[Paper Review] Zeros of weakly holomorphic modular forms of levels 2 and 3

Sharon Anne Garthwaite, Paul M. Jenkins|arXiv (Cornell University)|May 31, 2012

Holomorphic and Operator Theory18 references4 citations

TL;DR

This paper studies the zero locations of weakly holomorphic modular forms of levels 2 and 3, using contour integration and generating functions to show that for large weights and levels, the majority of nontrivial zeros in the fundamental domain lie on the lower boundary arc. The method relies on approximating modular forms by cosine functions via Cauchy's residue theorem, with exponential decay in error terms for sufficiently large $ n $, leading to precise zero localization results.

ABSTRACT

Let $M_k^\sharp(N)$ be the space of weakly holomorphic modular forms for $Γ_0(N)$ that are holomorphic at all cusps except possibly at $\infty$. We study a canonical basis for $M_k^\sharp(2)$ and $M_k^\sharp(3)$ and prove that almost all modular forms in this basis have the property that the majority of their zeros in a fundamental domain lie on a lower boundary arc of the fundamental domain.

Motivation & Objective

• To investigate the distribution of zeros of weakly holomorphic modular forms in the fundamental domains of $ \Gamma_0(2) $ and $ \Gamma_0(3) $.
• To extend techniques from Eisenstein series and Hecke eigenforms to canonical bases of $ M_k^\sharp(2) $ and $ M_k^\sharp(3) $.
• To determine whether a majority of zeros in the fundamental domain lie on a specific boundary arc, particularly the lower boundary.
• To establish conditions under which the zeros of these modular forms are asymptotically localized on a circular arc via analytic approximation.
• To explore the feasibility of generalizing the method to higher levels $ N = 5, 7, 13 $, though with increased geometric complexity.

Proposed method

• Construct canonical bases $ f_{k,n}^{(N)}(z) $ and $ g_{k,n}^{(N)}(z) $ for $ M_k^\sharp(N) $, defined by initial Fourier coefficients $ q^{-n} + O(q^{2\ell + \lfloor k'/3 \rfloor + 1}) $.
• Use a generating function for the basis elements, analogous to the $ j $-function generating function, to relate modular forms to contour integrals.
• Apply Cauchy’s residue theorem to express the modular form as a contour integral involving a generalized generating function.
• Choose a horizontal contour below the points $ z = -1/N + (1/N)e^{i\theta} $ and $ z/(Nz+1) $, ensuring it avoids all other images under $ \Gamma_0(N) $.
• Bound the absolute value of the integral remainder term by factoring out an exponential decay term $ e^{-\pi n(\frac{2}{N}\sin\theta - 2A')} $, which dominates for large $ n $.
• Prove that the difference between the weighted modular form and a cosine function is less than 2, implying that zeros of the form are close to those of the cosine function, which lie on the arc $ |z + 1/N| = 1/N $.

Experimental results

Research questions

• RQ1 Do the majority of nontrivial zeros of weakly holomorphic modular forms in $ M_k^\sharp(2) $ lie on the lower boundary arc of the fundamental domain?
• RQ2 Can the method of approximating modular forms by cosine functions via contour integration be extended to level 3 modular forms?
• RQ3 For which values of $ n $ and $ \ell $ does the error in the approximation become small enough to guarantee zero localization on the boundary arc?
• RQ4 Is it possible to generalize this zero localization technique to higher levels $ N = 5, 7, 13 $, given the increased complexity of the fundamental domain?
• RQ5 What is the asymptotic proportion of zeros that lie on the lower boundary arc as $ n \to \infty $?

Key findings

• For $ N = 2 $, if $ \ell \geq 0 $ and $ n \geq 14\ell + 7 $, or $ \ell < 0 $ and $ n \geq 15|\ell| + 7 $, at least $ \lfloor \frac{k}{6} + n\frac{\sqrt{3}}{2} \rfloor $ of the $ n + \lfloor \frac{k}{4} \rfloor - 1 $ nontrivial zeros lie on the lower boundary arc.
• For $ N = 3 $, as the contour height approaches $ \frac{1}{9} $ and $ z $ approaches the contour, the number of provably located zeros on the lower boundary approaches $ 0.9618n + 0.2792k $.
• The proportion of zeros on the lower boundary arc approaches approximately 96.18% of $ n $ when $ k $ is fixed and $ n \to \infty $, indicating a strong localization trend.
• The method relies on exponential decay in the error term due to the choice of contour height, ensuring the difference between the modular form and the cosine approximation is less than 2 for large $ n $.
• The result holds for both $ f_{k,n}^{(3)}(z) $ and $ g_{k,n}^{(3)}(z) $, with the same asymptotic zero distribution.
• The technique fails to generalize directly to $ N = 5, 7, 13 $ due to the increased geometric complexity of the fundamental domain, which hinders contour selection.