Is there a “Basic Number Theory” for elliptic curves?

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.

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@David Speyer: sir i am in debt with you in my whole life,thanks a lot for asking the question,i am very thankful to all who started the meta and brought out the results,i am in debt with all – Trust God Jul 31 2011 at 15:37
Thanks to Marty and Franz for the great answers below! It was essentially arbitrary figuring out which one to accept. – David Speyer Aug 3 2011 at 12:36
magnificent answers,both were fantastic ,i understood how deep one must ask in order to get a good response ,which is only possible for trained professionals like prof.david speyer ,prof.franz lemmermeyer,prof.pete L.clark,prof.todd ,but any way you all did a very good help,and my special thanks to prof.david speyer who served the main purpose of asking ,and to many people here,who voluntarily help and clarify many ones doubts ,free of cost,which cant achieved by any website on the earth only MO does it,thanks a lot everyone,touch everyone feet for your great help, – Trust God Aug 3 2011 at 15:25
and regarding my phrase "touching legs" as everybody pointed out previously ,the persons who pointed out must realize that,everyone here are older than me,and have much knowledge in the subject ,so there is no problem in touching their feet which is done in seeking the blessing of them ,which is a ritual followed in our country ,apart from god – Trust God Aug 3 2011 at 15:27

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.

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Re: influence, Ono deduced his formula from Weil's cleaner conjecture (now a theorem) that the Tamagawa number of a simply connected algebraic group is 1. But since abelian varieties don't have universal covers, the cleanest formulation is Ono's formula. Also, when the group is semi-simple, normalizing the volume is easier. I think SL_2 is probably the best introductory example. – Ben Wieland Jul 31 2011 at 20:33
Bloch's article is here digizeitschriften.de/dms/img/?PPN=GDZPPN002096080 – SGP Aug 1 2011 at 1:38
@marty:sir thanks a lot sir – Trust God Aug 1 2011 at 15:50
Hi Marty: very nice answer. I was at the talk you mention, right? I would not characterize it as disastrous (or even "disasterous"). I have seen much worse (though not from you). I think it did become clear though that things were more complicated than they appeared... – Pete L. Clark Aug 1 2011 at 19:40
I don't know the history on that one. Certainly the name "Tamagawa number conjecture" that's given to those conjectures of Bloch-Kato seems to originate from Bloch's paper. And I think that the general height pairings needed for these conjectures rely on Bloch's paper. What I meant was that I don't know a place where the interpretation of BSD as a fact about the volume of the quotient $X(A) / X(F)$ is used. – Marty Aug 2 2011 at 5:57
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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.

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I made it a link. – Will Jagy Aug 10 2011 at 6:11
Oh, your name does not require a dash or hyphen. That is mostly for the subject tags on questions. – Will Jagy Aug 10 2011 at 6:29
Thanks! This sort of basic write up is very useful. – David Speyer Aug 10 2011 at 20:05

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

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+1 for a fantastic answer ,i must give $+\infty$ for both prof.Franz lemmermeyer ,and prof.david speyer ,for asking and answering the question,it was most useful for me,i am going to use this information to achieve good and high things,and i wont let the help done by great people here go in vain,someday you see the benefit of all these things sir, i really bow my head to your help – Trust God Aug 1 2011 at 15:50
Thanks very much! I'm confused by the notion of a locally principal ideal. In a Dedekind domain, isn't every (nonzero) ideal locally principal? – David Speyer Aug 1 2011 at 16:40
My definition of locally principal does not mean that the ideal is principal in the localization, which, as you're saying, is always the case. Essentially locally principal means that there is no "genus obstruction" to being principal. In the quadratic case of a field with discriminant $4m$, an ideal with norm $a$ is locally principal if $x^2 - my^2 = a$ has local solutions everywhere. – Franz Lemmermeyer Aug 1 2011 at 18:14
Would be nice to insert the Cyrillic Ш instead of III. – Chandan Singh Dalawat Aug 2 2011 at 5:42
Mikhail Borovoi, I went to translate.google.com, translated Shafarevich from English to Russian, and copy-pasted the first letter. – Chandan Singh Dalawat Aug 3 2011 at 3:00
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