Algebraic number theory

The Ramanujan-Nagell Theorem: Understanding the Proof

The pdf version is A crash introduction to BSD conjecture . We begin with the Weierstrass form of elliptic equation, i.e. look it as an embedding cubic curve in $latex {\mathop{\mathbb P}^2}&fg=000000$. Definition 1 (Weierstrass form) $latex {E \hookrightarrow \mathop{\mathbb P}^2 }&fg=000000$, In general the form is given by, $latex \displaystyle E: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6 \ \ \ \ \ (1)&fg=000000$ If $latex {char F \neq 2,3}&fg=000000$, then, we have a much more simpe…

The pdf version is A crash introduction to BSD conjecture . We begin with the Weierstrass form of elliptic equation, i.e. look it as an embedding cubic curve in $ {\mathop{\mathbb P}^2} $. Definition 1 (Weierstrass form) $ {E \hookrightarrow \mathop{\mathbb P}^2 } $, In general the form is given by, $  E: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6 \ \ \ \ \ (1)$ If $ {char F \neq 2,3} $, then, we have a much more simper form, $ y^2=x^3+ax+b, \Delta:=4a^3+27b^2\neq 0. \ \ \ \ \ (2) $ Remar…