Marvin Knopp

Generalized modular forms

Abstract 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 .

On the Signs of Fourier Coefficients of Cusp Forms

Let Γ be a discrete subgroup of SL(2, ℝ) with a fundamental region of finite hyperbolic volume. (Then, Γ is a finitely generated Fuchsian group of the first kind.) Let

On Dirichlet series satisfying Riemann's functional equation

f\left( z \right) = \sum\limits_{n + \kappa > 0} {a{{\left( n \right)}^{e2\pi i\left( {n + \kappa } \right)z/\lambda }}} ,z \in H.

Polynomial automorphic forms and nondiscontinuous groups

be a nontrivial cusp form, with multiplier system, with respect to Γ. Responding to a question of Geoffrey Mason, the authors present simple proofs of the following two results, under natural restrictions upon Γ.

PARABOLIC GENERALIZED MODULAR FORMS AND THEIR CHARACTERS

Emil Grosswald was a mathematician of great accomplishment and remarkable breadth of vision. This volume pays tribute to the span of his mathematical interests, which is reflected in the wide range of papers collected here. With contributions by leading contemporary researchers in number theory, modular functions, combinatorics, and related analysis, this book will interest graduate students and specialists in these fields. The high quality of the articles and their close connection to current research trends make this volume a must for any mathematics library.

Some New Old-Fashioned Modular Identities

SummaryAccording to convention, Hamburgers theorem (1921) says-roughly-that Riemanns ζ(s) is uniquely determined by its functional equation. In 1944 Hecke pointed out that there are two distinct versions of Hamburgers theorem. Heckes remark has led me, in examining just how “rough” the convention is, to prove that, with a weakening of certain auxiliary conditions, there are infinitely many linearly independent solutions of Riemanns functional equation (Theorem 1). In Theorem 1, as in Hamburgers theorem, the “weight” parameter is 1/2. In Theorem 2 we obtain stronger results when this parameter is ≧2: a “Mittag-Leffler” theorem for Dirichlet series with functional equations.

Sums of Squares and the Preservation of Modularity under Congruence Restrictions

By using Stokess theorem, we prove an Eichler cohomology theorem for generalized modular forms with some restrictions on the relevant multiplier systems.

Eichler cohomology for generalized modular forms ii

is said to be an automorphic function with respect to F. (The usual definition of automorphic function imposes a growth condition on f(i) as z approaches certain points of the real axis from within an angle. The same remark is applicable in connection with the definition of automorphic form, given below. It will be apparent from the nature of the results given here that this growth condition plays no role in the present context.) Let r be a real number. A functionf(z), meromorphic in X, is said to be an automorphicform of dimension r on F, vith multiplier system v, provided