Full Modular Group Research Articles

The square mean of Dirichlet series associated with cusp forms

Letbe a cusp form of even integral weight k > 2 for the full modular group. Then the Dirichlet seriesis absolutely convergent for σ > ½(k + 1). Hecke showed that LF is an entire function of s satisfying the functional equation

The 𝐿²-norm of Maass wave functions

Let D D denote the fundamental domain for the full modular group. Suppose that f ∈ L 2 ( D ) f \in {L^2}(D) satisfies the wave equation Δ f = λ f \Delta f = \lambda f , where Δ \Delta is the noneuclidean Laplacian, and further, assume that f f is a common eigenfunction for all the Hecke operators. Then upper and lower bounds for the L 2 {L^2} -norm of f f are determined which depend only on λ \lambda and the first Fourier coefficient of f f .

The vanishing of Poincaré series

Every holomorphic modular form of weight k > 2 is a sum of Poincaré series; see, for example, Chapter 5 of (5). In particular, every cusp form of even weight k ≧ 4 for the full modular group Γ(1) is a linear combination over the complex field C of the Poincaré series.Here mis any positive integer, z ∈ H ={z ∈ C: Im z>0} andThe summation is over all matriceswith different second rows in the (homogeneous) modular group, i.e. in SL(2, Z).The factor ½ is introducted for convenience.

Hecke operators and an identity for the dedekind sums

If h, k ∈ Z, k > 0, the Dedekind sum is given by s(h,k) = ∑ μ=1 k μ k hμ k , with ((x)) = x − [x] − 1 2 , x∉Z , =0 , x∈Z . The Hecke operators T n for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities ( n ∈ Z +) ∑ ∑ s(ah+bk,dk) = σ(n)s(h,k) , ad=n b( mod d) d>0 where ( h, k) = 1, k > 0 and σ( n) is the sum of the positive divisors of n. Petersson had earlier proved (∗) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (∗) when n is prime.

Eisenstein series on the symplectic group

Analytic continuation is proved for certain Eisenstein series on the symplectic group which are associated with nonparabolic forms. In the case of the full modular group an explicit functional equation is obtained, and the singularities of the series are completely described.

EISENSTEIN SERIES ON THE SYMPLECTIC GROUP

Analytic continuation is proved for certain Eisenstein series on the symplectic group which are associated with nonparabolic forms. In the case of the full modular group an explicit functional equation is obtained, and the singularities of the series are completely described.

Modular graphs

It is shown that a collection of vertex two-colorable graphs with certain degree restrictions is intimately connected with those subgroups of the full modular group Γ of finite index. These (modular) graphs are analyzed as to structure, being shown to arise from explicit “lifts” of (modular) trees, which are then enumerated.

Construction of modular forms

Modular forms arising from lattices are constructed and their transformation properties under the full modular group are obtained in explicit form suitable for calculation. The forms are obtained via specialization of the several variable theta function.

Upper bounds for modular forms, lattices, and codes

Let W(z) = 1 + Ad ezmidr + AdCl ewrim+lM + . . . be a modular form of weight n for the full modular group. Then for every constant b there exists an n, = n,,(b) such that if d > n/6 b and n > n, , then one of A@, A,, ,..., has a negative real part. This implies that there is no even unimodular lattice in E”, for n > n, , having minimum nonzero squared length > n/12 b. A similar argument shows that there is no binary self-dual code of length n > n, having all weights divisible by 4 and minimum nonzero weight > n/6 6. A corresponding result holds for ternary codes.

PERIODS OF PARABOLIC FORMS AND $ p$-ADIC HECKE SERIES

The author proves an algebraicity theorem for the periods of parabolic forms of any weight for the full modular group, gives explicit formulas for the coefficients of the forms, and constructs -adic analogs of their Mellin transforms in the manner of Iwasawa and Mazur.Bibliography: 13 items.

Some Remarks on Symplectic Groups, Modular Groups and Poincare's Series

This paper may be considered as a preparation for later investigations. I shall prove some theorems and lemmas which are of general interest to the theory of modular functions of several variables. Some of the results were already known but it seemed desirable to find better proofs. The part about Poincare's series is closely connected with some recent work of H. Klingen [19], [20], [21]; a more detailed explanation about this connection may be found at the end of Chapter III, ? 3. The paper consists of three chapters, the contents of which is the following. The first chapter deals with the symplectic group. Some inequalities and the boundedness of certain expressions will be proved. The second chapter is devoted to Hilbert-Siegel's modular groups. First we define arithmetically certain equivalence classes and prove that there are only finite many such classes provided some definite expression is bounded. These results are needed to prove the convergence of Poincare's series. Then we deal with elliptic fixed points of Hilbert-Siegel's modular groups and improve some result obtained in [11]. At the end of the second chapter I prove that the quotient spaces of the generalized upper half-plane with respect to the full Siegel's modular group and the so called group of integral modular substitutions have the first homology group zero provided n > 1. In chapter three we show that under certain assumptions a HilbertSiegel's modular form of weight zero is constant. Then we investigate the convergence of certain integrals and in connection herewith the uniform convergence of Poincare's series. Finally some theorem about separation of variables through modular functions is deduced. This paper was partially supported by an N. S. F. grant.

The coefficients of certain integral modular forms

The notation which we use is that of a recent paper by one of us, and we quote results from that paper as they are required. It is known (see R, Theorem 1, for example) that the vector space k of all cusp-forms f(z) of even negative dimension – k (k ≥ 12), belonging to the full modular group Γ(1), possesses a finite basis of formswhere k is defined by (2·10) of R and the coefficients possess the following properties:for a prime p, where p is a positive integer.

Remarks on some modular identities

Introduction. We shall consider a certain class of functions invariant with respect to the substitutions of the congruence subgroup Fo(p) of the modular group r. By specializing these functions, we shall obtain classical identities in the analytical theory of numbers: E.g., the Ramanujan identities for partitions modulo 5, 7 and Mordell's identity for r(n). We shall also derive some new identities. These functions bear some resemblance to those considered by Rademacher in his paper [1 ](1) to prove the Ramanujan identities, certain modular equations, etc. The type of function considered, however, seems first to have been studied by Watson in his paper [2]. 1. Definitions, notations. (1.1) r is the full modular group; i.e., the group of 2 X2 matrices of determinant 1 with rational integral elements. (1.2) Fo(m) is the subgroup of r characterized as follows: The element

Divisibility Properties of the Fourier Coefficients of the Modular Invariant j(τ)

The general idea behind the proof of this theorem is as follows. Consider the congruence (1. 2) as an example. By applying a certain linear operatordenoted by U5-to the right member of (1. 1) one obtains the series with v replaced by 5v. The same operation performed on the left member yields a modular function belonging to a subgroup of the full modular group. We shall try, therefore, to express that modular function in terms of the basic function of the subgroup, which is known to have integral coefficient Fourier expansions. The identity which results is of the form

On the Expansions of Certain Modular Forms of Positive Dimension

1. A definition of a modular form of positive dimension has been given in a paper by Rademacher and the author.2 In that paper we found the Fourier expansions of those forms F(r) which belong to the full modular group and which have only polar singularities at the parabolic point T == i when measured in the uniformizing variable x e2wriT. In the present paper we shall also restrict ourselves to functions which belong to the full group and which have only polar singularities at -r == io, but in the definition of a modular form we shall omit the restriction that F(T) be analytic in the upper half-plane and shall merely assume that F(7-) has, as singularities in the fundamental region,3 at most a finite number of poles and possibly a polar singularity at ioo. This problem of determining expansions of modular forms having poles in the upper half-plane was partially considered by Hardy and Ramanujan.4 However they considered only forms o-t positive inltegral dimension which have no singularities at the parabolic points. The generalization to forms of real positive dimension presents no difficulties. We have only to introduce the roots of uiiity E(a, b, c, d) and e21iaL in the transformation formulas (1. 11) and (1. 12) below, and to carry them through the analysis. However in order to take care of forms having singularities at ioo we have to evaluate certain integrals which Hardy and Ramanujan were able to eliminate by means of simple estimates. This part of the work is contained in sections 3, 4, 5, and 6. In this paper we shall consider most of the integrals in the r-plane rather than in the x-plane, where x e2w1r. The original path owr of section 2 is taken in the i-plane so that we may avoid the poles of Fl(T) which are easier to treat there than in the x-plane. After this point much of the work could

On the coefficients of certain modular forms belonging to subgroups of the modular group

1. Introducton. The Fourier coefficients of modular forms F(r) of positive dimension have been determinedt for the case in which F(Q) belongs to the full modular group. It is the purpose of this paper to generalize some of those results to the case where F(r) belongs to an arbitrary subgroup of the modular group subject to certain restrictions. By F(r) belonging to a subgroup 7y we mean that F(r) is analytic in the upper half-plane I((r) >0 and that F(r) satisfies a transformation equation