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2 Spelling correction.

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 disasterous 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.

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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 disasterous 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.