Given an elliptic curve y^2 = x^3 + ax + b with 4a^3 + 27b^2...

Given an elliptic curve $y^2 = x^3 + ax + b$ with $4a^3 + 27b^2 \neq 0$ over $R$, derive the formulas of group operations. That is, given distinct $(x_0, y_0)$ and $(x_1, y_1)$ on the curve, derive the formulas of addition $(x_2, y_2) = (x_0, y_0) + (x_1, y_1)$ and doubling $(x_3, y_3) = 2(x_0, y_0)$.

Transcript

0:00 Hello there.

00:01 In this occasion, we are going to work with an elliptic curve that does this that we got this equation and the first thing that we need to do is prove that this point lies on the elliptic curves are really a really interesting field in mathematics.

00:17 If you are interested in this field of pure mathematics, to be more specific, there is an area in mathematics is called algebraic geometry.

00:29 A lot of people use these elliptic curves.

00:32 Also has some applications of this on tropical geometry, modular forms.

00:40 There is a deep connection with modular forms.

00:44 And actually, the ones that prove the last theorem of format use elliptic curves and modular forms to prove that theorem.

00:56 Okay, so elliptic curves are really important and have a lot of interesting properties one of those properties are the one that we are going to show or in this exercise okay, so let's suppose that you got some elliptic elytic curve that is in most of cases are like this the other depending on on the coefficients that form the elliptic curve are they have like this structure so what happened in is that if you choose two points on this elliptic curve that has a rational coefficients and then you draw a line through those points then you will find that another point that intersects this elliptic curve that also has a rational values okay so that points also are rational numbers okay so in this occasion you're going to do the same but just using one point so if you you have only one point and how to construct a line that pass through that point and also intersects the elliptic curve well we need to use the notion of derivative and tangent line okay so that's the procedure in this this occasion so the first thing is to prove that this point is lies on this elliptic curve so that is really simple this point is equivalent to say that x is equal to minus one and that y is equals to three.

02:33 So we replace these values on the equation and we obtain here nine is equal to minus one, plus six and plus four.

02:46 Okay, so this clearly is nine equals to nine.

02:49 So yes, this point lies on the elliptic curve.

02:52 So the first thing has been shown.

02:55 Now we need to find the tangent line to this point and for that we need to get the derivative.

03:00 Derivative of this function.

03:04 So to obtain the derivative we cannot do it explicitly.

03:08 We need to do it an implicit derivation, so that's the next step.

03:12 So if we derivate this function, we obtain 2y, y, y prime equal to 3x square minus 6...