PREVIEW
... 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. ...
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