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Genus 0 And Genus 1
1. Genus 0 and Genus 1
I am going to prove that The Birch and Swinnerton-Dyer Conjecture exists under certain
Conditions for Genus 0 and Genus 1.
Case I
Genus 0
L(C,s) = c(s-1)^0 in this case Genus is 0. Therefore, c(s-1)^0. c is not 0 as stated in the
Conjecture and s>1. 1^0 = 1 therefore the answer is c.
If by chance that s=1 then c(1-1)^0 = 0, hence there are two possible solutions for Genus
0 (c,0).
Case 2
Genus 1
L(C,s) = c(s-1)^r in this case Genus is 1. Therefore, c(s-1)^1. c is not 0 as stated in the
Conjecture and s>1. c(s-1) is the result. On the contrary that at Genus 1, s = 1, then c(1-
1)^1 is c (0)^1 which leaves a second answer of 0. Therefore, we can expect two
possible solutions of c(s-1) or on the other hand 0.