PSE Lectures - Ep 19 - Historical remarks and Weierstrass Form

Transcript

so um I'm going to change a a bit notation ER and call our generic field K that's because I would like to reserve the H the letter F to to the specific gal Fields FP fq and so on um and I want to remind you the definition

um a polinomial in two variables over k a um is a formal expression uh say fxy equals um sum where I goes from 0 to n and J goes

from Zer to say k a i j x to the i y to the J um where we assume that the um the coefficient of the the highest monomial namely a n k is non zero so you can write [Music]

um uh and and of course um for any I between zero and n and any uh J between zero and K the the coefficient a i j is an element of the field K okay so you can uh I I'll soon give an example and with Den

note um by K XY ER the collection of all such polinomial now um kxy you can think of it as a an inductive

definition I mean we know that for any actually pols er do not require a field er er as the the the set of coefficients H but but rather you you can do it everything with with a ring because pols I mean in order to add and and multiply pols we only need to add and multiply ER the coefficients and we don't need a a Inver I mean multiplicative inverse for for

elements so so in fact kxy is simply equal to you take K and er you look at pols in one variable over K this is this is a ring let's call it maybe R and then you you can look at pols over r with a new variable Y and that's KX

y okay H so that's that's one thing and of course as a result kxy is a ring and you can add and multiply pols into variables so this is just to set set the stage I mean what happens in I mean the the complexity ER of the theory of polinomial in a in two variables is a far more far far bigger

than ER the case of polinomial in in one variable ER one main difference is that we don't have a a division with remainder a for pols in two variables ER H how do we see this well

for a long division if you remember I need to have a notion of degree and there is a natural notion of degree ER of of a polinomial in two variable ER namely the degree of f as written here is simply n+ K so it's it's the the degree of the the highest monomial whose coefficient is non zero

and I claim that we cannot have a long division in polinomial in two variables that is if um for any f and g in kxy h we we had a a polinomial Q and R in kxy such that I can write f as

GQ + r with the degree of R strictly smaller than the degree of G right that's that's long division um I can take fxy to b y gxy to b x uh and then if I had long division for

these two polinomial H then there would would exist h a q and R such that Y is equal to QX plus r and the degree of R of course needs to be smaller than the degree of G but the degree of G is one so it means that the degree of R is zero which of course means that that R

is an element in the field it's simply a constant polinomial but but then this is a polinomial equation it needs to be true for any substitution ER of X and Y so I can substitute I I can substitute um y equals Z and also x equals

z and I get that R actually needs to be zero but then I have an equation of the form y equals QX and this cannot be possibly true for any for any substitutions uh X and Y I mean for example you can take a I mean of course Y is a why is simply our now annotation why is FX y right so um if I do

F x0 I get something that is is that is zero or rather uh yeah if I take evaluating at x equal Z we get a F of 0 comma y needs to be needs to be zero but F of0 comma Y is is y so it

cannot that's a contradiction this is all to say um we have a much higher complexity for polinomial in in two variables and furthermore um we now have um additional component in in the picture ER that is we have um geometry

for um for the field the ground field ER being being the real numbers or the complex numbers the set of all solutions H let's say of fxy equal z um the solutions that I remind you is is the set of all uh let's say x0 y z such that if you you substitute it in the

polom if you substitute it in the polinomial you actually get zero and and the solution set of of a polinomial over the the real numbers for example it has a it has a geometry for example you take fxy

equals y - x² let's say over R then I can picture the the solution set of f this is simply the set of all XY in in the plane r² such that y minus or y equal x² so this is a parabola okay and and this is in in

in sharp contrast to the solution set of a of a polinomial in one variable because we know that a polinomial in one variable can have at most a a n distinct Roots where n is the the degree so we cannot possibly expect any meaningful geometry and and what happened his story Ally with I mean how did we how did mathematicians get into elliptic curves so historically it

wasant studied a integer solutions to various polinomial polinomial equations so you can I mean the the first example is um an equation of the form a X plus b y equals c let's say A B and C maybe maybe I should write g a b and G G are integers um

find all integer solutions to a to this equation so if you remember we talked about it when we did some some Elementary number Theory this is called I mean Solutions and why to this equation are the So-Cal bazo coefficients and they exist

um if and only if G is the gcd of a of a and b but um typically they are not unique they're they're many bazook coefficients for a given a given triple or a given pair A and B and there is a method modernly you you do it with the an integer version of gaas elimination for those of you who know there is a simple method and also dantus

apparently knew of a simple method to ER to find all the possible integer Solutions ER to uh this equation so this is a a solutions for a for polinomial in two variables of degree one and the next ER the next thing to ER to look for is is pols of I mean into variables of degree two ER and I will not get into the the details but H it turns out that you can also ER study I mean you can also

completely classify the solution integer solution for a a polinomial in a two variables sorry in two variables of of degree two and then then the next thing to to look at is is a a degree three polinomial so degree 3 polinomial have the general form let's say a x

Cub + B x² y + C X y² plus d y Cub + e x² Plus fxy plus uh G y² plus h x plus I Y plus h some free coefficient J equals zero and here the the

complexity ER is much more significant than ER than in degree one and two ER and theant to started to study I mean this this was the the place where he kind of got stuck I mean he did have some H some insights ER but it it was apparent to to him and and to later mathematicians like Oiler and Hilbert ER that you you actually need Theory to ER to deal with this

case and I mean over the years in order to simplify you can do this this this is true also in degree one and two I mean you can do a change of variables so for example so to simplify let's say h you have a X plus b yal c you can do uh let me just pick up the the details

here um let's say that b is is non zero then I can write it as yals um minus A over B X plus minus c/ B is is a or rather well okay let's say Plus minus C over B and this this is the familiar ER The Familiar form of of a

line I mean you call this guy M and this guy b or maybe B Prime and you get y equal MX plus b Prime and you can do similar erh similar substitutions in in degree two ER to get into well in in degree two and and also in degree three er in degree three the substitutions ER lead you to so in degree

three um yes there shouldn't be minus uh before uh the second term or should it h because we have a X plus b y so um yes C divided by B yeah yeah yeah yeah think yeah you're right thanks yeah um but okay so so in degree three I I will not give you the details but it can be found ER some of it is in the notes and other you can look in the web um in

degree three the first simplification H gives you an equation of the form y y 2 plus A1 XY + A3 y = x Cub plus

A2 x² plus A4 X Plus A6 and and this is called the generalized V form so degree 3 by the way is called degree 3 polom in in two variables is called a cubic so by a change of variables I can

um turn any any cubic into into a generalized vus form and of course since I only care about we only study solutions for H these these equations the change of variables is I mean does not change the the set of solutions so we are we are not losing any generality in this change of variables and um it turns out that um if the characteristic of K is not a two or three

then there is a further simplification to um to the form y² = X Cub + a x + b okay so an elliptic curve uh over over a

field okay is a a an equation of the form e is Y 2 = x Cub + a x + B where of course the A and B are a Scholars in in the ground field K here we also denote e over

K so one thing to notice in the the definition is that elliptic curve ER comes um kind of equipped with a a um a declaration over which H which field it it is defined over because of course you could have a um I mean if let's say k Prime is is a field containing K fi

extension we can consider the same equation but over the larger field K Prime and this this will be a a for us in the formal the formal aspect it will be a different elliptic curve even though it is the same equation the same coefficients everything but if it is defined over a over a different field we consider it as a different object and the reason is that again we are interested in the

solution set and the solution set of this equation over a k Prime is typically will be will be different than H the solution set over over K I mean namely because K Prime has more points then there will be an inclusion I mean the solution set over K will be a subset of the solution set over K Prime okay so H so an elliptic curve is a a you always have to specify over which which field you are considering

it ER and you have this equation that as we said this is um this can be this can be a simplified um from the the generalized V form if we have a characteristic different than uh than two or three and I didn't finish the definition um the additional

condition is that a the quantity 4 a cub plus 27 b² is not zero okay so that's a that's the definition of an elliptic curve um let me just I mean henceforth um we simply

assume that um all our Fields have a a characteristic that is not h two or three and this is this covers the the vast majority of cryptographic applications I mean typically ER we take I mean in cryptography we take um an elliptic curve defined over FP where p is a very large Prime let's say a 200 digits or something like that so it is very safe to to assume that the

characteristic is not two or three there are applic ations in cryptography in which ER people work with with the characteristic two or three er there are not many of them and they have a security issue there are specific specific attacks that that work in this in this characteristic so H for us it will be it will be convenient to to Simply assume we work in a in characteristic that is not two or three because the equation is much is

much more convenient and as a result all the stuff we build on it all the calculations and so on will be will be simpler any questions so far maybe one question uh when we went from generalized um where transform to uh simplified uh we had uh X and Y in the generalized and x and y in the simplified but I guess these are like different X and Y right yeah you do a change of variables I mean you you

replace x i I mean again for Simplicity I I just use the same notation but if you want to be formal you you can call these guys for example X Prime y Prime yeah okay okay that's all yeah and this by the way this is called this thing is called the the short virus to for okay so we we will focus on on the short virus transform and er the next thing in order

is to to explain this uh strange condition right this uh this is called the discriminant Delta and it is analogous to what you know from high school is the discriminant of a qu quadratic polinomial in one variable ER and uh okay so but before we go into this uh into this condition let's H let's draw something so typically say over

r ER the elliptic Curve will look like like this my my drawing skills are limited but should be something like this like this fish and sometimes it could a ER it could have another another part of the the curve being a being this this lake

so you see of course this this um you can see algebraically the equation y² = x cub plus a X plus b this is symmetric along along the x axis right I mean for every H for every X we we possibly I mean we have exactly a I mean if we have a solution y then also minus y is is a solution so we have symmetry along the the

x-axis and these are the the intersections of the um of the polinomial x s sorry X Cub plus a X plus b with the with the x axis and there is a there is a proposition it's not hard to see that um over the real numbers Pol of a polinomial in one variable of degree three can either have one root or three Roots but it cannot have two Roots so so either we we have

this H this full picture of the the fish in the lake or we have the only the fish okay and of course you can um I mean you can you can do a translation of the graph and so on but we don't care that much about all these change of variables we have this canonical virus transform so okay so so this is this is the picture and the the main theme of um

well at least one major way to to reason about elliptic curves at least in the the early stages is to to use the ER the geometry over R and to deduce deduce something about the elliptic curve when it is defined over a over a finite field so another warning take for example a the elliptic

curve y² equals um X Cub + uh 2 x + three and let's say over uh over the real numbers I I could think of a um we could consider um some large large Prime p h and and look at at the equation the

same equation e but now over over FP so this will be again Y2 = X Cub plus 2 x + 3 and let's let's suppose that um over r i I didn't check it but it's it's just one of these I mean one of these cases so uh for Simplicity let's let's suppose it it only has the the fish component

and now one thing you you could you could maybe try I mean of course if p is large then uh you could you could try to Simply plot the solution set of this of this equation H in a discrete in a discrete manner right this this would be P minus one and so on this would be one and you do the same in in the y- axis and you plot the ER the graph I mean sorry the the discrete ER

version of the solution set over FP it would not by any means be a discrete version I mean some discrete subset of the graph er er of e over R not in any I mean not even remotely can anybody say why mean how come I simply took a

subset a subset FP is a subset of R and I looked at all solutions I mean here the graph was the the set of I mean here the graph was the set of all solutions to this equation now I look at the set of all solution why don't I get something like like this I want to say something related to sub Fields but yeah so

say I mean if we can assume that FP is a subset of all the real numbers uh everything that it's clo I mean it will share at least from zero to p uh all of the values right so according to the image that you show all of the values between zero and B should still be a straight line once you arrive to B you have the modulo so you get back and at the end you cannot have a drawing that it's just dotted lines so the reason is that the the solution set highly depends on the arithmetic of the of the

field that we work over right and FP is indeed a subset of of of the real numbers but it's not a sub field I mean the arithmetic on of on FP is not induced by the arithmetic of of the real numbers so the solution set is completely different okay and yes Lisa uh but if it uh will be be uh another function uh that another equation that uh on a real numbers is defined uh exactly on the

same uh like if it is defined uh exactly on the FP of R like on FP uh area of R then it will be the same I'm not sure I understand the question I mean the issue is of course if if all the all the values of x x cub and y s ER are smaller than P me if you take XY such that both X Y X Cube and y s and also X Cub + 2 x and x

Cub + 2 x + 3 are all smaller than than P then the arithmetic is the same as the arithmetic of R and this will be a subset yeah this is my question okay but but but of course once you go typically once you you take High values of X then X cube is a in in in FP I mean X cube is is not the same as as X cube in R

so it will be right I mean you you can H if you want you the same principle of course works for a for any prime number so take a take P equals 5 P equals 5 once you do I mean once you take X to be even even two then X Cub is going to be eight which is which modu of five is is three so it's not going to be I

mean it's not the same arithmetic as as taking X Cube as a real number okay and typically and that's that's really the the the heart of the the the difficulty I mean typically over over a field of finite characteristic we don't have any geometry I mean sure you you could arrange change the the numbers one up to P minus one in the xaxis in the y axis but you will not get this H this nice picture of a fish or a fish in the

lake and so on you will get a a you can try to plot it for large primes you will get some a some junk I mean there is no no nobody at least h until now was able to to make sense of the the the geometry if you try to plot ER the the solutions of a of an elliptic curve over a finite field so a main theme in the in the study of of elliptic curves is to use

analogies that is a I I have the intuition of geometry from the elliptic curves of the over the real numbers and I can I with this geometry I could do different constructions like I could look at the tangent line and now when I when I work over a finite field it turns out that a lot of the constructions like the the tangent line h make sense in over any finite field so I simply call them the tangent line even though there is no real geometry for which this is a tangent line okay so

this this this kind of a a method of reasoning is called algebraic geometry yes um would FP be called the base field of thetic curve ER yes we say that the elliptic curve is defined over I mean this it's kind of in your declaration you need to declare over which field your elliptic curve is defined yeah it's a bit like a a a a function you need to declare what is the

the the domain and the range what is the input set and the output set like in a a a typed programming language when you when you write a function you have to declare what is the type of the input and what is the type of the output and if you declare different input type and output types it's it's a formally speaking it's a different function thank you okay so um so this kind of reasoning is algebraic is is generally called algebraic geometry algebraic geometry studies a a solution sets of pols in

a I mean typically in many variables polinomial in two variables I mean the solution set of polinomial in two variables result in to a to a a a onedimensional graph that that's that's the reason for the name curve ER of course the the one onedimensional graph depending on the degree of the the two variable polinomial the onedimensional graph could be could have a different shape um and uh you what people do in algebraic geometry I mean they study a a

for example if you take polinomial in three variables and you look at the solution set then then you have like two degrees of freedom I mean right if you have a polinomial in two variables and one equation then you have one degree of Freedom so that's why you get one dimension if you have three three variables you have two degrees of freedom so you get surface and in general polinomial in N variables I mean the solution set of polinomial in N variables will be a a an N minus one dimensional it's called a variety ER and you I mean just like just

like I mentioned here with this H ER I mean the the geometry over R can serve as as analogy for a constructions over over a finite field this is something typically people do in algebraic geometry a lot but we so from the perspective of algebraic geometry we are on the grass level we study the onedimensional ER I mean basically the simplest simplest object it's like a curves onedimensional

varieties ER but as you will see it's still it's still a very rich object ER and um we will use a little bit of tools from algebraic geometry but I will not I will not get into the the actual Theory so maybe it's a good time to take a break and and after that I want to H come back to this uh this condition on the discriminant 10 minutes Madame as usual yeah yeah okay cool

okay uh can we start again yep all right so remember we we have a the typical equation Y2 = X CU plus a x + B and we said that an elliptic curve is such an equation with the condition that for

a for a cube sorry yeah 4 a cub plus uh 27 b² is non zero by the way I will often denote the elliptic curve as EAB so this is a a short annotation to to talk about the parameters A and B now suppose you have a um

suppose um fxy is is an equation um over over the real numbers a tangent line in to F or to to the graph of f to be

precise at a point in the graph so at uh x0 y z I mean such that F of x0 y0 is equal to zero H is is given by let me denote it as T x0 y0 this is an equation of the form Yus

y - yals m x - x0 where m is the derivative of y with respect to X at the point x0 y0 now the tangent line

yes I have one question so the tangent line is is going to be a a straight line if I'm not Mist yeah a straight line so in regards of that depending on the point where you are doing the tangent or maybe not even depending this line will collide twice with the curve or collide with the c at least I don't know yeah yeah it will intersect but a a a tangent at a point let's let's take the example of a of of an elliptic curve ER let's say you do you do tangent here this is x0 y

z so it is true that the tangent will intersect the the elliptic curve in at least another Point possibly even further if you have this Lake right but a tangent means that there is a a neighborhood let's say on the x axis around x0 for which if you take the the part of the the graph above the the

neighborhood the the line intersects the the curve in exactly one point in this neighborhood Okay and like so can I say that I mean what does this imply in the sense of because we the neighborhood just appeared now but um I'm I'm wondering in respect to the solutions to the polinomial and all this kind stuff does it have any effects that this tangent line collides in in places it doesn't really matter for anything it doesn't it doesn't matter as long as we again we are interested in the the property that it

ER it touches the ER it touches the curve in exactly one point in some some neighborhood around around the the point x0 y z and it's true it it may intersect the the curve in in other points but it's a this this kind of intersection in x0 y0 is a is of a special characteristic right it's it's it just like a you know you have a parabola then the line that is tangent

to the the base of the parabola or even the line that is tangent here has different characteristics then a line that simply intersects the parabola right it this this kind of line the the tangent line you can take the slope this is a I mean let's call this this angle Alpha then Al then then tangent Alpha is a what we denote as the slope

and and M is as I said it's it's the derivative of the the function with respect to the the variable X and it has a special characteristics precisely because of the of the the relation between the slope and the derivative okay so it's like it's a a it's a line that touches the the graph of the function but does not does not intersect it I mean it it may intersect it in a a later right but in some

neighborhood Let Me Maybe draw it again in a parabola you take a you take a point and this is x0 this is x0 y0 and you take the tangent line H then it yeah it's a um it kind of it touches the graph of the

function in exactly one point but it's it's either outside of the graph or inside but it's not in both ER in both sides I don't know if if that that's yeah that explains it thank you very much so it's it's like a just like for a circle you I mean there is a difference between h i mean there is a different characteristic between the tangent line to a circle and simply a line that a generic line intersects the circle in in two points but the issue is not the the the number of intersection points but rather

that the line is is is completely on one side either inside or or outside but okay so so this is the tangent line it's a we have an equation for it I'm not going to develop it completely this is something you typically do in high school mathematics but for us okay so we have a tangent line for for

any any graph of a function if the derivative exist but we are dealing with pols Pol for pols the derivative is very it's very easy to to calculate I mean unless you're dividing I mean unless the derivative ends up a a in a formula that I mean when you substitute x0 y0 you suddenly divide by zero then the derivative always exist and my first claim is that um the the condition on the

discriminant so let let me let me be formal H suppose e over R is an equation of the form Y2 = X Cub plus a X plus b then e is an elliptic curve I mean elliptic curve of course it

means that in addition to the equation we have this discriminant being non zero if and only if for every a a point x0 Y zero on E there there is a well defined tangent line and just to give you an example of

kind of a example when when the the tangent line does not H is not defined uniquely H not an example you take a let's say y equals a x Cube then this is something like like this and the tangent line or maybe this

it would answer your question Carlos a bit better why in this case we don't have a well-defined tangent line I mean I would typically want to take the tangent line to be the the purple let's say it's zero I mean x x0 y0 equals 0 0 and I mean I I would want if I had a tangent line in this H in this case then I would expect it to be the the purple

line but it I mean one property that I expect from tangent line is a kind of continuity so if I take the tangent if I take the tangent line H at this point and then I take the tangent line at this point and I get closer closer to ER uh to zero then I mean I get a right then I I get this this tangent line here but on the other hand if I if

I do it from this direction then I get the the blue line so one property that I I demand from a tangent line is that if I if I kind of take a a family of tangent lines at a points getting closer closer and closer to to x0 y y z then I get kind of a sequence of tangent line tangent lines that converges to uh

to the resulting tangent line at at x0 y0 yes I mean from like if if I analyze what you just did from the limits perspective like if you were doing a limit from one side and from the other one and getting close to zero you will end up in the same value if you go from both sides but in the case that you did now which is taking the tangent light at points that are closer to zero coming from one side and the other one how can you end up with different uh axis lines like how is that from the right you get the purple and and the blue from the other one yeah yeah yeah sorry I'm I got a bit confused ER I think let me take it back I I do not get

the the blue I'm not correcting you in any sense about that you you right to uh you're right to the issue here is I mean this is not a this is not considered a tangent line I mean we H we we do have the derivative ER okay but know what let me let me take it take it back I

mean what I want in the CA in the case of elliptic curves ER let me take back the non example for now ER because I don't want to to get you into this H calculus issues of limits and so on it's it's too complicated I think for for our purposes but a well okay you you you will see in the proof of the proposition what I mean by a Well define tangent line okay okay fair

enough so so of course um we can write y equals plusus square root of um X Cub + a X Plus B and H so it has two parts right I mean h in this example it would it would be

like the the first half and then the second half okay and each of these is an ordinary function I mean it's an ordinary function in one variable so I could think of it as a y equals a y of X it's a function of x and um when y when Y is non zero then I can simply take the

derivative ER then um Dy to DX is going to be uh plusus uh 3 sare that's a over uh 2 square < TK of X Cub + a x + b equal Z right so um

okay so I need to I need to correct er sorry this m is not I apologize this I did it a bit fast um so going back to the definition the tangent line is a line that ER passes ER

passes through through the point x0 y0 ER and is given by this H by this equation m is called m is the the slope of of the tangent line T and what I know I mean in the ER in the tangent line is that H the the derivative okay

so uh in this H in this case where H ER Y is non zero then I want the derivative of y with respect to X to be zero and this happens ER if only if I mean this this equation is is satisfied ER and and then you we will have a h

I mean a well defined tangent line if um so we will have a well- defined tangent line um in order to have a well defined tangent line I need that X Cube plus a X

plus b is a bigger than zero yes do we need any quadratic residue proper this like how can we be sure that the square root of of this number exists with yeah so as long as as it is positive then the square root exist now okay so now if a x Cub + a X plus b is negative then

and there is no point on the elliptic curve for which the ER I mean but mad positive or negative here is might just be a convention right because we are in a cyclic uh like we're in a prime field where we have a modulus and we go from from zero to P so vales are positive no the the entire proposition is is over the real numbers oh oh oh fair enough fair enough fair enough okay apologies apologies H so so if if the expression X Cub plus a X plus b is bigger than zero or

even yeah if it's bigger than zero then the square root exist and this expression here in the Box H is Meaningful I mean I'm dividing by ER by two two * the square root of something and something is non zero if it if this expression is negative then there is no XY on E ER with this H with this value

because because remember to to to be a point on the curve you need to you need to satisfy y² equals x Cub + a x + B if this thing is negative if this thing is is negative then it cannot be y Square I mean of any y right cu cu y square is always non negative so so this this case we can ignore ER and then we are left with the

case um of X Cub + a x + b equals zero and here we need to take a slightly different approach we we do implicit a implicit differentiation and with the simplicit differentiation we take um I

mean we simply derive the the the equation y s equals x Cub + a x + B with respect to X so we do D of DX and what we get is a d of y² DX equals um I mean here here you do 2 y times Dy DX and of course if you do you do D of X Cub + a

X plus b with respect to X then you get a a 3X s plus a so what we get is that 2 y Dy DX = 3x² plus a or in other words Dy to DX is 3x² + a over 2

y so okay so now we analyze this expression suppose Y is equal to Z but a a 3x² + a is non zero yeah so if Y is equal to Z then this this expression is not really defined because I I would need to divide by zero but I could do H we can

interchange and um the roles of of the variables X and Y ER so that the ER numerator is zero and uh the denominator is non

zero so this means the the the slope is a after we interchange between X and Y Dy to DX which is which is the slope is is a a so the the slope ER will be zero and and then I I can I can interchange again

we get a tangent line parallel to the the Y AIS so this this the picture here is really I mean if y equal Z but this uh this expression 3x² + a is non zero then this is in fact the point

here or it could be one of the points here and in each of the cases we have a tangent line that is parallel to the to the Y AIS I mean the slope is a is kind of Infinity but the tangent line is is well defined and then finally I need to take care of the the case where why equal 0 and 3x² + a is is also equal

Z if um y = 0 and 3x² + a equal Z ER then necessarily a is a is a negative number right I mean a a equals minus 3 x square H so we H substitute

instead of a I substitute minus a h to [Music] get so now the equation looks as y² = X Cub - a x + B with a and now a is is a positive number in that case 3x² - a = 0 implies x equals to

plusus square root of a over3 and now I I can substitute it so I have I have this equation it's called it star a substituting that in star gives us the following I have let's say I substitute a A plus square root of a over 3 instead of

X and I know that Y is equal to zero so I I get that um square otk of a over3 cube minus a square Ro TK of a over 3+ B is equal to zero that's the the equation that that comes from the fact that we assume that XY is a point on the curve and this is equivalent to simply saying a 2 a a 3 over

2 divided by a 3 S < TK 3 equals B and from here I simply get a a 4 a cub over M 27 equals b² this is simply by a exponentiation of the the previous equation and and now I get um 4 a cub minus 27

b² equal z um when I substitute back again a equals minus a I get that uh um this is equivalent to 4 a a cub plus 27 b s b² equal Z so we have a so H so we have a well

defined a tangent line at this point if and only if Delta is non zero so this is a okay I I think it was a bit of a a mess um but I wanted to kind of give you at least the the

intuition of where this the geometric intuition of where this uh this condition comes from so what it means geometrically at least over R is that ER I mean we can h we can talk about the the tangent line ER at every point and it will be a a well defined I mean a well- defined notion coming from a equation yeah coming from this equation

ER written written here and M M needs to be needs to be the the derivative at the point but it has to be non zero otherwise we get a well at the generate version so when I I mean when I divide into all the cases and so on ER what I get is is that I need this condition ER Delta being non zero in order to H to have tangent

lines defined at all points H any any questions about this uh this proposition um is there a link between this and the characteristic not being equal to two or three so the characteristic ER not not being two or three is H is simply because we want the short vir form instead of the general vir form yeah okay so there's no thank you it in a way you could you could say it is effect

because when you do the the I mean Mark yeah yeah if we had car two or three we didn't have the short and so we wouldn't have had the same equation at the end yeah but also Delta would be different I mean Delta depends on the type of the equation yeah so in general virus transform a Delta of e is defined differently but but still if you would

like to Define elliptic curve in a general virus transform then once you know what is the the formula for the discriminant of a o of of your equation then your condition would be that the discriminant is is non zero thank you so okay so this is the geometric intuition between for the condition of the discriminant being non zero and I want to to finish today's session by by the algebraic interpretation of this of this condition on the discriminant and this

is the the last proposition which is is true actually um for all all the fields we care about so let K be either the field of real numbers or um uh a field of positive character positive characteristic that is different than

two or three right um a cubic um FX = X Cub + a x + B over K admits a repeated root over K

Bar if and only if the discriminant is is non zero discriminant again let me just remind you this is uh 4 a cub plus 27 7 b² and this this needs to be non zero so in another way of saying the the the polinomial cubic polinomial has a three distinct roads if the discriminant

is not zero correct exactly so this is this is analogous if you remember from high school H when you when you have a quadratic er a x² + b x + C then somehow in the the solution formula you have a a square root of the discriminant or something like that right and so you have plus minus I mean I don't want to write the the whole formula but the discriminant in this case would be h

um what is it I already forgot but b b squ 4 a b square minus 4 a c and of course you have no simply no minus or plus I forgot minus minus um so of course you have no solution if the discriminant is is negative because then you cannot take you have no real solution you cannot take square root of a of negative but you will have complex Solutions um

but you can ask okay I mean the the algebraic closure of the real numbers is the complex numbers so I can always h i mean I can always look look for Solutions of this quadratic over the complex numbers and what I want to to know at least sometimes is when do I have a um repeated repeated complex roote repeated complex root exist if and only if the discriminant is is exactly equal to zero because then the plus minus cases coincide

and what I'm saying is that the the discriminant that we have here has the same property so I I take ER I take any field ER that is not ER I mean either the real numbers or one of our finite characteristic fields and I look for all the all this all the roots over the the algebraic closure so over the algebraic closure I have exactly three Roots but possibly with the repetitions and the condition that Delta is

a is non zero is is completely equivalent to the condition that when I look at the roots in the algebraic closure then I have no repetition okay is this uh the statement of the proposition clear all right so the proof H the proof is very simple um over over K Bar and we can

write FX equals x - alpha 1 x - Alpha 2 x - Alpha 3 H and but of course this is also equal to X Cube plus a x + B so I can open up and and I get a I get three equations one of the equations this you'd have to believe me for now but you you can do it in your spare time one of

the equation is the equation for the coefficient of x square the coefficient of x square is um Alpha 1 plus Alpha 2 plus Alpha 3 and this this has to be zero then the coefficient um and the coefficient of x in this right hand side expression is a alpha 1 Alpha 2 plus alpha 1 Alpha 3 plus Alpha 2 Alpha 3 equals a and a

a and the last equation is a alpha 1 Alpha 2 Alpha 3 equals minus B okay so if F had a repeated root let's say that alpha 1 is equal to Alpha 2 so I will denote both of them as Alpha then I can simply write the third

root Alpha 3 is is minus 2 Alpha yes um sorry I maybe I missed this step I don't understand uh how do you come towards thinking alpha 1 plus Alpha 2 plus Alpha equal Z I don't see the link between the line before and this one I I mean I have the the the right hand side and I know that it must be equal to the left hand side y so I open up the right hand side and this is an equation of two pols equation of two pols means that the coefficient of all

the monomials are are are equal okay so you just expand all the monomals at the right hand side and say because there's no x square then must be equal to zero or something yes exactly okay I I I I mean I'm a bit short of on time so I'm yeah yeah I understand I'm but but it's straightforward y thanks ER okay so so we conclud that Alpha 3 is minus 2 Alpha and then I can substitute it back to this this system of equations H and I get that a is equal to

um Alpha Square - 2 Alpha squ and minus 2 Alp Square which is umus 3 Alpha squ and I also get that b is equal to -2 Alpha Cube okay so that's that's my system now and I can exponentiate I want to get rid of the alpha so I exponentiate the first equation

by by three and the second equation by two so I get a cub equal - 27 Alpha to the 6 and a b² equals 4 Alpha to the 6 ER and now I I can simply H divide H divide the the the second the first equation by the second equation and I get a a

cub over b² = - 27 over 4 h which is of course equivalent to to Simply say that uh for a Cub + 27 b square um equal Z so if there are repeated roots

repeated roots of H of FX equals x Cub + a x + B in K Bar then um the discriminant is is zero and if the discriminant is non zero then I cannot possibly have repeated Roots because that's a a I mean we our reasoning led us led us to H to see that if you have repeated

Roots You ER you cannot have a so if if the discriminant is non zero there are no repeated notes okay so that's H that's that's the proof you can H fill in the details it's it's also in the notes um but but basically what I want you to take from this uh this session is that um the condition of the discriminant

being nonzero in the definition of elliptic curves has a two two justifications if you wish one justification is the geometry and that's we want a tangent line to be well defined in in all points and the second one is algebraic and this is a a the fact that the this polinomial does not have a repeated roots in the in the algebra closure questions requests for next

time uh it's maybe not a question I don't know but like I'm a bit confused with uh the tricks you use I mean uh when uh it's it's not like maybe uh like specifically here but in general like uh when uh we substitute whatever we want by whatever we want and like okay if if that I understand but uh here like when we just like uh take to the third power and to the second power something just because we want to take it to some power it feels like there should be some um uh some reasoning why why we are allowed to do that well I

mean the the reason for why we are allowed to do it is is because it's a the basic rules of Algebra I mean you have an equation if if the left hand side is equal to the right hand side then you can H you can exponentiate at any ex exponent if you exponent the right hand side and the left hand side then you you must get still an an equation in equality but but the reason that we we specifically did exponentiation here is that we want to cancel out the

Alpha and the Mak sense the only way to cancel out the alpha is to have Alpha of the same of the same power in in both you know in in the first equation and in the second equation then I can divide and it doesn't matter that we for example can't go back from this exponentiation or all I wanted to I mean what I wanted to reach is that if I have a repeated root then the discriminant is zero and and vice versa but that's easy I mean what I wanted here is that if I

have a repeated root you you should think of it as a a chain of reasonings so I start like with the Assumption of I have a repeated root I write all the equations and then I manipulate them in a way so that I get an expression depending only on the the coefficients of the original the original polinomial and not depend depending on the on the The Roots themselves Alpha and Alpha 1 Al okay that's that's kind of the the goal that I wanted I wanted to reach a condition for whether I have

repeated roots or not only using the coefficients of the the original polinomial not not depending on the the specific properties of the roots makes sense ER but in in general I mean what we had in the the maybe in the first part of this session ER there are a different substitutions that lead you to simplifying the equations this took a mathematicians many years to discover and so you should think of it

as kind of we we now know how to do it so we have I mean we can teach it very quickly but to actually think about it this requires a a lot of a lot of effort okay got it h there was another question from Ox maybe no I was just uh trying to say the reason we could divide those two equations because we are trying to eliminate Alpha to and in what condition What relation that Alpha will G that's what I was say yeah yeah so so we we are interested in in a in a

condition depending only on the coefficients A and B and not on the not on the roots Alpha because we don't have access to typically we don't H we don't have access to the roots alpha 1 Alpha 2 and Alpha 3 we we have access to the the original coefficients okay so uh let's stop here and uh we continue next time