Birch and Swinnerton-Dyer conjecture

Birch and Swinnerton-Dyer conjecture

 

 

Conjecture (Birch and Swinnerton-Dyer). The Taylor expansion of L(C,s) at s=1 has the form
 
L(C, s) = c(s − 1)r + higher order terms
 
with c/=0 and r = rank(C(Q)).
 
In particular this conjecture asserts that L(C, 1) = 0 ⇔ C(Q) is infinite.
 
 
But complex function z=x+yi can‘t construct the taylor expantion because complex number z=x+yi isn't function, so L(C, 1) don't construct the taylor expantion. Thus Birch and Swinnerton-Dyer conjecture don't establish.
 
 T.mitui