Automorphic functions on the upper half plane, especially modular functions

The goal of this paper is to construct infinite-dimensional Lie algebras by using infinite product identities and to use these Lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and Lie algebras. We expect the Lie algebras that we construct to manifest as algebras of physical states in an orbifold conformal field theory (yet to be fully constructed) with symmetries given by the monster simple group.\n¶\nWe introduce vector-valued modular functions attached to families of modular functions of different levels, and we prove infinite product identities for a distinguished class of automorphic functions on a product of two half-planes. We recast this result using the Borcherds–Harvey–Moore singular theta-lift and show that the vector-valued functions attached to completely replicable modular functions with integer coefficients lift to automorphic functions with infinite product expansions at all cusps.\n¶\nFor each element of the monster simple group, we construct an infinite-dimensional Lie algebra, such that its denominator formula is an infinite product expansion of the automorphic function arising from that element’s McKay–Thompson series. These Lie algebras have the unusual property that their simple roots and all root multiplicities are known. We show that under certain hypotheses, characters of groups acting on these Lie algebras form functions on the upper half-plane that are either constant or invariant under a genus zero congruence group.

In this thesis, we introduce an SL(2,R)-action on the chiral de Rham complex on the upper half plane, and study the vertex subalgebra Ω<sup>ch</sup>(H,Γ) of Γ-invariant global sections which are holomorphic at all the cusps, for any congruence subgroup Γ. The thesis can be divided into three parts. The first part includes a recollection on the theory of vertex algebras and modular forms. The second part consists of a brief review of the construction of chiral de Rham complex by Malikov, Schechtman and Vaintrob, which will be applied to the upper half plane. We consider the vertex algebra Ω<sup>ch</sup>(H,Γ) associated to an arbitrary congruence subgroup Γ, and compute its character formula. We also give an explicit formula for the lifting of modular forms to Ω<sup>ch</sup>(H, Γ), and the lifting formula is essentially unique and universal. The last part discusses the relations between the modified Rankin-Cohen bracket and the elements in Ω<sup>ch</sup>(H,Γ), and some further properties about Ω<sup>ch</sup>(H,Γ).

7. Modular functions. In this section necessary facts concerning the modular group and modular functions will be briefly summarized. More complete expositions of this material may be found in [3], [5], [6], [9], [15], [16], and [19]. Let r be the modular group, the group of all 2 X 2 matrices with rational integer entries and determinant equal to one. The elements S = (0 1) and T = (I 1) generate r. To an element A = (b b) E r, one associates a fractional linear transformation A(z) == (az + b)/(cz + d). The matrices A and -A give the same transformation. Let 3C* = 3C U {ioo} U Q where 3C is the upper half plane and Q is the set of rational points on the real axis. A fundamental domain is the set

Fourier series of modular graph functions

The purpose of this paper is to construct modular forms, both for SL27Z (and certain of its congruence subgroups) and for the Hilbert modular group of a real quadratic field. In w 1 we fix a real quadratic field K and even integer k > 2 and construct a series of functions co,,(Zl, z2) (m=0, 1, 2, . . .) which are modular forms of weight k for the Hilbert modular group SL2(9 ((9=ring of integers in K). The form co o is a multiple of the Hecke-Eisenstein series for K, while all of the other co,. are cusp forms. The Fourier expansion of co,. (z 1, z2) is calculated in w 2; each Fourier coefficient is expressed as an infinite sum whose typical term is the product of a finite exponential sum (analogous to a Kloosterman sum) and a Bessel function of order k 1 . The main result is that, for any points z 1 and z 2 in the upper half-plane .~, the numbers m k-1 co,,(z 1, z2) (m= 1, 2, ...) are the Fourier coefficients of a modular form (in one variable) of weight k. More precisely, let D be the discriminant of K, e.=(D/ ) the character of K, and S(D, k, ~) the space of cusp forms of weight k for Fo(D ) with character e; then for fixed z 1, zzc .~, the function

Tests are described for determining whether certain functions are the boundary values of analytic functions taking the upper half plane into itself. In one theorem the functions are real-valued and defined on an open interval and a single point. In other theorems, the functions are defined on a real interval taking values in an arc γ in the upper half plane. In every case these results are closely related to Loewner's theorem in which the putative boundary values are required to be real-valued. As in Loewner's theory, the quadratic inequalities are finite sums involving at most the value of the function and its first derivative at points. These results are closely related to the Pick-Nevanlinna inequalities for functions defined on a sequence of points in the upper half plane; the Pick-Nevanlinna inequalities characterize those functions that are restrictions of analytic functions taking the upper half plane into itself.

Ramanujan famously found congruences like p(5n+4) ≡ 0 mod 5 for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Г1(4) that is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored F-partitions.

Finite upper half planes are finite field analogs of the Poincaré upper half plane. Herein, we review the definition of vector-valued modular forms on finite upper half planes, and present the results and examples of these modular forms for certain subgroups of GL(2,𝔽q).

The following is an expanded version of the notes for a survey lecture on the theta correspondence given at Trichy in January, 1996. The aim of this lecture was to give an idea of this theory and some of the directions in which the theory has been developing. Perhaps some of the material contained in sections 2,3 and 9 is new. The theory of theta correspondence gives one of the few general methods of constructing automorphic forms of groups over number fields, or admissible representations of groups over local fields. The method has its origin in the classical construction of theta functions which are modular forms–perhaps of half integral weight on the upper half plane. Here is a special case of this: Let q be a positive definite quadratic form on a lattice L ⊆ IR, q : L→ Z . Let p be a homogeneous polynomial on IR which is harmonic with respect to q, i.e. ∆qp = 0 where ∆q denotes the Laplacian with respect to q (∆q is the unique homogeneous differential operator of order 2 invariant under the orthogonal group of q). Then,

In this paper, we prove a generalization of a correspondence between holomorphic Jacobi cusp forms of higher degree (matrix index) and elliptic cusp forms obtained by K. Bringmann [Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms, Math. Z.253 (2006) 735–752], for forms of higher levels (for congruence subgroups). To achieve this, we make use of the method adopted by M. Manickam and the first author in Sec. 3 of [On Shimura, Shintani and Eichler–Zagier correspondences, Trans. Amer. Math. Soc.352 (2000) 2601–2617], who obtained similar correspondence in the degree one case. We also derive a similar correspondence in the case of skew-holomorphic Jacobi forms (matrix index and for congruence subgroups). Such results in the degree one case (for the full group) were obtained by N.-P. Skoruppa [Developments in the theory of Jacobi forms, in Automorphic Functions and Their Applications, Khabarovsk, 1988 (Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, 1990), pp. 168–185; Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms, J. Reine Angew. Math.411 (1990) 66–95] and by M. Manickam [Newforms of half-integral weight and some problems on modular forms, Ph.D. thesis, University of Madras (1989)].

We know that the elliptic modular function j(z) gives the birational invariant of the elliptic curve with the analytic modulus z, and that the elliptic modular functions belonging to the congruence-subgroups are obtained from the values of elliptic functions at the points of finite order on elliptic curves. This is not only the origin of those functions, but one of the most essential points to which we may ascribe the significance of elliptic modular functions in number-theory. It is important to generalize these facts, namely, to investigate in a more general case the relation between automorphic functions with one or more variables and systems of algebraic varieties, especially, systems of abelian varieties. The object of this paper is to give some results on this subject. We shall deal with certain systems of polarized abelian varieties parametrized by holomorphic functions and show that there exist meromorphic functions whose values are considered as of the members of the systems (Theorem 1). The theory developed here will be chiefly concerned with systems of abelian varieties with non-trivial endomorphisms, since I have already given elsewhere a theory for Siegel's modular functions [11]. I am particularly interested, in the present paper, in the determination of the fields of definition for fields of automorphic functions. This is the first problem which confronts us when we proceed beyond a formal treatment in the arithmetic theory of automorphic functions. We obtain a criterion (Theorem 2) applicable even in the case of compact fundamental domain where one can not employ Fourier expansions. The last part of the paper is devoted to the theory of a certain type of automorphic functions of one variable known in the literature as functions belonging to indefinite ternary quadratic forms [8], [5]; they occur as moduli of abelian varieties of dimension 2 whose endomorphismrings are isomorphic to an order of an indefinite quaternion algebra. We get in this case the field of rational numbers as a field of definition for the function-field. We can also prove the congruence formulae for the modular correspondences, which give a generalization of what is obtained in [3], [10]. We shall treat these formulae in a subsequent paper together with the theory of the functions belonging to congruence-subgroups. Our method can be applied to other types of automorphic functions, for example, to Hilbert's modular functions, by considering a system of

We give explicit representation of Neumann boundary value problem for Bitsadze equation on the upper half plane. We will also give solution of the inhomogeneous polyanalytic equation arising from Neumann and (n-1) Dirichlet boundary conditions on the upper half plane H.

Let $\{\alpha _k\}_{k=1}^\infty$ be an arbitrary sequence of complex numbers in the upper half plane. We generalize the orthogonal rational functions $\phi _n$ based upon those points and obtain the Nevanlinna measure, together with the Riesz and Poisson kernels, for Caratheodory functions $F(z)$ on the upper half plane. Then, we study the relation between ORFs and their functions of the second kind as well as their interpolation properties. Further, by using a linear transformation, we generate a new class of rational functions and state the necessary conditions for guaranteeing their orthogonality.

We consider complex-valued modular forms on finite upper half planesHqand obtain Fourier expansions of Eisenstein series invariant under the groupsΓ=SL(2,Fp)andGL(2,Fp). The expansions are analogous to those of Maass wave forms on the ordinary Poincaré upper half plane —theK-Bessel functions being replaced by Kloosterman sums.

A griffith crack at the interface of two bonded dissimilar elastic half planes in which body forces are acting