Tamagawa number conjecture

Question

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

Answer 1

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).