# Intuition behind the Tamagawa numbers

i have read many books concerning the definition of tamagawa numbers ,but none of the books explained an intuition behind the concept , i mean what could be the intuitive definition of tamagawa number

i am expecting some other explanation ,other than the ones present in the textbooks,

i wanted to know the reason why the "prof.Peter swinnerton dyer " used the tamagawa number as a central part in conjecturing his work about elliptic curves, what is the reason behind that ???

what do we get when we calculate the tamagawa numbers of the elliptic curve ,i mean i want a logical imaginative definition of it,

hope you understood,

• Try reading at the introduction of the Bloch-Kato article in the Grothendieck Festschrift. Jul 23 '11 at 4:27
• thank you for your suggestion ,i will surely read it @S.carnahan
– Trust God
Jul 23 '11 at 4:31
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Jul 23 '11 at 6:51
• "i have read many books concerning the definition of tamagawa numbers" :-) which ones? Jul 23 '11 at 13:37
• Just one stupid remark: the L-series of an elliptic curve depends only on the isogeny class of the elliptic curve. But the tamagawa numbers do not, so if you want to have a formula which is consistent, the numbers you add to it should be put in such a way that they cancel while changing an elliptic curve by an isogenous one. If I am not mistaken, it was Casselman who proved that the extra factors (including the Tamagawa, Sha and so on) in the way they are added are independent of the isogeny (although I might be giving the credit to the wrong person) Jul 24 '11 at 22:00

The Euler factors in the $L$-series of an elliptic curve at non-singular primes can be defined as integrals over the $p$-adic points of $E$. When one does the analogous integral over $E(\mathbb{Q}_p)$ for singular primes, then one gets the number of components, which is $\#E(\mathbb{Q}_p)/E_0(\mathbb{Q}_p)$, multiplied by the integral over the identity component $E_0(\mathbb{Q}_p)$. The integral over the identity component gives the Euler factor at the singular prime, i.e., one of $1$, $(1-p^{-s})^{-1}$, or $(1+p^{-s})^{-1}$. So there's this extra factor given by the number of components. Of course, the number of components is the Tamagawa number in this setting. This is all explained in detail in Tate's article in Antwerp IV (Springer Lecture Notes in Mathematics 476), which is where I learned about it. I expect this will be easier to read than the Bloch-Kato article.

BTW, in the original formulation of Birch and Swinnerton-Dyer, they didn't know where the extra factors came from, so they just said that they were small integer "fudge factors". I believe it was Tate who indicated why they should be Tamagawa numbers.

• @joe silverman,fantastic ,thanks a lot,for your good help
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Jul 23 '11 at 14:46
• @joe silverman:sir i cant find that article anywhere,and i am not in a sophisticated place,so please send me the link if you are free sir,but dont waste your valuable time for me if you are busy
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Jul 23 '11 at 15:09
• Here's the book: amazon.com/Modular-Functions-One-Variable-International/dp/… You may want to see if a local university library carries it. Jul 23 '11 at 16:39
• @stankewicz: Thanks for providing the link. @trust god: I don't know if it's freely available online. Jul 23 '11 at 16:40
• @stankewicz:thanks a lot for your help sir, @joe silverman:thanks a lot sir, touch both of your feet,who provided me the path
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Jul 24 '11 at 12:17