Elliptic Curves, the Finiteness Theorem of Shafarevič

Abstract

Instead of giving an introduction we refer to an arithmetic-geometric part of the theory of elliptic curves. Let ⋀ be a lattice in ℂ, that means a discrete additive subgroup of (ℤ)-rank 2. Two lattices ⋀ and ⋀′ in ℂ are said to be equivalent, if there is a complex number α ≠ 0 such that ⋀′ = α⋀. Each of our lattices is equivalent to a lattice ⋀_τ = ℤ + ℤτ with

$$\tau \in \mathbb{H} = \left\{ {z \in \mathbb{C};Im z > 0} \right\}.$$