Bound on Torsion Points on Elliptic Curves over Number Fields

Abstract

We propose a modified Rankin Selberg convolution, since the functional equation of Rankin-Selberg convolution for arbitrary cusp form doesn’t respect critical line s = 1/2. We extend a result of Goldfeld and Hoffstein about the congruence of cusp forms in ’new’ space under the assumption of the Riemann Hypothesis for modified Rankin-Selberg convolution.We prove Merel’s conjecture which states that the Hecke operators act linearly independently on the winding cycle in the homology group H1(X0(N), Z). We also provide an improvement on the bound of number of Hecke Operators which acts linearly independently on the space of cusp forms using estimates on Kloosterman Sums. It also gives linear independence of Poincare series.