PSE Lectures - Ep 20 - Elliptic Curve Tangent Line & Point at Infinity

Transcript

lete go all right so um and recall that um an elliptic curve over field ER k is an equation um of the form y² = X Cub + a X plus

b such that the discriminant of the equation H which is um 4 a cub + 27 b² is non zero okay and last time last time we showed that um the addition Delta Delta e being non zero is equivalent to say

that um the polinomial let's call it f has um or has no repeated roots in the algebra closure of K but we also had so this this is the the so-called algebraic

ER equivalent formulation of um of this of this condition but there is also an equivalent geometric a condition that says at least what we showed last time that over over the real numbers um saying that Delta Delta of e is non zero is the same as um it is

saying that e has a tangent line at every point and and this um this is usually called in in analysis real valued function analysis it's called smoothness now actually when I looked when I looked

back at the the last session I I saw there were ER some small mistakes um and um moreover what I want to demonstrate today is um how to take this geometric condition and make it meaningful over any field instead of the real numbers um and this would I hope demonstrate the the way of reasoning in algebraic geometry and along the way I will repeat

ER repeat the ER the part we did last time on on this uh um equivalence to the geometric condition and correct ER correct the the mistakes [Music] um what I want to start with is what is um what is the tangent line so in calculus you have um let's say a function the graph of a function it doesn't really have to be um

a actual function it could be a curve H like we have H that continues I don't know like this um but once you once you pick a point the whole discussion is local in the sense that you can you can look at the the graph of the resulting function over some small interval in in the x axis um below or above the the point and um whatever you get I mean whether there is a derivative or not does not depend

on the entire on the rest of the um the rest of the curve it only depends on a small neighborhood as small as you like um around this point so you know you could take a this this purple purple part and uh um if you you don't start with a function then you can you can even artificially just complete it to a to a trivial function like like so and this trivial F this resulting pink function in this this point we are interested whether it would have a

derivative or not um would capture all the situation we we look for I mean the the perple F function will have a derivative at the point P if and only if the original curve here on the white um will have derivative at p and derivative of course it has a a formal definition that people doing calculus um but for us you can think of it and this I think ER is is discussed in in

school mathematics um the the derivative at p is like a is like a vector that signifies the direction of the of the function at this point and if we accept so this this Vector this is the the derivative Dy of DX at p and if we accept this definition of derivative then it is natural to um to

define the T line as um as follows so so let F XY be um maybe we start with just a polinomial um polom say um over K oh you know what um yeah okay

um we have a polinomial in two variables over a general field K and I would like to say that um if you give me a um in K * K so it has it has an x coordinate and and a y-coordinate um such that F of p is zero so p is a point on the

curve then if Dy to DX at P exists we h define the tangent line to m f at p is uh the equation um y sorry Yus

y0 equals M * x - x0 so what do what do we make of this uh definition this is this is a general field K remember and of course the the the case you you should always have in mind is is when K is equal to to numbers and then what we get with the picture is is this uh blue line

that so you could say that the tangent line over the real numbers is is the unique line that passes through through the point p and has a slope that is a and goes exactly in the direction of the function um ATP so you can think of it as the the best the best H linear approximation of the of the function in a neighborhood of p is this clear is this h i I can explain a bit more if um if

not yes I'm not sure I understand why we want uh F at Point P to be equal zero because on the graph above it is not zero no no I I did not say zero ah we we say h we say f f at p h when I say f p equals to zero it means that F that P is in the solution set of f so the curve maybe if you want we can distinguish we can can call the curve C but the curve C is the solution set of a polinomial in two

variables fxy equal to zero oh that means that just p is a point in the curve yes so P belongs to C all right P belongs to is is is is saying that F of p is equal to Zer by definition yeah so but of course the derivative does not have to be zero the derivative is zero if you probably remember it from school when when we have a um a local minimum or a local maximum right

like at this point H the derivative would be I mean the slope of the of the T tent line will be zero so when I when I try to make this definition I actually ER presupposed H some some uh um some basic framework I mean namely what um what i presuppose is called a

aine geometry ER and in Aline geometry we it's like we replace um the real numbers with an arbitary field could be in our case we would be interested in in in a fields of positive characteristic and the afine space a n with sub index K is simply K times itself n times this is also k k to the

n and in our case since we we only deal with curves we need the the aine plane this is A2 A2 is simply K Square this this would be analogous I mean this would replace the ambient ER space r r square that we pictured the the curves we we have over the over the the real numbers or the general field line in

A2 is the solution set of um an equation of the form y equal MX plus b where and now m and b are in the H in the field C so in particular if K is finite then of course K square is is also finite and a line in a in the Aline plane this is the Aline

plane a line in the Aline plane will be necessarily A a finite collection of points okay so what we're doing here is we're taking kind of a a leap of faith that um things that we want to say about lines only depend on the algebraic description of what a I mean as of a line is the solution set of a certain polinomial um but as you will see this this can take us a quite far now in

particular when we um when we say the derivative Dy to DX the derivative over the real numbers it can be defined in a in an analytic way as a certain limit and so on I don't want to get into it but um what we need here is an algebraic variant and this is possible due to the fact that we are dealing with pols pols we can always do formal derivative so

um for um maybe example I have let's say y squal um X Cub plus uh 2 x then if I want Dy to DX what I need to do is to H to do what is called implicit

derivation so implicit derivation I um I think of Y as a function of and I derive the left hand side and the sorry Lisa you wanted to ask a question uh yeah I want you to ask about line definition does it mean that a line in the definition is a set of Point pairs but geometrically it can be not align there is no geometric it's a set of pairs yes just like if you think of it also a line in the

plane is is a set of pairs right it's the collection of all all pairs right right ER so yes the um but but it has no geometric interpretation at least not not anything er er meaningful or straightforward be once once the field K is is a I mean is of positive characteristic okay okay okay I mean of course you can H you can you can order this we discussed it last time you can

order the points 0 up to P minus one on the x axis 0 up to P minus one on the y axis and you will have but you will not get a line you will get a a you will get some discrete version of a line up to a certain point um and then things will start to get crazy because you you start doing modul p i mean whenever right so you will you will get some scattered the um scattered

plot and yeah okay so so for so for us a line is simply the the the solution set of um of this polinomial in two variables right so it's like the it's the collection of all X Y in okay a square such that sorry such that um y equals mx + b

um and the next thing I want to to talk about after the the solution set is is um the derivation because in the definition of a tangent line at least over the real numbers I use this notion of of a derivative and what I do in the in the the general case I simply I simply derive pols in a formal sense so to do Dy to DX in this example you think of Y as a function of X and you derive um both sides with respect to the

variable X when I derive the right hand side I simply get 3x² + 2 when I derive the H the left hand side well now I need to um this is like deriving the H the function y s is like 2 y times the derivative of y with respect to X and and now I can I can simply divide

by 2 Y and get that Dy to DX is H 3x² + 2 / 2 y okay so I can always I can always do it for any polinomial I mean I just picked the random numbers but I think you can you can easily believe that this can always be done for the for the polinomial we care about which are very very restricted right it's like just the cubics of a specific form but I can play the game also

differently I can so so this would be the the analog of the H the derivative in the sense we um we talked about over the real numbers but I could also do DX to Dy how do I do this well um I think of Y sorry I think of X as a function of Y and I derive the left hand side and the right hand side with respect to to Y so when I derive the left hand

side with respect to Y I simply get 2 y when I derive at the the right hand side with respect to Y I get 3x stimes DX to Dy and plus two H DX to so DX to dy I can isolate it and it would be 2 y over um

2 now when I discussed the Tang tangent line I left I left an edge case and the edge case is um if you have in our case you have an elliptic curve like this um then maybe let's let's let's suppose

that this is the y then this line is we would like to say that it is tangent to H the curve at but the slope of this line will be infinity and indeed if you try to to calculate the derivation of um of the curve at this point you will get um something divided by zero so it's not defined so how do we deal with this case we I mean of course it could be

that the derivative um doesn't exist for for for other reasons I mean the the derivative ER here is we would like to say that it's in Infinity um and that there is a tangent line but H if I take like a a curve like this this I think is um y² = um X Cub plus uh I think plus X but uh let me

Di and this is plus plus X squ so you see in this curve over the real numbers um we we would like to say that H there is no well defined ER tangent line or even welldefined derivative at H the origin and the reason is that I um I may come from this this direction then I think that the the tangent line is like this but if I come from this direction then the tangent line is is like

this so this this curve does not have does not have a derivative at um at the origin 0 0 and and in did if I if I do explicit sorry implicit derivation I get 2 y * Dy to DX equal um 2x and that is H Dy to DX is 3x² + 2 x /

by 2 Y and when I try to substitute 0 0 I get something that is undefined so maybe another example of of um a place where a tangent line does not exist um you take this curve and and what's the equation for

this this would be um just y^2 = X cub here um you you may at first glance like think that there will be a tangent line simply as the x axis because if I take a limit of a tangent lines coming from here then or or or from the um from the other side I I get the x-axis but when I when I examine it according to our definition I do 2y * Dy to

DX equal 3x² um it means that Dy to DX is uh 3x² uh ided by 2 Y and and that's undefined at a Z 0 and indeed we see this I think I mentioned last time the reason that this is not called a tangent line intuitively is that it crosses the the graph of the function um from two

sides so I would not get into the details of that but that's uh that's one way uh um one property of of tangent lines that can help you guess geometrically where H where they uh where you you have a problem and okay so how do we distinguish I mean we would like to say that here there is a tangent line but in this case and also in this case there are no tangent lines so to say that there there is a

vertical tangent line is equivalent to say that if I let me draw it again over R and the curve is C is a vertical tangent line say it at a point

P um if only if when we reflect C along the the line Y = X the resulting has um tangent

line with um slope zero at the reflected Point P bar and the picture I I I will have a hard time to draw it but in the notes if you open it it's at H at the top of page uh 68 so let me try nevertheless is it figure

two it is figure two yes exactly you see so so the you have the you have the blue c curve or whatever it is black and you have a point p in which the tangent line is um is vertical I mean parallel to the y- AIS and when I reflect it along the X the the Y equals X Line This is the Y 6 line right I get a let me try to um I get something like

H like this okay so it at at this this is the point p and this is the point P bar when I I it along the yal X and um if the if the purple curve has a um has a derivative zero basically I mean a tangent line with slope zero means that

the purple would have a derivative zero if the the purple line has derivative zero at P bar then I know that the original white curve has has deriv I mean has a um a vertical tangent line at the point okay is this this clear just the the geometric picture yeah there was a question which is can we say that we have a tangent if the limit on the left side is equal to the liit on the right side my inition was we can for the example you show

dou uh in the in the curve that is blue in the I uh can you repeat the question I I it's in the chat Rafael asked uh I tried to reply but I don't know if it's yeah no no it's good but H please H in the future just raise a hand um yeah I mean the definition of a tangent line via limit it's possible to to do it but I I'm trying to avoid it because this would uh to make it precise um would take time I mean I

would at least need to define the notion of a limit thanks lot it's just because you you show an example where you were going through the curve to uh towards the left and then towards the right and so it reminded me of the limits yeah yeah it is it's true but but this was just a an informal explanation yeah the formal explanation is in our case I did implicit derivation I isolated Dy to DX and I checked whether I can substitute um 0 0 at at Dy to DX okay if I if I

cannot yes yeah just one question so here I don't know because we have two variables but if if we had just one we can derive again such that we can cancel out the x that it's dividing and then we have 3x uh divided by two only if if those two where exes what I'm trying to say is can we look for inflection points which is the second derivative H those trying to how can I say it like trying to to remove this in determination because here you will be dividing Z by zero and this is not I mean it's true that um called we don't have you don't have to worry

about the the short answer is you don't have to worry about it because what we care about is only the first derivative ER whether I mean there are there are like a lot of kind of bations in calculus of trying to ER fix singularities I mean points in which the derivative does not exist but for us if it doesn't exist it's the end of story we say it's a you know it's a dead end okay okay so okay

so what I'm saying here if if I want to to formalize it um it would it would be like saying e of the general form let's say has uh a line and at

if or let me put it as a definition then if is is if and only if um if DX to Dy of P bar is okay and and and you see um this definition is applicable over any field right it it would be it's certainly a a definition that H we can

extract in over the real numbers over the real numbers it has a geometric meaning um of course the equation of this tangent line then ER the equation of the tangent line is um let's call it H we don't need to call it it would be

ex uh sorry it would be like a tangent sign uh like this it means that um X yes X is uh is equal to a constant which is in this case is x0 okay so this you can think of this equation as as a polinomial in two variables which is in a the generate form and the set of all pairs x x comma y satisfying this equation is is

precisely this uh this dotted line okay so this this is how we we Define vertical tangent lines I mean this is how we we check if a vertical tangent line exist to an elliptic curve over over an arbitary field is this uh is this clear all right so now that we have all the def

definitions in place ER let me just make maybe one more a curve C over smooth if it has a tangent and and of course this this definition

is completely meaningful ER over over any any field C and you can guess what is my next step I mean we kind of lifted the definitions to to the general case over an arbitrary field and now I want to lift the the theorem that told us that the discriminant is non zero if and only if the curve is is smooth okay but I will do it over over a general field and I think maybe it's it's a good

time to take a break here unless you have any any questions so see you in 10 minutes yeah see you in 10 minutes see you now hey um can we start

again all right so the proposition is now as follows um let K be a field and see maybe we call it e X Cub + a x +

b h a curve over over K of course and Delta e or rather e is an elliptic curve if on if it is

smooth in other words Delta e is non zero Delta e is this uh 4 a cub + 27 zero if and only if H if there is a tangent Point p on

E okay is the the statement clear so how do we how do we prove this remember to to have a tangent line there there could be two cases I mean either it's a um or the derivative at the point must be must be well

defined so I take a implicit derivation and I get the general formula for Dy to DX at er P Let's uh let's keep the same notation as before p is x0 y0 so Dy to DX in will simply be um 3

x s + a ided by 2 y this means that Dy to DX at x0 y0 is simply 3 x0 S + a divided by 2 y z so if if y0 is non zero then the derivative always exist and if the derivative exist then we know for sure that there is tangent line by by definition so if y z is non zero we

have a tangent at B suppose y z is e0 but um the uh the den numerator 3 x0 S + a is non zero and this

means that DX to Dy right the the derivative of the reflected curve and DX to d y is simply h 2 y over uh 3 x and do I want to say it like this yes so this is 2 y 3x² plus

a and since 3x uh x0 squ + a is non zero it means that um um that the the derivative at at the reflected Point P bar is is zero so e is a vertical tangent line ATP okay and now we need to handle the

ER the last case in which um both the numerator and the er er denominator um r r so if um two well y z is and 3x0 2 + a is also zero and

necessarily a must be um a negative number right because ER 3x x0 s + a is z um and uh we can substitute instead of a I I substitute minus a and I get that 3x0 minus a is equal to zero with with a now being a positive

number but of course this is equivalent to say that um X x0 is um a over3 square root and plus or minus now I also have a a the

equation um x 0 Cub plus sorry m in in our new substitution it's minus a x0 plus b equals to y0 but y0 is z so it's it's equal to zero um and now I can substitute x0 into this H into this equation and we square otk of a

over3 cub minus a square root of a over3 plus b equals z this is for the uh the Plus for example and uh and rearranging I I simply get a two a 3 / 2 divided by a 3 Square < TK of 3 equals to B which is equivalent to say that H

4 a cub ided by 27 b² this is by I take um I I I raise h both left hand side and right hand side to um to the power of two and of course this is equivalent to say that um 4 a a cub - 27 b² is equal to 0 now I back a equals minus

a and I get that this is like saying that um for a Cub + 27 b² is equal to Z so so you see and this is of course the discriminant of e so and do I want to say anything

else yes so okay so um so I claim that a there is a tangent line in this case if and only if Delta of e is non zero all right

so this this basically finishes the the proof what I I was a bit sloppy about um here in this in these equations um I wrote square root but of course over a general field we don't have necessarily square root of every of every number H but you could you could simply manipulate all the all the equations I mean what what you care about is a eventually this uh this expression

and this this finishes the proof right we we have um we wanted to show that H being being an elliptic curve is the same as saying that we have well defined tangent line at at every Point any questions about uh this proof I a bit confused with the last step uh with the conclusion of the last step when we say that uh if uh uh y0 is zero 3x0 SAR + a is also zero uh then the

discriminant is also zero and there is no tangent line at in this this case ER so yes so it's I mean to to have the the last step says basically that to get the condition that y0 is z and also 3x0 squ + a is zero is the same as saying that the discriminant is zero so if you you turn it over it means that if you know that the discriminant

is non zero then this case can never happen I mean this case the the case where y0 is zero and 3x0 2 + a is also zero is is the L case in which the the derivative is undefined right in all the other cases the derivative is defined either the derivative is or we have a tangent line so I'm taking the the the L possible case in which the derivative could be undefined and also that it is not a vertical tangent

line and I say that in this case I must have that the discriminant is is zero so we if I know that the discriminant is non zero then I know I can never I can never get to this case Okay Okay H so this this is the the proof over a general field and I think it's it's a it's a good demonstration of um of the principle of what we do here I mean we we take a um a geometric notion let's say from real

valued functions and we try to look for equivalent formulation of the the notion we we're talking about in a purely algebraic way and once we have the the purely algebraic definition then it makes sense over any field and we can we can then try to try to turn all the I mean at least some of the proofs about properties of this not notion that we had in the real numbers ER to approve over General field to and the way to do it is to to have

completely algebraic arguments so so this is this is kind of a a perpendicular way to H to analysis we we try to to evade all the analytical Notions that that depends on the the specific properties of the the the real numbers and we are left with kind of the algebraic content of it okay um other questions please raise a

hand because I I can't look at the chat all the time um okay so so this was um this was the ER the definition of analytic curve and the next thing in order is is the group LW uh on the set of points of an elliptic curve H and again we are going to follow a similar

principle I think I won't be able to do all the details today h but let me let me sketch the case and we we get into more details next time so let's say you have one of these elliptic curves over R and you take h two points p and uh Q

or maybe let's take a p here and Q here I I draw the line between p and now there is a general theorem that because this is a a cubic because e is a cubic then this line must intersect e in a third

point which we can call R H we we don't we are not going to rely on this theorem we are going to um show algebraically that there is is a third solution a third point which is a solution to the the line that passes through p and Q and uh and the curve e so this we can do purely algebraically and R let's say is is some x0 comma y0 and since e is um is symmetric along the x-axis

I can reflect the point R so this would be x x0 minus y0 and this I call p+ now this is not this is not um comp completely satisfactory because I have some edge cases I I didn't take care of the the most obvious Edge case is what do I do when

um well you could um that if if I want to add P to itself then I need the I take the line that passes through P this tangent line will intersect the curve in another another Point let's call it R and then I take the reflection and let's call it R Bar this is the

reflection along the along the x axis and I simply Define our p+ as as R Bar um a more interesting Edge case would be that well if now I take a point p in which the tangent line is um is vertical if the tangent line is vertical then it does not intersect the curve in

any other ER in any other point so for this I I need to um to have a a a an additional artificial point which I call the point at Infinity so this this is now the case now I have p and I want to do p plus P so to do p p+ p i I need to draw the line the tangent line at p and I need to look for the intersection

of this line with the Curve technically speaking it does not intersect the curve in any additional point but if I add artificially a point Infinity which I think is is lying H at the top of the um at the top of the picture and also at the bottom of picture then this this line will intersect and I and I now technically include e as sorry include Infinity as as a point is a point in the curve on

curve so now the tangent line will intersect the curve at infinity and according to my my recipe I let's say I take the tangent line I intersect it at infinity and then I need to reflect along the xaxis but to reflect along the x-axis is is also H is also Infinity so I get that a p+ p is going to equal Infinity um maybe it's good to make a formal

definition uh let yes yeah um so I understand the the point infinity and the point the previous case with the point R what I but do you have any intuition why we we are doing this thing like why are we taking the tangents and why are we taking the reflection on the XS is like how I guess it all works out in the end because it makes a group and this but it sounds like out of nowhere basically yeah I mean yeah yeah well I don't have a I don't have a short answer and the answer

is there's there are several types of answers basically there is there is an abstruct theorem that tells you that in this situation ER you always have the the structure of of an ailan group in this way but um for now you you can I mean let's say a more kind of naive motivation is you can always say I mean since I have a cubic it will

definitely let's say in the non- generic in the non Edge case I take two points I pass the line through these points and I um I I I know that I must get a third intersection point right this point R ER now okay it turns out that in order I I like the the what probably people did in in the early stages is said okay let's let's let's define p plus Q as as simply

R but that doesn't doesn't doesn't quite work but if you if you instead Define it as a as R Bar you you see that you get associativity let's I mean just as a I'm sure that some mathematicians reasoned about it in this way okay so it's more like a so your first answer says but was a theorem that guides how we Define the Operation and the second answer is more like try and errors and yes yes and I think I don't know for sure but I think historically it was a trial and error yeah kind of

thing and and and later people came with the the general theory thanks ER any other question I have one yeah so you you mentioned that um all of this is done such that you can kind of map the entire thing as if as if it was an ailan group um but this confuses me a little bit because whereas for associativity having identity and commutativity I do agree for the inverse element I'm not as sure okay so I I right I didn't Define the inverse but let me first let me first be formal sure

um so suppose we have um an the curve and the set o over over a general field K the set of K rational points um is um this is denoted as e of K it's the solution set to

e so y² = X Cub + a x + B Union the point Point infinity and the point at Infinity we actually Den note O because this may be counterintuitive but the point at Infinity is going to be our our zero element if you think of it according to the construction I

described um the when you when you want to add let's say you want to now add P to the point at Infinity then you take the the line that passes through p and infinity H you look for um for a third a third intersection so okay maybe yeah I I will I will not jump ahead um um what else I wanted to make formal

I wanted to make H the inverses ER the inverses formal so the inverse is very easy the inverse of um of R is is R Bar yeah this this I didn't uh yeah so the the inverse of R is is R Bar this is this is the natural thing you you would want to do but by this Logic the inverse of infinity is going to be Infinity because again Infinity we think of it as lying at the top and at bottom

so it means that Infinity is the is what we want to think about as the the zero element and and that's the reason we we denote instead of infinity we denote it as o um and now you see the importance of the the fact that we have a smooth because in this geometric definition I would I would need to have a tangent line at every point because if

I want to add the point to itself according to this geometric construction I I must take the tangent line to this point and then it would pass through another point I would take the and this would be Q + yes so what happens if the tangent line touches in two different points in the curve at the same time what do you mean will a tangent line to at more than two points apologies more than two points two of course it touches

okay so you when when you defined the first elliptic curve you draw one that that could have a shape of of like this plus a more circle on the left yes so ick I mean I could pick a point such that I I touch in this point I touch in the circle and I touch in the right hand side of the curve again so we will see algebraically that H there would be a unique solution this has a unique Sol this when you take the I mean I didn't draw it well okay that's that's my limitations but you take a tangent line right tangent line to Q so in the case of

a like this ER this thing um you we will see algebraically it's not hard that you you have I mean you take two points there would be a unique only a third Point only one other point that that intersect with it line and when you when you take the T line again my my

drawing is is very bad but the answer is that we will see algebraically that there is exactly okay one one solution to it okay let um okay so yeah so this is yeah so in a way this point at Infinity is kind of a neutral element that's kind ensuring the closure property on the group yes and we have to have it because we cannot have any any group

structure otherwise and and notice what what we say here in fact is that P plus p is equal to zero in for this P so and this will be true also for so it will be true for H all the points which have a vertical tangent line namely all all the roots these are the points in which you have a root of the polom X X Cub plus a x plus

b right so you write this polinomial as um xus Alpha 1 xus Alpha 2 po you know this this is over the closure in this this drawing over r h we some sometimes we have the circle and sometimes we don't have the circle this depends on how many uh real ER how many roots of the polinomial X CU plus a X plus b are over the real number and not not just over

the the complex numbers H but in any case so this this will be um no alpha 1 comma 0 and Alpha 2 comma 0 and alpha alpha 3 comma 0 and according to the the binary operation let's say that we defined all of these points let's say P1 sorry P2 and P3 will will H will be points such that a um if you add it to if you add them to

to themselves you get zero or you get the the point at Infinity are they the same point I mean of they're not the same point because they are not in the same place that's for sure but what I mean is it's it's a bit strange in the sense of it looks like like this identity point is not just a point but like the set of all of the combinations of values that have a particular X and all of the Y's because it's just this straight line of x equals whatever so all of the Y values are possible ah no no no not all the oh no because you're right on the curve only

this I should have said yeah this is P1 yes this is P2 and this is P3 that's and they all satisfy Pi I plus pi equals zero yeah fair enough fair enough thank yeah okay so this is this is the intuition behind the the the group law but of course um if you think of it the way for me to define the group law well I did we didn't prove it's a group law but the way for me to define the binary operation was basically through lines

and tangent lines that's all I needed and this is Meaningful over over field so you could say basically the same the same definition if you now replace the the meaning of a line over the real numbers with what we Define as a line over in the the the aine space over a general field ER then you know you need to work out the the equations to see that H some solutions are well defined but essentially you you you already have a recipe of how to define the binary of

operation over any field okay so this this is again repetition of this uh this type of reasoning that we do in in in algebraic geometry um maybe it's it's good to to stop here um we can have like a short discussion on whatever um questions you might have yeah yeah it's not a specific a question of that but about the topic

related to the limits that we discussed would it be possible to to have some I don't know a do section or call it what or do you think it's worth it at least to explain uh somehow a little bit more for the ones who are interested um about the topic of the limits and how to see I mean if it's po calculus like opal or stuff like this fair enough not need it but if it some some more details that are interesting maybe um you know I think many topics in mathematics it could be interesting but um but I think it would be too much of a diversion from

a ER from the material we want to cover I mean we we want to do algebra not not analysis fair enough and we if you if you notice I mean I I did mentioned limits but I did not rely on it because for us the derivative was was simply a formal er a formal thing I mean we only deal with polinomial and polinomial we know how to derive right so you don't need you know you don't need all the theory of limits when you do kind of formal algebraic derivations of a of

pols of course in order to um if I go back to to kind of justify this picture with the the direction vector and so ER you need to eventually you need to go deep slightly deeper to uh to to calculus and Define the derivative as a limit but that's that's the only thing it's um so if you accept if you accept this ER as as given then ER then I mean

you don't need anything else in this discussion makes makes complete sense you all right so yeah um what I'm I'm not sure how to proceed exactly um but I will I will reserve it for for next time the the issue is that to prove explicitly the the associativity of the group law would be a very tiresome task and ER what I think is we could do

some cases but not all of them but we we will see next time all right so um uh