3 added 565 characters in body

I apologize for in advance for making just a few superificial remarks. These are:

1. 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.

2. 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.]

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.

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.

2 added 1771 characters in body
• 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 curvescurves.]

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.

1

I apologize for in advance for making just a few superificial remarks. These are:

1. 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.

2. 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.