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I heard somewhere that the above formulation of conjecture is for predicting the exact leading term of a L-function at an integer. But i didnt find any reference about how it is stated, anyone please give me any reference or either state the conjecture precisely ,thank you

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up vote 2 down vote accepted

The 'Tamagawa Number Conjecture' is a bit murky. When I hear it, I think of two different things: the Bloch-Kato Conjecture, or the Birch and Swinnerton-Dyer Conjecture.

The Block-Kato Conjecture is badly named, because there are a couple different BK Conjectures on very different topics. So I have heard the one we are thinking of as the Tamagawa Number Conjecture because it deals with Tamagawa Numbers. More on this can be read, for instance, here.

The Swinnerton-Dyer Conjecture is one of the Millenium Prize problems, so lots can be read about it. But for the sake of completeness, I refer you to its wiki page and to this description by Andrew Wiles, at the Clay Institute's site. Look in particular at the first 'conjecture' of Wiles's summary.

For a general overview of the related ideas going into this conjecture, I recommend reading this abstract, or a copy of the lecture itself if you can find it (maybe not so unlikely, as Cambridge is good about those things).

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You are right, the "Block-Kato conjecture" is badly named ;) – Adrián Barquero Sep 25 '11 at 16:30
@mixed math:but what is the Difference between both of them ,i mean BSD and Tamagawa number conjecture – Iyengar Sep 25 '11 at 16:39

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