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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?

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I would juste like to advertise "Conjectures sur les dérivées logarithmiques des fonctions $L$ d'Artin aux entiers négatifs", Math. Res. Lett. 9 (2002), no. 5-6, 715--724, by V. Maillot and D.R., where the subdominant terms of Artin L-functions are (sometimes conjecturally) interpreted in terms of Arakelov-theoretic intersection numbers. –  Damian Rössler Jan 19 '12 at 8:54
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4 Answers

up vote 11 down vote accepted

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.]

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.

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Not a great answer, but some comments that hopefully push in the right direction.

For a number field $K$, there is naturally a finite-dimensional complex vector space associated to it, namely the space generated by the characters of the abelian group $Gal(H/K)$ with $H$ the Hilbert class field of $K$. Each character has an $L$-function, and one can use these characters all together to analyse the $L$-function of the trivial character (via an analysis of linear combinations of the $L$-functions of all the characters, something which might be more tractable). That is some sort of interpretation of your comments on the zeta function of $K$. EDIT: Crucially, this space has an interpretation in terms of automorphic forms on $GL(1,K)$, so the $L$-functions are very well-behaved and we know something about them.

For an elliptic curve $E$, even if its Tate-Shaferevich group has order bigger than 1, I can only see (in the theory of automorphic forms), a 1-dimensional space, namely the space spanned by the newform corresponding to $E$. One could also look at oldforms but these don't give any new arithmetic information.

In particular, although I can see class groups in the theory of automorphic forms, I can't see Tate-Shaferevich groups, and hence I can't see how one could formulate a good candidate for $L(E/Q,A,s)$. If anyone could explain that, it would be a great start.

Some more negative comments though---the two theories have other differences. For example the $L$-function of the curve is vanishing at $s=1$ whereas the $L$-function of the field has a pole. Arguments like "I have a pole, and all these other functions don't, so when you add us all up we have a pole" don't work if you replace "pole" by "zero". Also, in some sense the reason a function like $1^{-s}-3^{-s}+5^{-s}+\ldots$ doesn't have a pole at $s=1$ is because the sum is converging for the real part of $s$ bigger than 1 and you can just about see what is happening as $s$ tends to 1. The sums involved for elliptic curves only converge for the real part of $s$ bigger than $3/2$ so it might be more difficult to get from there to $s=1$ (although if you're "automorphic" then this problem might not occur).

As for the second question, I've seen decent mathematicians say that there is no arithmetical significance in higher order terms after the first. As you point out, this statement is wrong (even for the classical zeta function: $\zeta(0)\not=0$ but $\zeta'(0)=\log(1/\sqrt{2\pi})$) but I've never seen a fancy $K$-theory or whatever interpretation of this, so have no idea what's going on.

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Thanks for pointing out that the additive decomposition for Zeta_K(s) has a known automorphic interpretation. –  Jonah Sinick Nov 15 '09 at 21:31
    
I'm not really sure whether my observation is anything more than some bookkeeping device. All I'm saying is that I can see functions on the ideles that mirror what's going on on the ideals. I'm not really sure they're giving new information in any real sense. –  Kevin Buzzard Nov 15 '09 at 21:58
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Kevin: the Dirichlet series for $L(E/{\mathbf Q},s)$ actually converges for $s$ with real part greater than 5/6, so in particular it includes a neighborhood of $s = 1$. That delicate result comes from a theorem about the region of (conditional) convergence for Dirichlet series associated to modular forms. It's sort of like the Dirichlet series for nontrivial Dirichlet $L$-functions converging for $s$ with real part greater than 0, not just (absolutely) for $s$ with real part greater than 1. –  KConrad Apr 2 '10 at 22:45
    
I once asked an expert in automorphic L-functions about any arithmetic significance to higher order terms for elliptic curves. He said that Spencer Bloch had made some conjectures involving the dimension of some cohomology groups. I have some things I need to learn before I can understand the literature in this area, so I don't know much more than that. –  Jamie Weigandt Apr 4 '10 at 0:50
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The following observation suggests that perhaps the analogy between class groups and Tate-Shafarevich groups is not as close as one might think, and that, at least in the quadratic case, the right object is the group of ideal classes modulo squares.

Let $Q_0$ denote the principal binary quadratic form with discriminant $d$, and let ${\mathcal P}: Q_0(X,Y) = 1$ be the associated Pell conic. For each prime power $q = p^r$, denote the number of ${\mathbb F}_q$-rational points on ${\mathcal P}$ by $q - a_q$; it is easily checked that $a_q = \chi(q)$, where $\chi = (\frac{d}{\cdot})$ is the quadratic character with conductor $d$.

Define the local zeta function at $p$ as the formal power series $$ Z_p(T) = \exp\Big(\sum_{r=1}^\infty N_r \frac{T^r}r \Big), $$ where $N_r$ denotes the number of ${\mathbb F}_q$-rational points on ${\mathcal P}$. A simple calculation shows that $$ Z_p(T) = \frac{1}{(1-pT)(1-\chi(p)T)}. $$

Set $P_p(T) = \frac1{1 - \chi(p)T} $ and define the global $L$-series as $$ L(s,\chi) = \prod_p P_p(p^{-s}). $$ This is the classical Dirichlet L-series, which played a major role in Dirichlet's proof of primes in arithmetic progression, and was almost immediately shown to be connected to the class number formula.

Class groups do not occur in the picture above; like their big brothers, the Tate-Shafarevich group, they are related to the global object we started with: the Pell conic. The integral points on the affine Pell conic form a group, which acts (in a more or less obvious way - think of integral points as units in some quadratic number fields) on the rational points of curves of the form $$ Q(x,y) = 1, $$ where $Q$ is a primitive binary quadratic form with discriminant $d$. This action makes $Q$ into a principal homogeneous space ("over the integers"), and the usual action of SL$_2({\mathbb Z})$ on quadratic forms respects this structure. The equivalence classes of such spaces form a group with respect to taking the Baer sum, which coincides with the classical Gauss composition of quadratic forms.

Principal homogeneous spaces with an integral point are trivial in the sense that they are equivalent to the Pell conic ${\mathcal P}$. The spaces with a local point everywhere (i.e. with rational points) form a subgroup Sha isomorphic to the group $Cl^+(d)^2$ of square classes. Defining Tamagawa numbers for each prime $p$ as $c_p = 1$ or $=2$ according as $p$ is coprime to $d$ or not, we find that the usual class number (in the strict sense) is the order of Sha times the product of all Tamagawa numbers (the latter is twice the genus class number).

Now we can use Dirichlet's class number formula for proving the BSD conjecture for conics: $$ \lim_{s \to 0} s^{-r} L(s,\chi) = \frac{2hR}{w} = \frac{|Sha| \cdot R^+ \cdot \prod c_p} {| {\mathcal P}({\mathbb Z})_{tors}|}. $$ Observe that $R^+$ denotes the regulator of the Pell conic, i.e. the logarithm of the smallest totally positive unit $> 1$.

The proof of Dirichlet's class number formula uses the class group, which is a group containing Sha as a quotient, and a group related to the Tamagawa numbers as a subgroup. It remains to be seen whether such a group exists in the elliptic case.

It might be possible to make advance without having such a group: in the case of Pell conics, the zeta functions of ideal classes are, if I recall it correctly, closely related to the series defined by summing over all $1/Q(x,y)^s$ for integers $x$, $y$. The question remains how to imitate such a construction for the genus 1 curves representing elements in Sha.

Remark: The Tamagawa numbers may be defined as certain $p$-adic integrals; see an unpublished masters thesis (in Japanese) by A. Iwaomoto, Kyoto 2005. For more info on the above, see here.

For ideas pointing in a different direction, see

  • D. Zagier, The Birch-Swinnerton-Dyer conjecture from a naive point of view, Prog. Math. 89, 377-389 (1991)
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Hi Jonah,

The key point in proving the class number formula is a decomposition of the Dedekind zeta function as a sum over zeta functions of ideal classes; as well you know, these have residues independent of the ideal class and there are h(K) of them. By contrast, the definition of an elliptic curve's L-function makes no real reference to the Tate-Shafarevich group, even implicitly. Remember, the T-S group classifies "principal homogeneous spaces" for E, genus one curves C/Q with a freely transitive action of E but without any point defined over Q. I cannot imagine any way of breaking the L-function apart into pieces corresponding to elements of the T-S group. And ditto buzzard's comment on how zeros behave differently than poles when you add up functions.

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