CRITICAL POINTS OF THE CLASSICAL EISENSTEIN SERIES OF WEIGHT TWO

In this paper, we completely determine the critical points of the normalized Eisenstein series E-2(tau) of weight 2. Although E-2(tau) is not a modular form, our result shows that E-2(tau) has at most one critical point in every fundamental domain of the form gamma(F-0) of Gamma(0)(2), where gamma(F-0) are translates of the basic fundamental domain F-0 via the Mobius transformation of gamma is an element of Gamma(0)(2). We also give a criteria for such fundamental domain containing a critical point of E-2(tau). Furthermore, under the Mobius transformations of Gamma(0)(2) action, all critical points can be mapped into the basic fundamental domain F-0 and their images in F-0 give rise to a dense subset of the union of three connected smooth curves in F-0. A geometric interpretation of these smooth curves is also given. It turns out that these curves coincide with the degeneracy curves of trivial critical points of a multiple Green function related to flat tori.