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Elliptic Tales by Robert Gross, Avner Ash

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Chapter 9

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SINGULAR CUBICS

Road Map

Let E be an elliptic curve defined by a cubic equation with integer coefficients. The details of the BSD Conjecture require us also to work with singular cubic equations, because there will always be some prime numbers p dividing the discriminant ΔE. If we reduce the equation for E modulo such p, we will get a singular cubic, and we will need to count the number of points on these singular curves as well.

Our goal is to show that the group law on a singular cubic curve is always “the same as”—in technical terms, “isomorphic to”—a group law that can be defined in some other, easier-to-understand way. The reasoning ...

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