# Tamagawa numbers and Genus class numbers

I was reading the paper of Prof.Franz Lemmermeyer titled "Pell-conics" which is here, in that the author writes in page 9 that one can define Tamagawa numbers as $$c_p = \begin{cases} 2 & \text{ if } p \mid \Delta, \\ 1 & \text{otherwise}. \end{cases}.$$

Then he writes that Gauss-genus theory implies that $$\prod c_p=2(\rm{Cl^+(k):Cl^+(k)^2})$$ where $Cl^+(k)$ is the narrow class group of $k$, but I have referred to articles in the reference but can't find any such statement upto my sight. I know that one can write $\rm{Cl_{\large gen}}\cong \rm{Cl(k)/Cl(k)^2}$ where $\rm{Cl_{\large gen}}$ is the genus class group, I read the entire book of Reciprocity laws written by Prof.Franz Lemmermeyer but can't find any such notion.

After intensive search I found this one, even though Prof.Franz remarks that " latter is twice the genus class number ", he didn't point to any proof or reference pointing to that statement.

So I want any reference or proof of this statement

How can one say that product of Tamagawa numbers is equal to twice the Genus class numbers ? Is there any reference ?

Please answer this question as I dont have a proper access to materials neither know the current works.

Thanking you all.

Iyengar.

• Another version of Prof.Franz explanation can be seen here – IDOK Dec 9 '11 at 17:34
• Huh, I have been begging everyone to help me, but no one cared, my will and zeal helped me finding out the answer myself, after studying a lot I understood how it turns out to be, If anyone is looking for answer ask me to write, or else delete this question. – IDOK Dec 10 '11 at 11:35

This is normally phrased as $[Cl^+(k):Cl^+(k)^2] = 2^{r-1}$, where $r$ is the number of primes dividing $\Delta$. It is due to Gauss, and is proved in his famous book Disquisitiones Arithmeticae (which is available in English translation). A more modern treatment can be found in the book Advanced number theory, by Harvey Cohn.
I never know that there is such theorem but I understood it the other manner, we know that $$c_p = \begin{cases} 2 & \text{ if } p \mid \Delta, \\ 1 & \text{otherwise}. \end{cases}.$$
So if we consider $\prod c_p$ it would be $2*2*2*......$ r times ( since we take $p \mid \Delta$= $2^r$ ) , so it follows that $$\prod c_p = 2^r.$$ So we have $[Cl^+(k):Cl^+(k)^2] = 2^{r-1}$ so that probably $\prod c_p = 2[Cl^+(k):Cl^+(k)^2]$ , I was looking for that answer but as nobody answered it and as you remarked that you are a bit busy now and cutting of your participation further, I dont wanted to disturb you again sir. Your references were very useful and thanks a lot again sir !! . I think its the same version you posted here.