Since my name tends to come up regularly in this set of questions let me say a few things here even if it does not seem to directly answer the question posted here.

When I started thinking about conics at the end of the 1990s,
my motivation was not rewriting the theory of elliptic curves
but showing that certain conics have a structure that is
reminiscent of that on elliptic curves. By now, this project
morphed into one of rewriting algebraic number theory based
on notions coming from the theory of elliptic curves.

Here's the main idea. Let $K$ be a number field with
integral basis $\{\omega_1, \ldots, \omega_n\}$. The
main object is the norm form
$$ F(x_1, \ldots, x_n) = \prod (x_1\omega_1 + \ldots + x_n\omega_n)^\sigma, $$
where $\sigma$ runs over the $n$ embeddings of $K$ into ${\mathbb C}$.
It is easily checked that $F$ defines an irreducible variety $V_K$
over the integers. For quadratic number fields, $V_K$ is just the
conic defined by the Pell equation. Below, we will almost exclusively
work in the group $V_K({\mathbb Z})$ of integral points on $V_K$.

The reduction modulo $p$ of $V_K$ is smooth if and only if $p$ does not divide
the discriminant $\Delta$ of $K$. For such $p$, let $N_r$ denote the number
of points of $V_K$ over the finite field with $p^r$ elements. For
primes dividing $\Delta$, one can give an explicit
definition by ``omitting the repeated factors'' in the reduction
(take the radical of the $F_p$-algebra ${\mathcal O}_K/(p)$).
Define the Hasse-Weil zeta function $\zeta_p(s)$ as usual; since
the $N_r$ can be computed explicitly, it is quite easy to verify
the Weil conjectures for $\zeta_p$ and give explicit formulas e.g.
for the functional equation. By extracting certain factors from these
local zeta functions one can build the Dedekind zeta function for $K$.

Now let ${\mathfrak a}$ denote an integral ideal in the ring of integers
of $K$, and consider the variety
$$ V_{\mathfrak a} : F_{\mathfrak a}(x_1, \ldots, x_n) = N{\mathfrak a}. $$
The unit variety above is simply $V_{(1)}$. The integral points on
$V_K$ (corresponding to units with norm $+1$) act on $V_{\mathfrak a}$
via ``multiplication'' and make them into principal homogeneous
spaces for $V_K$. The Baer sum of two principal homogeneous spaces
corresponds to ideal multiplication. Call two varieties equivalent
if there is a unimodular matrix transforming the defining polynomials
into each other. Any variety $V_{\mathfrak a}$ with an integral point is
equivalent to the unit variety $V_K$.

An ideal ${\mathfrak a}$ is called locally principal if the equation
$$ F_{\mathfrak a}(x_1, \ldots, x_n) = N{\mathfrak a} $$
is integrally solvable in all completions of $K$. The equivalence
classes of principal homogeneous spaces $V_{\mathfrak a}$ then form
a group isomorphic to $D_K/D_{lp}$, where $D_K$ is the group of
fractional ideals and $D_{lp}$ is the group of locally principal
ideals (a fractional ideal $\frac1a {\mathfrak a}$ is locally
principal if ${\mathfrak a}$ is). This whole thing is essentially
some form of genus theory (it is genus theory if the extension is
quadratic; in general number fields I believe that ideals are locally
principal at all primes not dividing $\Delta$), and the class
groups to consider are class groups in a slightly stronger form than
the usual class groups: two ideals ${\mathfrak a}$ and ${\mathfrak b}$
are equivalent if ${\mathfrak a} = \xi{\mathfrak b}$ for some $\xi \in K$
with positive norm.

The equivalence classes of principal homogeneous spaces corresponding
to locally principal ideals form a group called the Tate-Shafarevich
group $Ш_K$ (this is the group of locally solvable varieties modulo those
with an integral point). In the quadratic case, this is the group of
square ideal classes in the strict sense.

In the quadratic case one can now define the Tamagawa numbers $c_p$
simply by setting
$$ c_p = \begin{cases} 2 & \text{ if } p \mid \Delta, \\\
1 & \text{otherwise}.
\end{cases}. $$
In a http://www.rzuser.uni-heidelberg.de/~hb3/publ/master.pdf">masters thesis written in 2005, M. Iwamoto has given a more
conceptual interpretation of these $c_p$ using p-adic integrals (this is the only adelic aspect of my answer). Perhaps someone on MO fluent in Japanese can tell me (or us) the basic results in this thesis - unfortunately I never got beyond kanji. For extensions of degree higher than $2$ I do not yet know what is going to happen.

The whole point of this exercise is that Dedekind's class number formula,
i.e. the formula for the residue of Dedekind's zeta function, can now
be stated in a form fully equivalent to the conjecture of Birch
and Swinnerton-Dyer: it is given by
$$ \frac{|Ш| \cdot \prod c_p \cdot R(V_K)}{|V_{tors}|}, $$
where $R(V_K)$ denotes the regulator (defined in terms of generators
of $V_K({\mathbb Z})$, and where $V_{tors}$ denotes the torsion subgroup
of $V_K$ (corresponding to roots of unity). More exactly we should say
that we can normalize the regulator (i.e. the canonical height) in such
a way that any additional constant factors of $2$ vanish.

Observe that the finiteness of the Tate-Shafarevich group follows
from the finiteness of the class group, which is a group built from
Tate-Shafarevich and a ``genus'' subgroup whose order is related to
the product of the $c_p$. One question is whether a similar group
exists on the elliptic curve side - but this is idle speculation in
absence of any hints in this direction.

The Weil conjectures for $V_K$ is a very special case of the Weil
conjectures for zeta functions of algebraic tori (see the book
"Algebraic Groups and their birational invariants" by Voskresenskii).
I do not know how much of the stuff on principal homogeneous spaces
for the action of $V_K$ is known.

Acknowledgement: Some of the ideas above emerged in discussions with
Jeff Lagarias and Samuel Hambleton.

Related questions: MO 61859 and
MO 60566