Infinite product exponents for modular forms

Asra Ali,Nitya Mani

doi:10.1007/s40993-016-0050-x

Asra Ali, Nitya Mani

Open Access

https://doi.org/10.1007/s40993-016-0050-x

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Abstract

Recently, Choi obtained a description of the coefficients of the infinite product expansions of meromorphic modular forms over $$\Gamma _0(N)$$ Γ 0 ( N ) . Using this result, we provide some bounds on these infinite product coefficients for holomorphic modular forms. We give an exponential upper bound for the growth of these coefficients. We show that this bound is also a lower bound in the case that the genus of the associated modular curve $$X_0(N)$$ X 0 ( N ) is 0 or 1.

Highlights

The study of modular forms is a rich and deep subject with connections to mathematics ranging from partitions to elliptic curves
Examinations of modular forms are often informed by their divisors, most determined from the infinite product expansion of the modular form
Consider the infinite product expansions of a family of such modular forms intimately connected to a modular form called the Dedekind η-function

Summary

Introduction

The study of modular forms is a rich and deep subject with connections to mathematics ranging from partitions to elliptic curves. The Dedekind η-function is a weight 1/2 modular form defined by η(z) = q1/24 (1 − qm). Some holomorphic modular forms can be expressed as an eta-quotient, giving the values c(m) in their infinite product expansion. Bruinier et al first derived an expression for c(m) as a function of m for an associated meromorphic modular form defined over SL2(Z) (see [3]).

Results

Conclusion