# Dirichlet's Class Number and its connections with the $GL(2)$

i posted the same question on MO,but cant get an answer so i am trying here

note:all those who answer my question just mention the question number in their reply so that i can tally them,thanks a lot for all those who help

When i was just reading the class number formula and thinking to interpret it with the Birch and swinnerton dyer conjecture,i found that the so called "class number formula" is only a volume computation of the $GL(1)$ ,

but here are my questions ,i try to post it as clearly as possible

1)Is there any way of extending the power of class number formula ,so that it can be a measure of volume of $GL(2)$ ?,is there any work done in this direction,if so provide me references

2)And probably the above answer is no,as everybody knows that $GL(2)$ is not abelian ,so is there any way of proving ,i mean Abelianization of $GL(2)$ group??

3)And how can one relate the $L$-function of an elliptic curve to the associated $GL(2)$ representation,so that one could compute the residue at $s=1$,

4)can anybody give me the link to Prof.John Torrence Tates,P.H.D thesis concerning the "fourier analysis and hecke Zeta function",i mean a open link,which doesnt involve money ,or logins

thanks a lot for anyone who has wasted their time in reading and took pains for answering them

any help is appreciated

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Tate's thesis is available in many books, for example in Cassels and Frohlich's algebraic number theory. –  Soarer Jul 2 '11 at 14:27