The Analytic Class Number Formula and the Birch and Swinnerton-Dyer Conjecture

Let K be a number field, let OK be the ring of integers, let K be an algebraic closure of K and let OK be the ring of integers of K. Let M 0 K be the set of finite places and let M∞ K be the set of infinite places. Let Kv be the completion of K at v and let Ov be the ring of integers of Kv. Let ℘v, kv, qv be the maximal ideal of Ov, the residue field Ov/℘v and the size of the residue field |kv|, respectively. The r real infinite places v correspond to embeddings ik : K ↪→ R and the s complex infinite places v corresponds to embeddings jk : K ↪→ C. For each finite place v, let v(x) be the valuation of x ∈ Kv at v. If v is infinite, let v = log |σ(x)|, where σ is the embedding associated to v. For finite v, let ev and fv be the ramification and inertia index of K at v. If v is real, let ev = fv = 1 and if v is complex, let ev = 1, fv = 2. Theorem 1.1 (Dirichlet Unit Theorem). The unit group O× K is a rank r+ s− 1 Z-module with O× K ∼= μ(K)× r+s−1 ⊕