2017/07/16 by Chen, Zhijie, Lin, Chang-Shou · 2 citations
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1707.04804
In this paper, we completely determine the critical points of the normalized Eisenstein series E2(τ) of weight 2. Although E2(τ) is not a modular form, our result shows that E2(τ) has at most one critical point in every fundamental domain of Γ0(2). We also give a criteria for a fundamental domain containing a critical point of E2(τ). Furthermore, under the Möbius transformation of Γ0(2) action, all critical points can be mapped into the basic fundamental domain F0 and their images are contained densely on three smooth curves. A geometric interpretation of these smooth curves is also given. It turns out that these smooth curves coincide with the degeneracy curves of trivial critical points of a multiple Green function related to flat tori.