On zeros of quasi-modular forms

Abstract

Several authors have studied the nature and location of zeros of modular forms for the full modular group Γ and other congruence subgroups. In this paper, we investigate the zeros of certain quasimodular forms for Γ . In particular, we study the transcendence and existence of infinitely many Γ -inequivalent zeros of these quasi-modular forms. We also estimate the number of such zeros in Siegel sets, motivated by a recent work of Ghosh and Sarnak.

Critical points of Eisenstein series

“…Remark 1. − These results are consistent with the following fact : since E k is a modular form of weight k which does not vanish at infinity, the weighted number of its zeros in H modulo SL 2 (Z) (counted with multiplicities, with weight 1 2 for those in the orbit of e πi/2 , 1 3 for those in the orbit of e iπ/3 , and 1 otherwise) has to be equal to k 12 . Remark 2.…”

“…Theorem 1.− The zeros of E k in D are simple, and have real part either 1 2 or − 1 2 . Those with real part 1 2 are the translates by 1 of those with real part − 1 2 .…”

“…The second assertion follows from the fact that E k (z + 1) = E k (z). So let us restrict our attention to the zeros of E k in D with real part 1 2 , i.e. located on the closed half-line 1 2…”

“…So let us restrict our attention to the zeros of E k in D with real part 1 2 , i.e. located on the closed half-line 1 2…”

“…2 is a zero of E k if and only if k ≡ 2 mod 6. b) The number of zeros of E k in the open half-line 1 2 + i] 4 6 ]. Assertion b) can be rephrased by saying that the number of zeros of E k in ].…”