Regulators of Number fields and Elliptic Curves
There is supposed to be a strong analogy between the arithmetic of number fields and the arithmetic of elliptic curves. One facet of this analogy is given by the class number formula for the leading term of the Dedekind Zeta function of K on the one hand and the conjectural formula for the leading term of the L function associated to an elliptic curve defined over a number field on the other. One can look at the terms that show up in these two expressions and more or less 'pair' them off as analogous quantities.
One of these pairs consists of the respective regulators. The regulator of a number field K is defined by taking a basis for the free part of the units of the ring of integers and then using the embeddings of K into C, taking logs of absolute values, etc. The regulator of an elliptic curve is defined by taking a basis for the free part of the K points of the curve and then computing the determinant of a symmetric matrix built out of this basis using a height pairing.
My question is: is there some way to view the number field regulator as coming from some kind of symmetric pairing on the units of K? Alternatively, just give some reasoning why these constructions appear so different.