"... Look, this is an elliptic curve. But it's not the ordinary ellipse from a conic curve, it's a smooth projective curve of genus 1 over a field. If the characteristic is not equal to 2, the affine equation is y^2 = x^3 + ax^2 + bx + c.
You surely remember the prerequisites for the BSD conjecture, right? An elliptic curve over the complex field is a Riemann surface of genus 1, and over a global field, it's a finitely generated abelian group. The Abelian variety is the higher-dimensional generalization of the elliptic curve.
So at this point, I feel the need to transform the elliptic curve into Weierstrass form. This is the method I found after studying a lot of related theories. Such a transformation is quite mechanical, assuming the equation has at least one rational point.
But obviously, this step holds, we've already proven it before, so we can derive these two equations..."
Qiao Yu was speaking while writing on the small table with a pen.
