Zeros of modular forms of half integral weight

Amanda Folsom,Paul Jenkins

doi:10.1007/s40993-016-0054-6

Amanda Folsom, Paul Jenkins

Open Access

https://doi.org/10.1007/s40993-016-0054-6

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Abstract

We study canonical bases for spaces of weakly holomorphic modular forms of level 4 and weights in $$\mathbb {Z}+\frac{1}{2}$$ and show that almost all modular forms in these bases have the property that many of their zeros in a fundamental domain for $$\Gamma _0(4)$$ lie on a lower boundary arc of the fundamental domain. Additionally, we show that at many places on this arc, the generating function for Hurwitz class numbers is equal to a particular mock modular Poincare series, and show that for positive weights, a particular set of Fourier coefficients of cusp forms in this canonical basis cannot simultaneously vanish.

Highlights

In studying functions of a complex variable, a natural problem is to determine the locations of the zeros of the functions
We show in that each of these basis elements fk,m(τ zeros in a fundamental domain for
Poincaré series of level 1 and even integer weight at least 4, and gave an explicit bound on the number of zeros lying on the intersection of the standard fundamental domain with the boundary of the unit disk

Summary

Introduction

In studying functions of a complex variable, a natural problem is to determine the locations of the zeros of the functions. For the spaces Mk (N ) of weakly holomorphic modular forms of integer weight k and level N = 2, 3, 4 with poles only at the cusp at ∞, showing that many of the zeros of the basis element fk(,Nm)(τ ) lie on an appropriate arc if m is large enough. A weak Maass form is a smooth complex function defined on the upper half of the complex plane which transforms like a modular form, but is not necessarily holomorphic Such a function must be annihilated by a Laplacian operator, and satisfy suitable growth conditions in the cusps. Poincaré series of level 1 and even integer weight at least 4, and gave an explicit bound on the number of zeros lying on the intersection of the standard fundamental domain with the boundary of the unit disk. The main argument in the proof of Theorem 1.1 appears in Sect. 3, with associated computations appearing in Sects. 4, 5, and 6

Definitions

Residue sums

The integral weight case