Zeros of $E_2$, quasi-periods, and Schwarz maps

Abstract

We study the zeros of the Eisenstein series $E_2$, their distribution near rational cusps, and the values of modular functions at these zeros. Using Watson's inversion formula, we prove that the number of zeros in $0\leq\operatorname{Re}τ<1$ with imaginary part at least $Y$ satisfies \[ N(Y)=\frac{1}{2πY} +O\!\left(Y^{-1/2}\log\frac1Y\right) \qquad(Y\to0^+). \] We also determine the limiting distribution of the scaled displacements from the cusps and prove that two zeros whose difference is rational differ by an integer. At every zero $ρ$, classical transcendence theorems imply that both $j(ρ)$ and $λ(ρ)$ are transcendental. We use this fact to obtain an exact multiplicity formula for arithmetic quasimodular forms at arbitrary congruence level and to show that all ordinary derivatives of arithmetic modular functions on congruence subgroups, evaluated at $ρ$, are algebraic over $\overline{\mathbb Q}(j(ρ),j'(ρ))$. We then relate Watson's formula to the Schwarz map of a rational transform of the Legendre hypergeometric equation. Finally, we give conditional results on the transcendence of $ρ$ under modular Schanuel and period conjectures, and determine why the associated analytic subgroup fails the rank condition in Waldschmidt's theorem.