elliptic curves - The class number formula, the BSD conjecture, and the Kronecker limit formula

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
$$ Z_p(T) = expBig(sum_{r=1}^infty N_r frac{T^r}r Big), $$
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)