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We have the formulation of B.S.D for modular forms, and if we can do this thing :

(prove b.s.d for modular forms)------------->(proof of b.s.d for elliptic curves)

via Taniyama-Shimura Theorem, as every elliptic curve has a modular form.

why cant the proving of B.S.D of a modular form imply the proof for elliptic curves.

Thank you Prof.Matthew Emerton.

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What exactly is your question? I don't see any question marks. The only sentence that is worded as though it could be a question is "why cant (sic) the proof of b.s.d...", which begs the counter question: who told you that it can't? –  Alex B. Apr 25 '11 at 7:00
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Now that there is a visible question, the answer is: it can and it would. Now, all you have to do is prove the Tamagawa number conjecture for eigenforms of weight 2 (feel free to do it in greater generality, of course). –  Alex B. Apr 25 '11 at 7:07
    
@ALEX;in fact your comment was very useful alex,thanks a lot –  Iyengar Apr 25 '11 at 9:07
    
@ALEX :can you give a bit elaborated answer like how would it relate the both things please,and how would proving the tamagawa number conjecture result in bridging both things –  Iyengar Apr 25 '11 at 15:43

1 Answer 1

up vote 21 down vote accepted

This is not a new idea; indeed, almost all work on the BSD conjecture has proceeded via a study of the associated modular form. In particular, one has the work of Gross--Zagier and Kolyvagin (see e.g. here), which verifies BSD for those $E$ such that $L(E,s)$ vanishes to order zero or one at $s = 1$, and the work of Kato (described very briefly here), which give another proof of BSD for those $E$ such that $L(E,s)$ is non-vanishing at $1$ (and much more besides).


The difficulty with proving BSD (for definiteness, let's consider the form which says that the order of vanishing of $L(E,s)$ at $s = 1$ is equal to the Mordell--Weil rank of $E$) is that there is no obvious relationship between the $L$-function and the points on $E$ at all. The modular parameterization provided by the modularity theorem for elliptic curves does provide a more direct relationship between the two sides of the conjecture, because the modular form associated to the elliptic curve has both a relationship to the $L$-function, via a Mellin transform, and a relationship to the elliptic curve via the modular parameterization, but it is still very difficult to extract anything related to BSD out of this relationship.

Gross and Zagier use Heegner points on the modular curve to generate an infinite order point on $E$ in the case when $L(E,s)$ vanishes to first order at $s = 1$. Kolyvagin then makes an elaborate argument (a so-called Euler system argument) to show that in fact in this case $E$ has Mordell--Weil rank exactly one. Gross and Zagier don't use the Mellin transform formula for the $L$-function, but a different formula, due to Rankin.

In Kato's work, he also uses a version of the Rankin--Selberg formula, and combines it with a different Euler system argument, generalizing a construction due to Beilinson.

In both cases, if you work through the arguments, you find that the relationship between the $L$-function and the group of rational points on the elliptic curve, while it is mediated by the modular parameterization, is nevertheless very subtle and indirect. Many experts have thought about how to push things further, with no success so far.

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sir i dont know you personally ,but the help you have done is unforgettable ,and you are doing a great service,may god bless you sir, –  Iyengar Apr 25 '11 at 9:03
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Sir, please answer this if you are free sir, infinitely many thanks. @Matt E –  Iyengar Jan 30 '12 at 17:03

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