Some notes on elliptic curves, L-functions and modular forms

Abstract

In this thesis, we mainly focus on three subjects: First, we shall be establishing the connection between congruent numbers and a certain family of elliptic curves, and describing some basic properties of elliptic curves. Second, we shall focus on determining "rank" which is much more difficult than finding the torsion group. Finally, we shall describe some basic properties of modular form and construct a quotient space $Γ_0(N)＼\mathfrak{H}^*$ and give a formula to calculate cusps and genera.