. We also classify solutions of “character type”, which are the ones that are expected to relate to characters of simple modules of vertex operator algebras and one-point functions of two-dimensional conformal field theories. Several connections to generalized hypergeometric series are also discussed.

We prove that zeros of each basis element of the space of weakly holomorphic modular forms of weight k for the Fricke group Γ 0 + ( 2 ) of level 2 interlace, extending the result for SL 2 ( Z ) of Jenkins and Pratt [4] .

This paper expands upon our results for the arithmetic of Drinfeld modular forms of level T [12] by providing an interpretation from a representation theoretic point of view. We identify GL ( 2 , F q ) -modules that arise naturally from the theory of Drinfeld modular forms of level T with classical GL ( 2 , F q ) -modules. Using the arithmetic of the so-called modified Eisenstein series all isomorphisms are stated explicitly. In particular, we examine the close connection between Drinfeld modular forms of level T and the theory of symmetric powers of the tautological representation of GL ( 2 , F q ) described in previous work [13] .

We determine the possible Serre weights associated to certain Hilbert modular forms when the rational prime p is totally ramified in the totally real field F. Our weight lowering method for arbitrarily large weight is applicable when the slope is sufficiently small, enabling us to compute the mod p reduction at inertia from the known results in the small weight range. In the case of elliptic modular forms (and for certain Hilbert modular forms in the p totally split case) we obtain a unique mod p reduction when the slope is in ( 0 , 1 / 2 ) .

Inspired by the recent work of Hirschhorn and Sellers, in this paper we present a unified proof of Ramanujan-type congruences modulo 5 for c ϕ 4 ( 5 n + 1 ) .

The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other meromorphic modular forms and quasi-modular forms which were subsequently established by Berndt, Bialek, and Yee. In this paper, we place these identities into the context of a larger family by making use of Poincare series introduced by Petersson and a new family of Poincare series which we construct here and which are of independent interest. In addition we establish a number of new explicit identities. In particular, we give the first examples of Fourier expansions for meromorphic modular form with third-order poles and quasi-meromorphic modular forms with second-order poles.

Let F f ∈ S k ( Sp 2 n ( Z ) ) be the Ikeda lifting of a Hecke eigenform f ∈ S 2 k − n ( SL 2 ( Z ) ) with the normalization 〈 F f , F f 〉 = 1 . Let E ( Z ; s ) denote the Klingen Eisenstein series. In this paper we verify that lim k → ∞ ⁡ ∫ Sp 2 n ( Z ) \ H n E ( Z ; n 2 + i t ) | F f ( Z ) | 2 ( det ⁡ Y ) k d μ = 0 which is predicted by the mass equidistribution conjecture of Cogdell and Luo [CL] .

Motivated by the problem of finding explicit q-hypergeometric series which give rise to quantum modular forms, we define a natural generalization of Kontsevich's “strange” function. We prove that our generalized strange function can be used to produce infinite families of quantum modular forms. We do not use the theory of mock modular forms to do so. Moreover, we show how our generalized strange function relates to the generating function for ranks of strongly unimodal sequences both polynomially, and when specialized on certain open sets in C . As corollaries, we reinterpret a theorem due to Folsom–Ono–Rhoades on Ramanujan's radial limits of mock theta functions in terms of our generalized strange function, and establish a related Hecke-type identity.

In the previous works, analytical exact and approximate formulas for the effective conductivity of two-dimensional random composites were deduced. These formulas contain the discrete convolutions of the Eisenstein series ( e -sums, generalized Rayleigh–Eisenstein–Mityushev sums). The considered objects become the classic lattice sums for regular doubly periodic locations of the centers of inclusions. In the present paper, we modify the above approach and extend it to Natanzon's series having applications in the elasticity theory. The conditionally convergent series are investigated by the Eisenstein summation method. We derive a computationally effective formula for conditionally convergent series. Numerical simulations suggest new formulas in which some lattice sums are equal to π .

Let U / Q be a unitary group of Q -rank one so that the group of real points U ( R ) ≅ U ( n , 1 ) . The group U is only quasi-split over Q if and only if n = 1 , 2 . The cohomology of a congruence subgroup of U is closely related to the theory of automorphic forms. This relation is best captured in the so-called automorphic cohomology spaces H ⁎ ( U , C ) , a natural module under the action of the group U ( A f ) . This paper gives a structural account of the U ( A f ) -module structure of that part of the cohomology which is generated by residues or derivatives of Eisenstein series. In particular, we determine a set of arithmetic conditions, mainly given in terms of partial automorphic L-functions, subject to which residues of Eisenstein series may give rise to non-vanishing cohomology classes. The main task is, although the usual method due to Langlands-Shahidi is not applicable, to analyze the analytic behavior of suitable Eisenstein series and to determine the location of their possible poles.

Several authors have recently proved results which express cusp forms as p-adic limits of weakly holomorphic modular forms under repeated application of Atkin's U-operator. The proofs involve techniques from the theory of weak harmonic Maass forms, and in particular a result of Guerzhoy, Kent, and Ono on the p-adic coupling of mock modular forms and their shadows. Here we obtain strengthened versions of these results using techniques from the theory of holomorphic modular forms.

In this paper we study the structure of the algebra of Drinfeld modular forms for the principal congruence subgroup Γ ( T ) of the full modular group GL ( 2 , F q [ T ] ) . To this end and based on work of Cornelissen, we introduce a new class of Eisenstein series and describe their fundamental properties. These so-called modified Eisenstein series allow for a simplification of prior results and are an indispensable tool for future work on Drinfeld modular forms of level T.

Following Bachmann's recent work on bi-brackets and multiple Eisenstein series, Zudilin introduced the notion of multiple q-zeta brackets, which provides a q-analog of multiple zeta values possessing both shuffle as well as quasi-shuffle relations. The corresponding products are related in terms of duality. In this work we study Zudilin's duality construction in the context of classical multiple zeta values as well as various q-analogs of multiple zeta values. Regarding the former we identify the derivation relation of order two with a Hoffman-Ohno type relation. Then we describe relations between the Ohno-Okuda-Zudilin q-multiple zeta values and the Schlesinger-Zudilin q-multiple zeta values., revised version, accepted for publication in Advances in Mathematics

The analogous theory for level 16 of Ramanujan's theories of elliptic functions to alternative bases is developed by studying the level 16 modular function h ( q ) = q ∏ j = 1 ∞ ( 1 − q 16 j ) 2 ( 1 − q 2 j ) ( 1 − q j ) 2 ( 1 − q 8 j ) .

Let E 2 ( z ) be the Eisenstein series which is quasi-modular of weight 2 for SL 2 ( Z ) . For each p = 2 , 3 , we show that there are infinitely many Γ 0 + ( p ) -inequivalent zeros of the quasi-modular form E ( z ) : = E 2 ( z ) + p E 2 ( p z ) of weight 2 for Γ 0 + ( p ) and give the estimates of their locations numerically.

Counting integral binary quadratic forms with certain restrictions is a classical problem. In this paper, we count binary quadratic forms of fixed discriminant given restrictions on the size of their coefficients. We accomplish this by investigating the analytic properties of a certain double Dirichlet series, which is a shifted convolution sum of certain classical automorphic forms.

Using certain representations for Eisenstein series and Ramanujan's quartic theory of elliptic functions, we establish some new series for 1 / π 2 .

We give algorithms for computing the singular moduli of suitable nonholomorphic modular functions F(z). By combining the theory of isogeny volcanoes with a beautiful observation of Masser concerning the nonholomorphic Eisenstein series E_2*(z), we obtain CRT-based algorithms that compute the class polynomials H_D(F;x), whose roots are the discriminant D singular moduli for F(z). By applying these results to a specific weak Maass form F_p(z), we obtain a CRT-based algorithm for computing partition class polynomials, a sequence of polynomials whose traces give the partition numbers p(n). Under the GRH, the expected running time of this algorithm is O(n^{5/2+o(1)}). Key to these results is a fast CRT-based algorithm for computing the classical modular polynomial Phi_m(X,Y) that we obtain by extending the isogeny volcano approach previously developed for prime values of m., Minor revision to reflect referee comments, 23 pages

In this paper, we compute pull-backs of Siegel theta functions to the Hilbert moduli space and consider their application to generating genus 2 curves for cryptography. We express invariants of genus 2 curves such as the Gundlach invariants and Rosenhain invariants in terms of these Hilbert theta functions. A result of independent interest is a simple formula in terms of these functions for the Eisenstein series of weight 2, which is not the pull-back of any Siegel modular form of level 1. We present an algorithm to compute minimal polynomials for the invariants, including a description of CM points and how to compute them, along with numerical examples.

By adding certain equianharmonic elliptic sigma functions to the coefficients of the Borwein cubic theta functions, an interesting set of six two-variable theta functions may be derived. These theta functions invert the F 1 ( 1 3 ; 1 3 ; 1 3 ; 1 | x , y ) case of Appell's hypergeometric function and satisfy several identities akin to those satisfied by the Borwein cubic theta functions. The work of Koike et al. is extended and put into the context of modular equations, resulting in a simpler derivation of their results as well as several new modular equations for Picard modular functions. An application of these results is a new two-parameter family of solvable nonic equations.

In this paper, we give an elementary proof of Ramanujan's Eisenstein series of level 7. In the process, we also prove four Eisenstein series of level 14 due to S. Cooper and D. Ye [4].