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 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 if you are interested, but feel free to ignore the past and just answer the question if you are not.

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 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 if you are interested, but feel free to ignore the past and just answer the question if you are not.