Mathematics

One of the most striking formulas one encounters in elementary number theory is Dirichlet's class number formula. This is a formula that relates the special value of an L-function (analytic data) to the size of the class group (algebraic data). The class group is a group that measures the failure of unique factorization in number fields. It turns out that this phenomenon of special values of L-functions giving arithmetic data is a very general one. We will discuss Dirichlet's class number formula, then move on to look at the analogous statement for elliptic curves, namely the conjecture of Birch and Swinnerton-Dyer. If time permits we will briefly discuss how these are both special cases of Bloch-Kato conjecture, a conjecture that relates special values of L-functions to arithmetic data in the context of motives.