Tate's thesis showed how to profitably analyze $\zeta$ functions of number fields in terms of adelic points on the multiplicative group. In particular, combining Fourier analysis and topology, Tate gave new and cleaner proofs of the finiteness of the class group, Dirichlet's theorem on the rank of the unit group, and the functional equation of the $\zeta$-function. Weil's textbook *Basic Number Theory* re-presented algebraic number theory from the adelic perspective, showing how adelic methods could provide simple and unified proofs of all the results proved in a first course in algebraic number theory (and perhaps in a second one as well.)


I have heard rumors that one can similarly rewrite the theory of elliptic curves in adelic terms, and that doing so gives intuition for the BSD conjecture. Franz Lemmermeyer's paper [Conics, a poor man's elliptic curves](http://www.fen.bilkent.edu.tr/~franz/publ/conics.pdf) provides a brief sketch. Is there a survey paper or textbook which lays this picture out in full, as Weil did for the multiplicative group, pointing out the connections between the adelic and the classical language at each step, and ideally discussing the connections with BSD?


**Note:** This question has a peculiar history. See this [meta thread](http://mathoverflow.tqft.net/discussion/1080/questions-posted-by-trust-god/) if you are interested, but feel free to ignore the past and just answer the question if you are not.



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


