Journal of the Ramanujan Mathematical Society
Volume 39, Issue 1, March 2024 pp. 21–31.
On the zeros of a certain family of weakly modular forms
Authors:
N. Saradha
Author institution:
INSA Senior Scientist, DAE-Center for Excellence in Basic Sciences, University of Mumbai, Kalina Campus,
Mumbai 400 098, India.
Summary:
Let ℓ ≥ 0, m ≥ 0 and k′ ∈ {0,4,6,8,10,14}. For h
≤ 1, let k{h} = 12h+k′. Let E{k}(z) denote the Eisenstein series of
weight k and F{k,D}(z) denote the generalized Faber polynomials of degree D =
ℓ + m if k = 12ℓ + k′. In this paper we consider the weakly
modular form G(t) {k,m} (z) = E{k′} (z) Δ t (z)F{k, D}(j(z)) + F{0,m} (
j(z)) ∑ {h=1} {t} a{h} E {kh} (z) Δ {t − h} (z), t ≤ ℓ
with a{h} ∈ ℝ. Under suitable conditions on ah, it is shown that all the zeros
of G(t) k,m(z), in the standard fundamental domain for the action of SL(2,ℤ)
on the upper half plane, lie on the arc A := {e{iθ} : π/2 ≤ θ
≤ 2π/3. Further, the arithmetic nature of the zeros are also
discussed.
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