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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.

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Let me first give you a heuristic "reason", why the regulator in the class number formula looks different from the regulator in the Birch and Swinnerton-Dyer conjecture. It is often more convenient (and more canonical) to combine the regulator and the torsion term in the Birch and Swinnerton-Dyer conjecture: one chooses a free subgroup $A$ of the Mordell-Weil group of $E(K)$ with finite index and looks at the quantity $R(A)/[E(K):A]^2$, where $R(A)$ is the absolute value of the determinant of the Néron-Tate height pairing on a basis of $A$. As you can easily check, the square in the denominator insures that this quantity is independent of the choice of $A$. Now, in the class number formula, the torsion term is replaced by the number of roots of unity in $K$ and that term is not squared. So for such a canonical formulation to be possible, it is reasonable to expect that not the regulator of the number field should be defined through a symmetric pairing on the units, but its square. Then you could make the same definition as in the elliptic curves case, and it would be independent of the choice of finite index subgroup.

Now, that we have established this, there are several ways to bring the two situations closer together. You could do the naïve thing: take the matrix $M=(\log|u_i|_{v_j})$, where $u_i$ runs over a basis of the free part of the units (or more generally $S$-units for any set of places $S$ which includes the Archimedean ones), and where $v_j$ runs over all but one Archimedean place (or more generally all but one place in $S$), v_0, say. The absolute values have to be suitably normalised (see e.g. Tate's book on Stark's conjecture). Now take the symmetric matrix $MM^{tr}$ and define this to be the matrix of a new pairing on the units. In other words, you would define your symmetric pairing as $$ (u_1,u_2) = \sum_{v\in S\backslash\{v_0\}} \log|u_1|_v\log|u_2|_v. $$ Then, it's clear that you have a symmetric pairing and that the determinant of that pairing with respect to any basis on the free part of the units is $R(K)^2$. As I mentioned above, the square was expected.

Depending on what you want to do, this pairing might not be the best one to consider. For example if now $F/K$ is a finite extension and you consider the analogous pairing on the $S$-units of $F$ and restrict it to $K$, then it's not clear how to compare it to the pairing on $K$. In the elliptic curves case by contrast, the former is $[F:K]$ times the latter. To fix this, we can make the following definition: $$ (u_1,u_2) = \sum_{v\in S} \frac{1}{e_vf_v}\log|u_1|_v\log|u_2|_v, $$ where $e$ and $f$ are the absolute ramification index and residue field degree respectively. With this definition, the compatibility upon restriction to subfields is the same as in the elliptic curves case. On the other hand, the relationship with the actual regulator is slightly less obvious. It is however quite easy to show (and I have done it in http://arxiv.org/abs/0904.2416, Lemma 2.12, in case you are interested in the details) that the determinant of this pairing is given by $$ \frac{\sum e_vf_v}{\prod e_vf_v}R(K)^2, $$ with both the sum and the product again ranging over the places in $S$.

So I guess, the moral is that one shouldn't seek analogies between the BSD-formula and the class number formula but rather between the BSD and the square of the class number formula (note that also sha, which is supposed to be the elliptic curve analogue of the class number, has square order whenever it is finite). There is also a corresponding heuristic on the analytic side.

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I don't think the two regulators are so different. Both the number field regulator and the elliptic curve regulator measure the volume of a parallelopiped. In both cases, the shape in question is the fundamental domain for a lattice made out of integral (resp. rational) points in a commutative group scheme. For the number field, the group scheme is the multiplicative group Gm, and for the elliptic curve, the group scheme is the elliptic curve.

In order to get a lattice in a real vector space with a quadratic form, we need to do some transformations on integral (resp. rational) points. In the case of the number field, one typically takes logarithms of all archimedean absolute values - this changes the multiplicative structure into an additive structure, which is what we want. With the elliptic curve case, one constructs a canonical height pairing in a rather elaborate way. One can also define the regulator for the multiplicative group using heights, but that level of sophistication is unnecessary due to the existence of an analytic "log" homomorphism, and the lack of O(1) error terms.

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I think the major difference between the BSD and the analytic class number formula is that BSD talks about zeroes whereas Class number formula talks about poles. Zeroes and poles behave differently. Also in number fields case we have the geometry of numbers that proves finiteness of class group. But if one takes the definition of the class group to be the kernel of $H^1 (G_K,E)$ to $H^1(G_(K_p), E_p) (just like sha) then its not clear how to show the finiteness. One can show in some special cases adapting the proof of Kolyvagin where the group of units in the number fields have rank 0 or 1. –  Arijit Apr 18 '11 at 4:03
    
Oops I have a typo the target of the map should be the product of all local cohomology groups. –  Arijit Apr 18 '11 at 4:04
    
That is a nice insight. Thank you. –  S. Carnahan Apr 18 '11 at 14:31
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First Lichtenbaum, and then in general Beilinson, have provided general conjectures about special values of motivic L-functions at integers relating them to so-called "Higher regulators". You could take a look at Nekovar's article "Beilinson's Conjectures" in volume 1 of the motives proceedings.

The answer to your question involves an answer to your "alternatively" part. The special values you are discussing are at points of a different nature: the point s=1 for an elliptic curve is the central point (i.e. the reflex point of the functional equation), whereas for the zeta function, the central point is s=1/2, so s=1 is not central. Because s=1 is a central point for an elliptic, the Beilinson regulator is more complicated and requires the construction of a height pairing. For other integers, the value L(n,E) (conjecturally) does not require a height pairing, as in the zeta function case.

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I wish I knew enough to understand your answer. I actually posted another question at mathoverflow.net/questions/2132/what-is-the-beilinson-regulator –  Ilya Nikokoshev Oct 23 '09 at 17:36
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