The Birch and Swinnerton-Dyer Conjecture

1. Power Series Analysis of the Birch and Swinnerton-Dyer
Conjecture
∞
∑ (n+1) cn+1 (s-1)(n+1)-1
n=0
=(1)c(1)(s-1)0
+(2)c(2)(s-1)(1)
+(3)c(3)(s-1)(2)
…
(n+1)cn+1 = cn, n= 0,1,2,3,…,
Or cn+1 = cn/(n+1)
Hence
c(1)=c(0), c(2)=c(1)/2, c(3)=c(2)/3
=(1)c(0)(s-1)0
+(2)/2 c(1)(s-1)1
+(3)/3 c(2)(s-1)2
…
∞
= c(0) (s-1)0
+ c(1)(s-1)1
+ c(2) (s-1)2
… = ∑ cn (s-1)n
n=0

2. ∞
∑ n cn(s-1)n-1
= (1)c(1)(s-1)0
+(2) c(2)(s-1)(1)
+(3)c(3)(s-1)2
n=1
c(1)=c(0), c(2)=c(1)/2, c(3)=c(2)/3
=(1)c(0)(s-1)0
+ (2)/2 c(1)(s-1)(1)
+ (3)/3c(2)(s-1)(2)
…
∞
c(0) (s-1)0
+ c(1)(s-1)1
+ c(2) (s-1)2
… = ∑ cn (s-1)n
n=0
Thus,
∞ ∞ ∞
∑ (n+1) cn+1 (s-1)(n+1-1)
= ∑ cn (s-1)n
= ∑ n cn(s-1)n-1
n=0 n=0 n=1

3. Hence,
∞
L´
(C,s) = ∑ (n+r) cn+1 (s-1)(n+r)-1
n=0
= (0+1) c1 (s-1)(0+1)-1
+(1+1) c2 (s-1)(1+1)-1
+(2+1) c3 (s-1)(2+1)-1
Therefore,
L(C,s) = c(s-1)r
s=1, r=2, & c≠0
= c(s-1)(s-1) = c(s2
-2s+1)

4. (1-s2
)y´´(s) - 2sy´(s) + λ y(s) = 0 (Legendre Polynomial)
[(1-12
)][c(2)] - 2(1)[c(2(1)-2)] + 6[(c)(12
- 2(1) + 1)] = 0
[(0)][c(2)] - 2(1)[c(0)] + 6[(c)(0)] = 0
∞
y1 = c0 + ∑ c2ks2k
k=1
λ = n(n+1) = 2(2+1) = 6
cn = (2n)!
2n
(n!)2
y1 = 1+ 3/2 (1)2
+ 4.375 (1)4
+…+cnsn