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Sometimes in order to understand a concept thoroughly we need to have a algebraic view ( in terms of equations ) and corresponding geometric view.

My interest always lies with understanding the Tamagawa Numbers from different view points. So to say something, I already know the algebraic version of Tamagawa numbers ( both local and global parts ) . It can be stated as follows " the semisimple linear algebraic group $G$ over $\mathbb Q$, the Tamagawa number of $G$ (which is the standard terminology for the volume of $G(\mathbb Q)\setminus G(\mathbb A)$ with respect to Tamagawa measure) should be equal to $1$, and when $G$ is an elliptic curve rather than a linear group, there are many interesting things that happened, and which gave rise to a series of seminal works, and the Tamagawa numbers are widely used for tori ( by T.Ono ) , and also later the Birch and Swinnerton-Dyer found an analogue of the tamagawa number of elliptic curve ( anlogous to the work of T.Ono in defining the tamagawa numbers of Tori ) that played a central part in defining the Birch and Swinnerton-Dyer conjectures."

In terms of measure the Tamagawa Number can be defined as $$\large \tau=\rho(G)^{-1} |\Delta_k|^{-\large \frac{1}{2} \rm { dim } G } \prod_{\nu \ | \infty} \omega_{\nu} \prod_{\mathfrak{p}} L_{\mathfrak{p}}(1,\chi_{G})\omega_{\mathfrak{p}}$$ where $\omega$ is the gauge form on $G$, $\chi_{G}$ is the character of the representation of the Galois group of $\bar{k}/k$ on the lattice $\widehat{G}$ and $\rho(G) = \lim_{ s \mapsto 1 } (s - 1)^{r} L(s, \chi_{G})$ , $\Delta_k$ be the discriminant of $k/\mathbb{Q}$. And in some sense $$\large \prod_{\nu \ | \infty} \omega_{\nu} \prod_{\mathfrak{p}} L_{\mathfrak{p}}(1,\chi_{G})\omega_{\mathfrak{p}}$$ can be taken as the Haar Measure on $G(\mathbb{A})$.

I am now very curious in hearing to alternate interpretations of Tamagawa Number in Geometric sense, as a intuitive way. If there is some geometric way of explaining what is the tamagawa number , and what impact does it give , I would be very happy in listening that. Please don't include the wikipedia article ( as I have read it already ).

P.S : I know the answer will express something related to the differential forms, but I am a bit confused. If there is some other intuitive explanation , I would be very happy. Even any article that does this job will be fine.

Thank you.

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Thank you Chandrasekhar for spotting out my typo and fixing it. @Chandrasekhar –  Iyengar Apr 14 '12 at 7:44
    
you are welcome. –  user9413 Apr 14 '12 at 7:47
    
I already know the differential geometric version of it. –  Iyengar Apr 30 '12 at 16:16
    
By the way, have you posted the question on MathOverFlow? Per chance you could get better answers? –  awllower Feb 5 '13 at 16:57

1 Answer 1

up vote 2 down vote accepted
+500

Here are my two cents: some references 1 and 2.

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The first paper is the one, which I have already got after searching in google, and the second one I don't know frankly. But I assume that if I had not seen the google and searched about this, your articles might have helped me a lot. So , why to wait, I will award a bounty to it ( even though the answer completely satisfy my requirements, at-least I must thank you for your helping nature ). I will wait for some time, and see if some one can add something, or else I award the bounty to you, instead of wasting it. –  Iyengar May 3 '12 at 5:29
3  
Those were some easy $\geq 535$ points. =) –  Rasmus May 4 '12 at 14:36
    
@Rasmus : Yes, but what to do, I didn't have any other alternative. No body cared to answer the question, so that was only the way, to save the bounty from getting wasted. At-least some body enjoy the reputation. –  Iyengar May 5 '12 at 10:13
    
@Iyengar, sorry for the lateness. My answer was just to save the bounty from waste. I hope you found it useful. Thanks for the bounty. –  Mathfun May 15 '12 at 8:15

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