Generalized modular forms

The theory of “generalized modular forms,” initiated here, grows naturally out of questions inherent in rational conformal field theory. The latter physical theory studies q-series arising as trace functions (or partition functions), which generate a finite-dimensional SL(2,Z)-module. It is a natural step to investigate whether these q-series are in fact modular forms in the classical sense. As it turns out, the existence of the module does not, of itself, guarantee that this is so. Indeed, our Theorem 1 shows that such q-series of necessity behave like modular forms in every respect, with the important exception that the multiplier system need not be of absolute value one. The Supplement to Theorem 1 shows that such q-series are classical modular forms exactly when the scalars relating the q-series generators of the module have absolute value one. That is, the SL(2,Z)-module in question is unitary. (There is the further restriction that the associated representation is monomial.) We prove as well that there exist generalized modular forms which are not classical modular forms. (Hence, as asserted above, the q-series need not be classical modular forms.)Beyond Theorem 1 and its Supplement, which serve to relate our generalized modular forms to classical modular forms (and thus justify the name), this work develops a number of their fundamental properties. Among these are a basic result relating generalized modular forms to classical modular forms of weight 2 and so, as well, to abelian integrals. Further, we prove two general existence results and a complete characterization of weight k generalized modular forms in terms of generalized modular forms of weight 0 and classical modular forms of weight k.     

Metacyclic groups and modular forms

In the paper we find the metacyclic groups of the form 〈a, b:a ^m=e, b ^s=e, b ⁻¹ ab=a ^r〉, wherem=3, 4, 5, 7, 11, 23, such that the modular forms associated with all elements of these groups by some faithful representation are multiplicative η-products. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 163–173, February, 2000.     

The joint weight enumerators and Siegel modular forms

The weight enumerator of a binary doubly even self-dual code is an isobaric polynomial in the two generators of the ring of invariants of a certain group of order 192. The aim of this note is to study the ring of coefficients of that polynomial, both for standard and joint weight enumerators.

     

On product expansions of generalized modular forms

We give examples of generalized modular forms with the property that their divisors are supported at the cusps and the exponents in their q-product expansions take infinitely many values.      

Siegel modular forms of degree 2 over rings

Based on moduli theory of abelian varieties, extending Igusa's result on Siegel modular forms over C, we describe the ring of Siegel full modular forms of degree 2 over any Z-algebra in which 6 is invertible.     

On convolutions of Siegel modular forms

In this article we study a Rankin‐Selberg convolution of n complex variables for pairs of degree n Siegel cusp forms. We establish its analytic continuation to ?ⁿ, determine its functional equations and find its singular curves. Also, we introduce and get similar results for a convolution of degree n Jacobi cusp forms. Furthermore, we show how the relation of a Siegel cusp form and its Fourier‐Jacobi coefficients is reflected in a particular relation connecting the two convolutions studied in this paper. As a consequence, the Dirichlet series introduced by Kalinin [7] and Yamazaki [19] are obtained as particular cases. As another application we generalize to any degree the estimate on the size of Fourier coefficients given in [14]. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Pareto equilibria in vector-valued extensive form games

In this paper we investigate the existence of Pareto equilibria in vector-valued extensive form games. In particular we show that every vector-valued extensive form game with perfect information has at least one subgame perfect Pareto equilibrium in pure strategies. If one tries to prove this and develop a vector-valued backward induction procedure in analogy to the real-valued one, one sees that different effects may occur which thus have to be taken into account: First, suppose the deciding player at a nonterminal node makes a choice such that the equilibrium payoff vector of the subgame he would enter is undominated under the equilibrium payoff vectors of the other subgames he might enter. Then this choice need not to lead to a Pareto equilibrium. Second, suppose at a nonterminal node a chance move may arise. The combination of the Pareto equilibria of the subgames to give a strategy combination of the entire game need not be a Pareto equilibrium of the entire game.     

Weight reduction for mod ? Bianchi modular forms

Let K be an imaginary quadratic field with class number one and ? be a rational prime that splits in K. We prove that mod ?, a system of Hecke eigenvalues occurring in the first cohomology group of some congruence subgroup Γ of SL₂(OK) can be realized, up to twist, in the first cohomology with trivial coefficients after increasing the level of Γ by (?).     

Petersson norms and liftings of Hilbert modular forms

We prove the algebraicity of the ratio of the Petersson norm of a holomorphic Hilbert modular form over a totally real number field and the norm of its Saito-Kurokawa lift. We prove a similar result for the Ikeda lift of an elliptic modular form. In order to obtain these we combine some results on local symplectic groups to generalize a special value of the standard L-function attached to a Siegel-Hilbert cuspform.     

A note on Siegel modular forms

This paper gives a new identification for Siegel modular forms with respect to any congruence subgroup by investigating the properties of their Fourier-Jacobi expansions, and verifies a comparison theorem for the dimensions of the spaces Skn (Γn) and J0k, 1 (Γn) with small weight k. These results can be used to estimate the dimension of the space of modular forms.     

Multiplicities of representations in spaces of modular forms

This paper shows that for a given irreducible representation of , the two functions dim( ) and dim( ) of are almost linear functions.

     

Coefficients of Drinfeld modular forms and Hecke operators

Consider the space of Drinfeld modular forms of fixed weight and type for Γ₀(n)⊂GL₂(Fq[T]). It has a linear form bn, given by the coefficient of tm+n(q−1) in the power series expansion of a type m modular form at the cusp infinity, with respect to the uniformizer t. It also has an action of a Hecke algebra. Our aim is to study the Hecke module spanned by b₁. We give elements in the Hecke annihilator of b₁. Some of them are expected to be nontrivial and such a phenomenon does not occur for classical modular forms. Moreover, we show that the Hecke module considered is spanned by coefficients bn, where n runs through an infinite set of integers. As a consequence, for any Drinfeld Hecke eigenform, we can compute explicitly certain coefficients in terms of the eigenvalues. We give an application to coefficients of the Drinfeld Hecke eigenform h.     

On singular modular forms for principal congruence subgroups

We show that spaces of vector–valued singular modular forms for principal congruence subgroups of the symplectic group Sp(n,ℤ) of integral weight are generated by suitable finite dimensional families of Siegel theta series. This is obtained as an application of some results concerning the action of trace operators on non–homogeneous theta series.     

Clifford algebras and vector-valued rational forms II

We demonstrate how the use of Clifford algebras in the theory of vector-valued rational forms leads to practical recurrence relations which do not involve representations of the algebras     

Spaces of vector-valued Dirichlet series in a half plane

Dirichlet series with real frequencies which represent entire functions on the complex plane C have been investigated by many authors. Several properties such as topological structures, linear continuous functionals, and bases have been considered. Le Hai Khoi derived some results with Dirichlet series having negative real frequencies which represent holomorphic functions in a half plane. In the present paper, we have obtained some properties of holomorphic Dirichlet series having positive exponents, whose coefficients belong to a Banach algebra.     

Lyapunov exponents of solutions to linear differential equations with periodic forcing functions

We give Lyapunov exponents of solutions to linear differential equations of the form x^′=Ax+f(t), where A is a complex matrix and f(t) is a τ-periodic continuous function. Notice that f(t) is not “small” as t→∞. The proof is essentially based on a representation [J. Kato, T. Naito, J.S. Shin, A characterization of solutions in linear differential equations with periodic forcing functions, J. Difference Equ. Appl. 11 (2005) 1-19] of solutions to the above equation.     

On some properties of polynomials related to hypergeometric modular forms

We find the discriminants, Galois groups, and prove the irreducibility of certain hypergeometric polynomials, which are closely related to modular forms and supersingular elliptic curves. 2000 Mathematics Subject Classification Primary—33C45; Secondary—11F11