A converse theorem for second-order modular forms of level N

… , Department of Mathematical Sciences, University of …, 2003

Nagoya Math. J, 2008

Abstract. We study the Fourier coefficients of generalized modular forms f(τ) of integral weight k on subgroups Γ of finite index in the modular group. We establish two Theorems asserting that f(τ) is constant if k = 0, f(τ) has empty divisor, and the Fourier coefficients have certain ...

Pure and Applied Mathematics Quarterly, 2008

Proceedings of the American Mathematical Society, 2012

Let ρ : SL(2, Z) → GL(2, C) be an irreducible representation of the modular group such that ρ(T ) has finite order N . We study holomorphic vector-valued modular forms F (τ ) of integral weight associated to ρ which have rational Fourier coefficients. (These span the complex space of all integral weight vectorvalued modular forms associated to ρ.) As a special case of the main Theorem, we prove that if N does not divide 120 then every nonzero F (τ ) has Fourier coefficients with unbounded denominators.

Mathematische Annalen, 1977

Archiv der Mathematik, 2014

In this note, we show that the algebraicity of the Fourier coefficients of half-integral weight modular forms can be determined by checking the algebraicity of the first few of them. We also give a necessary and sufficient condition for a half-integral weight modular form to be in Kohnen's +-subspace by considering only finitely many terms.

Mathematical Research Letters, 2000

It is a classical observation of Serre that the Hecke algebra acts locally nilpo- tently on the graded ring of modular forms modulo 2 for the full modular group. Here we consider the problem of classifying spaces of modular forms for which this phenomenon continues to hold. We give a number of consequences of this investigation as they relate to quadratic forms, partition functions, and central values of twisted modular L-functions.

The Ramanujan Journal, 2006

1997

We start with a brief overview of the necessary theory: Given any cusp form f=∑ n≥ 1 an (f) qn of weight k, we denote by L (f, s) the L-function of f. For Re (s)> k/2+ 1, the value of L (f, s) is given by L (f, s)=∑ n≥ 1 an (f) ns and, one can show that L (f, s) has analytic continuation to the entire complex plane. The value of L (f, s) at s= k/2 will be of particular interest to us, and we will refer to this value as the central critical value of L (f, s).