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can anyone please explain me in simple terms ,why cant the stuff done in the case of pell conics cant be done for elliptic curves,i mean we can prove the Birch and Swinnerton Dyer in a similar way by using the proof for proving the same for Pell conics ,i.e. $$ \lim_{s \to 0} s^{-r} L(s,\chi) = \frac{2hR}{w} = \frac{|Sha| \cdot R^+ \cdot \prod c_p} {| {\mathcal P}({\mathbb Z})_{tors}|}. $$

why does one refrain using the techniques used in the case above, sharply i mean,what are the things that prevent one from proving the conjecture by using the class formula ,in case of elliptic curves ,i mean can anyone enumerate the reasons why does one fail to prove Birch and Swinnerton Dyer conjecture by using the Class formula

your comment/answer is very valuable for me,thanks a lot

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The question is very difficult to read to put it politely. For example, you write "i mean we can prove the Birch and Swinnerton Dyer in a similar way ...". My understanding is that the Birch and Swinnerton-Dyer conjecture is an open problem. Could you please clarify what you mean by this? Instead of offering a bounty, I think it would be a better idea to write your question more clearly such that it is intelligible to people other than yourself (please see the faq). That 232 people have viewed this question thus far and that none of them have even commented here is quite ominous. –  Amitesh Datta Jun 13 '11 at 8:48
    
Page 10 of mat.unimi.it/users/mbertoli/report.bsd.pdf –  Samuel Hambleton Oct 10 '11 at 8:02
    
..talks about it briefly –  Samuel Hambleton Oct 10 '11 at 8:03

1 Answer 1

up vote 4 down vote accepted

I am not an expert.

The class number formula is a volume computation on $GL(1)$, and the L function of an elliptic curve is an $GL(2)$ object after using Wiles-Taylor theorem. So there are similar questions in spirit, but the main task is that if you have an L function of an elliptic curve, and know the associated $GL(2)$ representation, and can compute the residue at $s=1$, that the analytic data coincides with the algebraic data of the elliptic curve.

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i am in debt with you for the answer,but is there any work in that way i mean,representing $L$-functions in $GL(2)$ representations ,and can the dirichlet class number formula can somehow be extended to be valid for $GL(2)$ objects,is there any work in that side,any help is appreciated @late_learner –  Iyengar Jun 28 '11 at 10:01
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Even, if there is such an interpretation as a volume for $GL(2)$, which reasonably, the main task remains in showing that this volume is actually the data you get from the elliptic curve. So if you are really interested, you should read perhaps first read Tate's thesis to understand properly $GL(1)$, then proceed to Godement/Jacquet "Zeta functions of simple algebras", I guess. But that gives you only the automorphic side, and nothing about elliptic curves. –  plusepsilon.de Jun 28 '11 at 10:52

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