The Beilinson Conjecture for curves

Associated to a curve over a number field is an L-function, which encodes much information about the arithmetic of the curve. The Birch Swinnerton-Dyer conjecture relates the rank of an elliptic curve to the order of vanishing of the L-function at s=1 and gives its leading coefficient in terms of several important arithmetical invariants of the curve. The Beilinson conjecture suggests that there is a similar expression for values of the L-function at other nonnegative integers s, as well as for curves of higher genus, in terms of an invariant coming from algebraic K theory. We'll look at the case s=2 here and see a possible approach for numerical verification in specific cases, using torsion divisors on curves.