Executive Summary : | The Eisenstein series E_k, a modular form of weight k for the full modular group SL(2,Z), has been studied by numerous mathematicians. Wohlfahrt conjectured that the zeros lie on the unit circle of the standard fundamental domain F for any weight. This was proved by F.K.C. Rankin and H.P.F. Swinnerton-Dyer in 1970. Kohnen in 2003 proved that all the zeros of E_k are transcendental other than \rho and i. Gekeler observed an interlacing property between any two zeros of \cos(k \theta/2) and vice versa, leading to the prediction that the zeros of E_k and E_{k+12} interlace on the open arc on the unit circle. A natural extension of the Eisenstein series is the Poincaré series G_k(z,m), where the index 0 Poincaré series G_k(z,0)=E_k(z). For positive index m, G_k(z,m) is a cusp form, while for negative index m, it is a weakly holomorphic modular form. R.A. Rankin in 1982 proved that for non-positive m non-positive, G_k(z,m) has all its zeros in the standard fundamental domain lying on the unit circle. However, finding the zeros of cusp forms, particularly for Poincaré cusp forms, is difficult. A philosophical reasoning can be given in light of quantum unique ergodicity, which states that the zeros of cuspidal Hecke eigenforms get uniformly distributed in the fundamental domain as weight increases. Studying the zeros of Poincaré series is a delicate matter, and this project aims to unearth many unknown facets of the zeros of Poincaré series. |