2025/06/04 by Zhijie Chen, Chen, Zhijie
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Advanced Mathematical Identities
paper · pdf · doi:10.48550/arxiv.2506.03475
In previous works joint with Lin, we proved that the Eisenstein series E4 (resp. E2) has at most one critical point in every fundamental domain γ(F0) of Γ0(2), where γ(F0) are translates of the basic fundamental domain F0 via the Möbius transformation of γ∈Γ0(2). But the method can not work for the Eisenstein series E6. In this paper, we develop a new approach to show that E6'(τ) has exactly either 1 or 2 zeros in every fundamental domain γ(F0) of Γ0(2). A criterion for γ(F0) containing exactly 2 zeros is also given. Furthermore, by mapping all zeros of E6'(τ) into F0 via the Möbius transformations of Γ0(2) action, the images give rise to a dense subset of the union of three disjoint smooth curves in F0. A monodromy interpretation of these curves from a complex linear ODE is also given. As a consequence, we give a complete description of the distribution of the zeros of E6'(τ) in fundamental domains of SL(2,ℤ).