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I have a function y^2 = x^3 + Ax + B mod p. I know the curve has a singularity as the discriminate is zero mod p. I'm trying to isolate the non-singular points of the curve by mapping the singularity to infinity. I've read a lot about the steps I'm supposed to follow - find the singularity S, calculate the distinct tangents at S (y = a1x + b1 and y = a2x + b2) and then apply the mapping, but I can't find any practical examples on how to actually do this.

Can anyone please point me in the direction of a similar calculation using actual numbers, instead of a proof?

Thanks very much!

user3081739
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1 Answers1

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I think that what you want is described in Proposition 2.5 of Silverman's Arith. of Elliptic Curves: If your elliptic curve $E$ has distinct tangent lines at the singularity, then $E_{\rm ns} \to \bar K^*$ given by $(x,y) \mapsto \frac{y-a_1x-b_1}{ y-a_2x-b_2}$ is an isomorphism of groups. If $E$ has a cusp (the tangent lines are coincide), then $E_{\rm ns}\to \bar K^{+}$ given by $(x,y) \mapsto \frac{x-x(S)}{y-a x-b}$ is again an isomorphism (where $S$ denotes de singular point, while $x(S)$ is the coordinate of $S$)

ADDED: If you really want to work with numbers why don't you construct some of them? Take the simplest example of singular cubic curve you know, i like $C:Y^2 =X^3$. Find the tangent lines, that is, only $Y=0$ since $(0,0)$ is a cusp. Now observe that the map I described above is given by $(0,0)\neq(x,y) \mapsto x/y$, which gives you the bijection between $C$ and $K$ 'sending' $(0,0)$ to infinty.

Vinicius M.
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  • Thanks very much for the reply! I've been using Silverman as a guide - great book, but a lot of it goes over my head. I guess my actual problem is probably a bit simpler than this... assuming the first case (node), I'm trying to find (a) the position of the singularity and (b) the values of a1,b1,a2 and b2 for the distinct tangents. Using y^2 = x^3 + ax + b, I get y = 0 but I can't determine a generic formula for solving x = 0 in this case. Then I'm not sure how to find the distinct tangents at the singularity... probably really simple questions that I'm overthinking! Thanks again! – user3081739 Dec 21 '13 at 00:17
  • @user3081739 You'll have to solve the equations given by $F = y^2-x^3-ax-b$, and it's partial derivatives (I guess you did that to conclude that the singular point has $Y$ coordinate $0$). For the tangent lines, I recommend that you read some notes on algebraic curves, they'll tell you how you can find the tangent lines of a curve at the singular points. – Vinicius M. Dec 21 '13 at 01:14
  • Thanks again! I realised I've been leaving off the mod p in my calculations and found A = 0 mod p, B = 0 mod p, which gives me exactly the situation you've described. This question is for a crypto problem... I've updated the main thread at http://math.stackexchange.com/questions/606188/attacking-elliptic-curve-cryptography-problem-with-a-bad-reduction-pmod-p.... Thanks! – user3081739 Dec 21 '13 at 06:48