Title: A Cohomological Interpretation of the Class Group
Abstract: The celebrated Class Number Formula, which dates from the nineteenth
century, relates the residue of the Dedekind Zeta function of a number
field to the order of its class group, among its other main invariants.
According to the Birch and Swinnerton-Dyer conjecture, one of the
Millenium Problems, there should be a notably similar formula relating the
main arithmetic invariants of an elliptic curve to a certain derivative of
its L-function. One of these is the (conjecturally finite) order of its
Tate-Shafarevich group, which arises from considering the cohomology of
the Galois group acting on the points of the curve.
In this talk we will compare both formulas term by term, focusing on
giving a cohomological interpretation of the class group, which allows us
to match it with the Tate-Shafarevich group.