Zeros of weakly holomorphic modular forms of levels 2 and 3

Paul Jenkins; Sharon Anne Garthwaite

arxiv: 1205.7050 · v2 · pith:RTNYP7XKnew · submitted 2012-05-31 · 🧮 math.NT

Zeros of weakly holomorphic modular forms of levels 2 and 3

Sharon Anne Garthwaite , Paul Jenkins This is my paper

classification 🧮 math.NT

keywords formsholomorphicmodularsharpbasisdomainfundamentalweakly

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Let $M_k^\sharp(N)$ be the space of weakly holomorphic modular forms for $\Gamma_0(N)$ 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.

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Zeros of certain combinations of Eisenstein series of weight 2k, 3k, and k + l
math.NT 2019-07 unverdicted novelty 4.0

For sufficiently large k and l, all zeros of E_k² + E_{2k}, E_k³ + E_{3k}, and E_k E_l + E_{k+l} in the fundamental domain lie on the arc A = {e^{iθ} : π/2 ≤ θ ≤ 2π/3}.