If K is a number field then the Dedekind zeta function Zeta_K(s) can be written as a sum over ideal classes A of Zeta_K(s, A) = sum over ideals I in A of 1/N(I)^s. The class number formula follows from calculation of the residue of the (simple) pole of Zeta_K(s, A) at s = 1 (which turns out to be independent of A).
Let E/Q be an elliptic curve. One might try to prove the (strong) Birch and Swinnerton-Dyer conjecture for E/Q in an analogous way: by trying to define L-functions L(E/Q, A, s) for each A in the Tate-Shafarevich group, writing L(E/Q, s) as a sum of zeta functions L(E/Q, A, s) where A ranges over the elements of Sha, then trying to compute the first nonvanishing Taylor coefficient of L(E/Q, A, s) at s = 1.
Has there been work in the direction of defining such zeta functions L(E/Q, A, s)? If so, what are some references and/or what are such zeta functions called?
Also, taking K to be quadratic, there is not only a formula for the first nonvanishing Laurent coefficient of Zeta_K(s) (the class number formula), but there is a formula for the second nonvanishing Laurent coefficient of Zeta_K(s) (coming from a determination of the second nonvanishing Laurent coefficient of Zeta_K(s, A) – something not independent of K – this is the Kronecker limit formula). Does the Kronecker limit formula have a conjectural analog for the L-function attached to an elliptic curve over Q?
A less sharp question : are there any ideas whatsoever as to whether any of the Taylor coefficients beyond the first for L(E/Q, s) expanded about s = 1 have systematic arithmetic significance?
Best Answer
I apologize for in advance for making just a few superificial remarks. These are:
The question is not uninteresting. Just because Sha doesn't appear in the definition of L easily, there's no reason one shouldn't ask about manifestations more fundamental than the usual one.
An approach might be to think about the p-adic L-function rather than the complex one. I'm far from an expert on this subject, but the algebraic L-function is supposed to be a characteristic element of a dual Selmer group over some large extension of the ground field. The Selmer group (over the ground field) of course does break up into cosets indexed by Sha. Perhaps one could examine carefully the papers of Rubin, where various versions of the Iwasawa main conjectures are proved for CM elliptic curves.]
Added, 8 July:
This old question came back to me today and I realized that I had forgotten to make one rather obvious remark. However, I still won't answer the original question.
You see, instead of the $L$-function of an elliptic curve $E$, we can consider the zeta function $\zeta({\bf E},s)$ of a regular minimal model ${\bf E}$ of $E$, which, in any case, is the better analogue of the Dedekind zeta function. One definition of this zeta function is given the product $$\zeta({\bf E},s)=\prod_{x\in {\bf E}_0} (1-N(x)^{-s})^{-1},$$ where ${\bf E}_0$ denotes the set of closed points of ${\bf E}$ and $N(x)$ counts the number of elements in the residue field at $x$. It is not hard to check the expression $$\zeta({\bf E},s)=L(E,s)/\zeta(s)\zeta(s-1)$$ in terms of the usual $L$-function and the Riemann zeta function.
The product expansion, which converges on a half-plane, can also be written as a Dirichlet series $$\zeta({\bf E},s)=\sum_{D}N(D)^{-s},$$ where $D$ now runs over the effective zero cycles on ${\bf E}$. This way, you see the decomposition $$ \zeta({\bf E},s)=\sum_{c\in CH_0({\bf E})}\zeta_c({\bf E},s), $$ in a manner entirely analogous to the Dedekind zeta. Here, $CH_0({\bf E})$ denotes the rational equivalence classes of zero cycles, and we now have the partial zetas $$\zeta_c({\bf E},s)=\sum_{D\in c}N(D)^{-s}.$$ It is a fact that $CH_0({\bf E})$ is finite. I forget alas to whom this is due, although the extension to arbitrary schemes of finite type over $\mathbb{Z}$ can be found in the papers of Kato and Saito.
It's not entirely unreasonable to ask at this point if the group $CH_0({\bf E})$ is related to $Sha (E)$. At least, this formulation seems to give the original question some additional structure.
Added, 31, July, 2010:
This question came back yet again when I realized two errors, which I'll correct explicitly since such things can be really confusing to students. The expression for the zeta function in terms of $L$-functions above should be inverted: $$\zeta({\bf E},s)=\zeta(s)\zeta(s-1)/L(E,s).$$ The second error is slightly more subtle and likely to cause even more confusion if left uncorrected. For this precise equality, ${\bf E}$ needs to be the Weierstrass minimal model, rather than the regular minimal model. I hope I've got it right now.