Analogies between the BSD Conjecture and the Analytic Class Number Formula
Summary
One of the central objects in number theory are the so-called L-series. For instance, on the rational numbers we have the famous Riemann zeta function. This function has a generalization to an arbitrary number field K, which is called the Dedekind zeta function. This function has an analytic continuation to the entire complex plane with a simple pole at s=1. Its residue at s=1 involves many of the basic invariants of the number field. For instance the regulator, the class group, the finite torsion subgroup of the ring of integers, and the discriminant of the number field appear in the residue. In other words, this function encodes a lot of fundamental information about the number field.
The L-series of an elliptic curve defined over a number field is the analogue of the Dedekind zeta function for a number field. Both series can be defined by an Euler product, i.e., a product indexed by the primes, on a part of the complex plane. It is conjectured that this function also has an analytic continuation to the entire complex plane. It is moreover conjectured by Bryan John Birch and Peter Swinnerton-Dyer that it has a zero of order equal to the rank of the elliptic curve at s=1. The first non-zero coefficient of the corresponding Taylor expansion at s=1 is conjectured to consist of multiple basic invariants concerning the set of global points on the elliptic curve, in particular the regulator, the Tate-Shafarevich group, the torsion subgroup of the global points, and the period of the elliptic curve.
In this thesis we will provide the required theory for understanding the two formulae, and to obtain as many similarities as possible between them. The obtained analogies could function as a helping hand for a better understanding of the BSD conjecture.