Class number Formula and Birch and Swinnerton Dyer conjecture can anyone please explain me in simple terms ,why cant the stuff done in the case of pell conics cant be done for elliptic curves,i mean we can prove the Birch and Swinnerton Dyer in a similar way by using the proof for proving the same for Pell conics ,i.e. $$ \lim_{s \to 0} s^{-r} L(s,\chi) = \frac{2hR}{w} = \frac{|Sha| \cdot R^+ \cdot \prod c_p} {| {\mathcal P}({\mathbb Z})_{tors}|}. $$ why does one refrain using the techniques used in the case above, sharply i mean,what are the things that prevent one from proving the conjecture by using the class formula ,in case of elliptic curves ,i mean can anyone enumerate the reasons why does one fail to prove Birch and Swinnerton Dyer conjecture by using the Class formula your comment/answer is very valuable for me,thanks a lot

I am not an expert.

The class number formula is a volume computation on $GL(1)$, and the L function of an elliptic curve is an $GL(2)$ object after using Wiles-Taylor theorem. So there are similar questions in spirit, but the main task is that if you have an L function of an elliptic curve, and know the associated $GL(2)$ representation, and can compute the residue at $s=1$, that the analytic data coincides with the algebraic data of the elliptic curve.