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I suspect that they knew that the $L-$function is defined only for $Re(s) \gt 3/2$. Did they attempt to evaluate the $L-$function at $s=1$ by plugging $s=1$ in the infinite product $\prod_p (\frac{1}{1-a_pp^{-s}+p^{1-2s}})$? I think that it gives $\prod (\frac{N_p}{p})^{-1}$, which does not make sense under the usual definition of infinite product.

Note: I have posted the same at math.stackexchange but I would like further explanation beyond what was given there.

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I still don't think using MO to discuss very advanced number theory is a better approach to learning some basic algebra and number theory –  Yemon Choi May 31 '11 at 17:20
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iyengar, my (lack of) proficiency is not the issue. My concern is that since you are asking about very technical mathematics, it is not clear how much of the answers you actually understand. If you understand all of it, then great; but myself, I am trying to give advice that may help, believe it or not. You would be better off looking up material in Davenport's Higher Arithmetic, or Stewart and Tall's Algebraic Number Theory, and working through those and asking questions when you get stuck. –  Yemon Choi May 31 '11 at 17:47
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@iyengar: you are missing my point. If you don't have access to more basic mathematical texts, the solution is not to ask specialists on the internet about some of the most intricate or deepest mathematics currently known, which moreover depends on understanding a whole bunch of other things which you also haven't learned. You seem to believe that one can somehow avoid doing this. If you do not have access to mathematical education, asking people about BSD, Hodge, etc is not going to be a substitute. You will, IMHO, benefit more from learning basic things than by asking about cool things –  Yemon Choi Jun 1 '11 at 9:17
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Put it this way: why do you think your time is better spent "cracking your brain" trying to understand very hard problems in modern number theory, rather than learning the prerequisites? If you have no education and are reading mathematics yourself then it is even more important that you learn to do the basic things well. Try reading the online notes of J. S. Milne or Paul Garrett, for instance. There is plenty of interesting mathematics out there which is not as hard as the BSD conjecture... –  Yemon Choi Jun 1 '11 at 9:24
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What's wrong with reading the original articles by Birch and Swinnerton-Dyer? And I am repeating myself when I say that I miss some civility in trust god's comments. –  Franz Lemmermeyer Jun 1 '11 at 10:35
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up vote 37 down vote accepted

For what it's worth, here are some historical comments.

Both Birch and S-D spoke in Cambridge a few weeks ago, on the history of their conjecture. To my surprise, both of them emphasized the role not of the $L$-function, but of the Tamagawa number, in their comments on how it all came about.

Tamagawa had introduced this invariant associated to a semisimple algebraic group over a number field, and one can interpret it adelically or as a product of local factors. B and S--D were trying to "compute the Tamagawa number of an elliptic curve" -- or more precisely, to see what the analogue should be in this situation. The local factors came out to be $N_p/p$, at least at the good primes. Historically, the $L$-function came later. I asked S--D why this might have been, and he said something like "Weil was pushing the $L$-function as being of central importance, so, naturally, everyone else was avoiding it like the plague". I am not sure this comment is to be taken so seriously -- this is perhaps more a reflection of S-D's sense of humour (he'd made some rather caustic comments about Mordell's bridge playing skills earlier, again probably just to get laughs, and he succeeded admirably in getting them). But you have to remember the resources available to them at the time: they were initially not thinking about the $L$-function, but they did have this access to this gigantic machine, the size of a lecture theatre, that was capable of computing the product of $N_p/p$ for all primes less than 1000, and this for them was basically a miracle, because ten years prior to that if you wanted to do this calculation then you'd better have a lot of pencils/paper handy.

So they used what they had, they were focussed initially on Tamagawa numbers (this is I guess the reason that the fudge factors at the bad primes became known as Tamagawa factors?), they had access to a computing machine and S--D knew how to use it, and perhaps crucially one should stress that whilst we now know the $L$-function to be of central importance, it was perhaps not so clear at that time. There was no Langlands programme, there was no converse theorem -- this was the late 50s. It was only a couple of years later, when Birch was talking to Shimura, that Shimura told him that in the CM case one could actually evaluate the $L$-function at 1 in concrete cases and get concrete numbers, and then Davenport told Birch a concrete algorithm which would work and could be done by hand. I have seen with my own eyes the piece of paper in which Davenport sketched the idea to Birch, and Birch has written "keep this" on the top and underlined it! Birch then proceeds to compute various explicit examples of special values of $L$-functions on the next few pages, in the CM case, but maybe this was already after the first work had been published.

I mentioned all this to Rene Schoof yesterday and he claimed that there were pictures of the pieces of paper on youtube of all places (I know William Stein was taking pictures frantically -- my eyes were just popping out of my head -- all these really important historical documents, that Birch claimed were just gathering dust in a wardrobe at home!).

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I am surprised no one has mentioned that Dirichlet's class number formula must have been had some relevance in suggesting a conjecture to B+S-D especially in the form $L^{(r)}(0,\chi)$ where $r$ is the unit rank. It is though not the same as the critical strip is shifted. –  Junkie Jun 2 '11 at 19:55
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@trust god: your comments indicate precisely why you should follow Yemon's advice and learn some basic undergraduate number theory before you start trying to understand profound conjectures which are generalising things you don't know. –  Kevin Buzzard Jun 3 '11 at 12:24
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My gauge is the fact that you seem to be claiming that you don't understand Junkie's comment. My humble suggestion to you is that you don't ask Junkie to elaborate, but that instead you pick up a book which is at an appropriate level for you. –  Kevin Buzzard Jun 3 '11 at 18:17
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@trust god: Junkie's comment is perfectly clear as is. Just because you vaguely know what some big words mean, doesn't mean you understand things. I used to spend large amounts of time "thinking" about hard problems whose subtleties I didn't understand at all; I now regard this as wasted time. Don't waste your time like that too. –  David Hansen Jun 5 '11 at 15:54
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@Junkie, Kevin: my reading is that the complete statement of the BSD including the regulator part, as we know it today, is due to Tate. In "Notes on elliptic curves II" in Crelle, B and S-D briefly say that "there was some tendency for $\prod N_p/p$ to be large when the generators of $A$ were small", but they don't pursue this further. Tate in "On the BSD and a geometric analogue" gives the formula we know today, including the regulator. In "Conjectures concerning elliptic curves", Proc. Symp. Pure Math. Vol VIII, Birch explicitly credits Tate with this formulation (penultimate paragraph). –  Alex B. Jun 7 '11 at 3:16
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