Is there a "Basic Number Theory" for elliptic curves?

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 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

I don't think that such a survey paper or textbook exists, but the closest thing I know of is "A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture" by Spencer Bloch, Invent. Math. v.58, no.1, pp. 65-76, 1980.

Here's an abbreviated history, picking up where you left off: Takashi Ono wrote a paper "On the Tamagawa number of algebraic tori", Annals of Math., v.78, no. 1, July 1963. In that paper, Ono computes the volume of $T^1(A) / T(F)$, where $T$ is an algebraic torus over a number field $F$, and $A$ is the adele ring, and $T^1(A)$ denotes the intersection of kernels of $\vert \chi \vert$ as $\chi$ ranges over $F$-rational characters of $T$. Ono's formula states that this volume (called a Tamagawa number, but not to be confused with the local Tamagawa numbers $c_v$) equals $ \vert Pic_{tor}(T) \vert / \vert Sha(T) \vert$.

The numerator is the order of the torsion subgroup of the Picard group of $T$. The denominator is the order of the Tate-Shafarevich group of $T$. Most of the arithmetic is contained in the normalization of the measure on the quotient space $T^1(A) / T(F)$ -- this normalization of measure uses the L-function (an Artin L-function) of $T$, and the special case $T = G_m$ corresponds to the Dirichlet class number formula for $F$.

From looking at Ono's paper (an earlier Annals paper from 1961), it appears that Weil and Tate were influential in his work.

Fast forwarding to 1980 (skipping lots of great things for reductive groups), here's a brief summary of what Bloch does (in the Inventiones paper mentioned above). He begins with an abelian variety $E$ over a global field $F$ (I already used $A$ for the adeles). Using the fact that the dual abelian variety $\hat E$ can also be viewed as $Ext(E, G_m)$, Bloch uses the Mordell-Weil lattice $L$ of $F$-rational points on $\hat E$ to construct an extension of algebraic groups over $F$: $$1 \rightarrow T \rightarrow X \rightarrow E \rightarrow 1$$ in which $T$ is an $F$-split torus with character lattice $L$.

Remarkably, Bloch proves that $X(F)$ is discrete and cocompact in $X(A)$. Moreover, most suggestively, Bloch proves that the BSD conjecture for $E$ is equivalent to the conjecture that the volume of $X(A) / X(F)$, with respect to a suitably normalized measure, equals $\vert Pic_{tor}(X) \vert / \vert Sha(X) \vert$.

Of course, the meat of Bloch's approach is in the normalization of measure, which uses the L-function of $E$. I once gave a truly disastrous talk as a graduate student about Bloch's paper, in which all this normalization of measure stuff completely escaped me. I still find Bloch's paper very difficult and mysterious. It seems that it is mostly cited for its novel construction of height pairings, but not much has been done (publicly) with its interpretation of BSD.

Hello, It's nice that there has been some more interest in Pell conics. I decided a couple days ago after reading a few of the interesting discussions here to post on arXiv what I've learned, arXiv:1108.1610, about Franz Lemmermeyer's result that Sha for conics is isomorphic to a subgroup of the narrow class group of a quadratic field.