PSE Lectures Ep 38 - Summary

Transcript

so so as I said I want to to use this last session for um an overall summary of what we did um and of course the the main focus will be elliptic curves but I also want to to talk a little bit about groups and Fields so that you you will I mean to refresh your memory and also um to give you something to um I mean kind of uh the main the the main highlights of each topic that I think it's is good for

you to take to take with you and to to remember so we started with with groups and H here the Focus was a billion groups and for billion groups we had a um what is called the classification which means that um every finite bilan

Group G is isomorphic um to well a product of cyclic groups but it has two um two ways to to write it um so one i''s say finite um of order n so um one way to write this isomorphism is

um integers modulo P to the E P1 to the E1 direct sum integer um p k to the EK K where it is the the product of the the pi to the AI are the the prime decomposition of N and and the second the isomorphism is is written like this

um wait just so I don't make a so it's basically integers modul N1 direct sum integers modulo and um l such that for any I bigger bigger equal to one and smaller equal well smaller than l n i divides n i + one and we we use both of

these um both of these forms kind of canonical formed for for a billion groups to to reason about um a billion groups that we we come across mainly a billion groups that appear as elliptic curves and then we we went to fields and here as well we had a classification result which means that um for any Prime

p and a a positive integer n there exist um a field H which is denoted f P to the N um with P to the N elements um and any other field H with um p2n elements is

isomorphic to uh to this field FP to the n h so this is a very strong uniqueness property um and using this um we we can define concretely a what is called the algebraic closure so the algebraic closure I take um well there is a a small fact that um if h n divides M then M we have a sub field

inclusion or field extension FP to the n as a subfield of f p to the m so so so what we can do now is H we we start with FP and then we go to for example F FP square and now if we want to keep a a

field extension then we need for example FP um h four er and so on and we can take the union of all this so I take Union of FP ER of 2 to the K and then I'm guaranteed that I have a field extension in each step and this I call the algebraic closure of FP and it's also as a set you can you can simply think about

it as just the union of all FP to the to the N where n is big big bigger or equal to one but um but the field structure ER on FP bar requires that ER that you you you talk about field extension so you need a tower like this and FP bar has this property that um every po bomal over FP bar

splits to linear factors okay so that's that's groups and Fields ER the takeway from from these sections um and then we move to H to elliptic curves so we we have this a a group LW on the elliptic curve that that is inspired by this geometric

picture over the real numbers ER that you you take two points you draw the line p and Q and this is say p p star q and this is called p+ q and the formulas can I mean can be presented in a in a purely algebraic way and that's why they they are meaningful over um over FP over finite

Fields um and um this we didn't prove but in fact it ER it follows from something that uh that we proved in in later sessions and that e over um over K if for any field K ER the the the K rational points um is an aan group and of course if we take this

tower that we had of of fields before um then I I can apply and let's say e e is defined over FP then I can take the FP rational points and this would be a subgroup of the FP Square rational points and all of it is going to be a subgroup of e over FP bar so the FP bar rational points of

e this is a tower of group Tower of groups then we went into this this H business of of um rational Maps so I mean we we want we we want to be able to understand maps from the the the algebra geometric object

that we care about in this case this elliptic curve in into the field um but the maps that we we want to consider are pols or quotients of pols in fact so we we do some algebraic construction namely um we take the the the the Ring of pols in two variables and we qu by a um it's in fact an ideal I didn't mention I didn't talk about

ideals in in the course um but we we questioned out by by the relation let's say y² - x Cub - a x minus b equal Z so it's like we we take pols ER and declare that a certain polinomial is going to be zero and this is a a analogous to you take let's say FP you take pols in FP and you you mod out by some some given

polinomial fee and this means that you you look at the pols over FP with one variable and you whenever you do operation like addition and multiplication you do as you are used to like operations on on polom but you do mod it's like and this in turn is analogous to what what we do in integers right we I mean when we do integers modu n we um we do addition and multiplication as we are used to of integers and then we we take modul n so here I I can Define this this I call

pols um over e okay it's like pols in which I identify the the the specific polom that defines the the elliptic cve and already here I have I mean pols over E I have the the algebraic um description so a polinomial over e is is is going to be some some fxy that is polinomial over K but if I if I add two of them or let's say I

multiply two of them then um basically I I can I can replace all the power of of what well I guess um in pols over over E I have a canonical form which means that I take fxy and I replace all the powers of Y by um by x x Cub plus a X plus b and then I get um

some polom in in the X variable plus y * another polom in the X variable and and I can think about pols over e as just um elements of this form and I do addition multiplication and then I I mean the the reduction modu this this polinomial is simply replacing all the all the instances of Y squ by um by X CU plus a X plus b so this is the algebraic description

but it's not just an algebraic description I can actually think of a polom over e as a function namely I take the collection of all K rational points and I it to to the the field K in fact it's not exactly the field K because I might here I have the finite points plus the point at

Infinity H so I mean obviously what what I want to do if I have such an f i i want to say that F of p i if if p is some x0 y0 then F of p is f of x0 y0 and this I mean this is always in K unless p is equal to O So when p is equal to O I say that F of f of p is is infinity and this we called the projective line so a polom in over

e is basically um an algebraic creature but it induces an actual function now not every function from um from E of K to to the projective line is a polom over e but uh I mean obviously like you can do you can do many many nonp [Music] polinomial functions but but the this this algebra creature that we care about always induce um a function like

so is this okay now are you following yes thank you okay so so that's polinomial and the intuition I mean let's say again we we go back over R I have this H this curve e e of R and if I have some polinomial fxy it basically it defines a map I mean typically fxy defines a map from R square to

R right you you you take a point x0 y0 and you substitute in F but when I take f um in this in this uh ER a ring of pols over E I think of f as as restricted to to only the points on on E and and then I I do whatever whatever F does as a polinomial I just substitute

in um in the polom so so a a polom over e is like a function from the green h part to the green uh part here this is like R and of course the the same thing applies to um to to any field but this but then I don't have like a nice picture to to draw um and then we can say okay we we anyway start getting Infinity when we talk about pols over

e ER this because we have this point at Infinity so we might as well talk about quotients of polinomial one motivation you can say you can you can say I have this this guy polom over e what what is it as an algebraic structure it's it's a ring a ring means that I don't have inverses to to every element I mean some elements have inverses but some elements do not have inverses multiplicative inverses so I I would like to embed this ring

into into a field kind of a minimal field so um so the natural way to do is is just is to imitate the construction of the the rational numbers I mean what what do I do with the integers I have I don't have multiplicative inverse for n so I I I look at at the collection of all fractions and in the collection of all fractions the the element n will have an inverse one / n so here as well I can do this this construction and that's why it's called rational functions so

rational functions let's call it rational of e we had a better notation for this this is like K of e um this is I take quotient of FNG G such that f and g are pols over [Music] e so they already are need to come in either they come in this canonical form or I need to think about them as

equivalence classes um such that a g is non zero is not the zero polinomial just like irrational number numbers I don't allow the the the numerator to be to be zero and here I have basically an equivalence relation so I write it like this where the equivalence relation is that F over G is equivalent to um I don't know H over K if and only if

FK is equal to GH but equal equal means in pols over okay and just like in pols we we have the algebraic description of rational functions H but we also have you can think of them as a as a geometric function I mean if R um is a rational map a rational function we

have um a function from the collection of all K rational points to the projective line um that as as with pols I mean I take a point p and I do um R of P so if if p is equal to x0 y0 then I do h r of P this is just substituting x0 y0 in in the polom or in

the the the quotient of pooms are and sometimes I I I get Infinity but I anyway had to introduce Infinity as a a possible value of a of polom over e so I don't care that much um and so so again we have this algebraic description it's like a rational function on e is a quotient of polinomial but it it also induces an actual function so we can think about it and eventually we want sometimes to think about it as a function because we

want to compose like let's say I have some map between the K rational points to the K rational points this could be multiplication by N I would like to compose the the the two things and so this this point of view as of a function is important ER is this okay so now there is this a a a a business of zeros and poles so we have a notion I mean the set of

zeros of R is the collection of all points in E of K such that R of p is equal to zero and the polls of R are the collection of all points P such that R of p is equal to infinity and we we we started to um to unravel a a a very strong symmetry property of a rational

functions um that well I mean it has several implications but the the the the bottom line implication is that a rational function on an elliptic curve is determined up to a up to a product by a constant by the the set of zeros and the set of poles if you if you know the multiplicities of these zeros and PS um and so so we went we went into a into a detour of trying to Define what is the

multiplicity of a of a zero or a pole of a rational function this is this business of uniformize so a uniformize let let me see if I remember the definition it's a I mean say R is a rational function and a p is a point on the curve um let's say we only care about

zeros and polls so um um such that a p belongs to the set of zeros and poles of of R um and now I I I want to Define what is the multiplicity of p with respect to r or what is the multiplicity of r at p so I say um

if there exist a rational function U such that one you at p is zero and two R can be expressed as U to Su D times s where s is another rational function um that doesn't take funny values at P so it's not zero and not

Infinity then and U is a uniformize and um and D is the multiplicity the multiplicity of m r p and we had a theorem that uniformes always exist they exist you

you you pick a point um yeah uh I need to say to be more precise um I I need to remove the the quantification of rational points sorry of rational function so I want to say suppose you are given a point P if there exist U such that for any rational function R

um you you can Express let let me put it here so if there exist a a rational function U that is being nullified at p and for every other rational function R um R can be written as U to the D * s where s is is um does not have a zero or a pole at P then we say that U is a uniformize at p and in this case for this a specific r

that has a specific D we say that D is the multiplicity of r at p and this this is before the theory maybe this is analogous to what we do with in pols in one variables so so I mean if I have a FX is a polom in one variable over a field um and x0 um is a zero of f then how do I determine the

multiplicity of x0 as a a zero of F I say um I I can write f as x - x0 to the um D times G of X where G of x0 is non zero right so the the the D the D that that allows me to do that um is the is the then the multiplicity of um of r at x0 sorry of

f of F at x0 is defined to be this this uh this number D and the definition of uniformize is is supposed to to imitate this procedure but because we are talking about pols in two variables it's a bit harder um so we need I mean here polinomial is in one variable the uniformize is always is always of the form xus x0 for some Z but um in pols in two variables we have

a a more difficult time so we has this we have this definition of a uniformis um and there is a theorem that for any P um there exist a uniformize at P the terminology uniformer is because because of this quantification here um that is a uniformize is some rational function that is suitable to um I mean

with which I can extract the zero or or a pole of rational function R so in kind of a uniform manner is this okay okay so uniformis at P always exist they have a I forgot the so we have three cases I mean h H we have it a at a point that is not of order two P does not belong to e

of2 or or Infinity then I think the uniformer was a I mean let's say p is is a b then the uniformer was xus a and then um if p is is a point of order two but not the point at Infinity um um I I forgot what was the form let's see

maybe it was I think it was it's um it's y so U XY is just y it's it's not a because it's actually a z a point of order two has to be of the form a z right and then in the third case if if p is equal to O and then U needs to be U XY is X over Y and this is a a one possible choice of uniformis er you can you can choose

others I mean first you can multiply by any constant it will stay a uniformize um but you you have other choices as well um but the important thing is that the um in a d is uniquely determined um so for any rational function R the the D or D of r

is uniquely determined for any rational function and any and any point well um so all in all we have um um if R is a rational function we Define d I mean sorry we Define the the if R is a rational function and and P is some point on the curve we Define the order of of r at P

to be this this number D and this is well defined and for this I mean it would follow from the definition um if a p does not belong to Zer or P then um R of P sorry the order of of of r at p is is going to be zero um if p is a zero of

R then the order of P at R is positive and if it is a pole of R then the order of P at R is is negative um questions so far ER you can also H ask for ER things

I didn't talk about that you want to to cover AR you specifically wanted to know a bit to to refresh your memory on on this part no but I think this part it's fine but thanks yeah I mean so far so would okay so so then we went into this um this business of a trying to to show the symmetry of zeros and PS of of a rational function um the the main theorem here is that if you take a rational

function and you you take the sum of all orders of um of points with respect to R so I run over all points of e there is a um a prerequisite result that we need to know that the the the number of zeros and poles of a rational function is always a finite set so I can always take this this sum I mean

for for for for points that are not zeros or po the order of of this point with respect to R is is zero so it's a sum with only finite many non zero um summons so I can take the sum it's a finite sum and the theorem says that it's always zero this this is a very in my opinion it's a it's a very surprising result um I mean if you don't know what is going on and this this means that there is a a very powerful symmetry right it's like I have a rational function I don't know anything

about it I look at the the number of poles and the number of zeros and I count each with their multiplicity and I get zero ER and so um this this will eventually will be will um be one of the pillars that that we use in order to construct the well pairing and so on um but to show this you you you can first go to pols over so let's say a stepping

stone um you take pols let's say a polom over e this polom I kind of know I mean what is the possible poll of a polinomial over E I claim that um the only poll of a polom is the point at infinity and this is because I mean what is the polinomial this F is like a I can put it in canonical form for example

and I I need to find a point P x0 y0 such that if I substitute it in F I get infinity well if I if I take x0 and y z I mean if if p is a finite point then x0 and Y Z are element in the field and I substitute it into a polinomial I always get just another element in the field not Infinity so the only way I can get infinity is if I substitute in a polom I substitute um the point at Infinity which I remind you it's useful in this part of the the the course to think

about the point at Infinity as tically as being Infinity comma infinity and then it's it's immediately clear that if you substitute Infinity comma Infinity in F you get infinity so so so o is is a is a pole of F and it's the only pole um but in fact you can you we we had like a a proposition that that showed that the order of a polom at the the pointed Infinity which we know is a pole is exactly minus the

degree of f the degree of F this I skipped but we had like a a a we had a notion of of degree that is special for pols over e because in po nomials over E I have this relation y s - x Cub - a x - b = z and this relation if I want to keep a consistent way of a a degree then I need I need to declare

um let's say there are there are many options I need to keep the proportion between X and Y the degree of X and the degree of Y to be 2/3 so I I take the the Natural Choice is that the degree of Y is three and the degree of of of X is two okay and then and then this and then this equation um is also is also compatible with degree I mean the the the degree degree let's say I move this equation I have y equals y^2 = x Cub + a x +

B then I need that this this will um will will also be equation of degrees but if you look at this choice I mean why if I take the degre of Y to be three and and that of x to be two then I also have equation equality of degrees so that's the motivation but then with this notion of degree of a polom I I find out that the the the multiplicity of the unique pole of any polom is minus the degree of

this this polom and the multiplicity of all the zeros um is going to be the sum I mean is going to be if you sum all the multiplicities of of the zeros I guess I can write it um the sum of all order of um P at F where p is a zero of F this equals to the degree of F this this is like in in ordinary one variable pols I mean so so Al together we get the the big result that we

want we get it for pols I mean the the sum of all um of all multiplicities of of zeros and po is at least in the case of a polom um is zero so the sum of all order p r p in E not r f um is going to be zero but then I mean what is a rational function it's just a quotient of polinomial so once you have the stepping stone you um you can uh it's not hard to

to prove the claim for a general Russian function ER I guess now would be a good time for a break but uh do you have a questions request for the session the part after the break I would love to ER revisit devisors yes because it is still for me something that is like a bit of foreign thing okay yeah sure we we will definitely get to

devisors you see in one in one hour we covered like a half the course or something yeah this is very useful uh so thank you very much yes okay H so see you see you soon e

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so just a a small correction um everything that I said here about this this big Theory and the stepping stone stone is assuming that K is algebraically closed otherwise this symmetry ER doesn't quite hold and we had a few examples you you can cook easy examples with points of order to because you know that if I don't go to the algebraic closure I don't have all

the points of order two like there should be always a Four Points of order two and um and I don't always have them if if I'm not algebraically closed ER so this this symmetry really assumes that that K is is algebraically closed and otherwise ER you know you can have many many examples that do not satisfy it so and this brings us to to D resource I mean what I want to do is I I I take a

rational function and I mean we already saw that it's very useful to to to look at the zeros and the poles of of a rational function and and also the multiplicity so to every rational function I can associate um something I I will eventually a post prior I will call it or let's say I can already call it now the divisor of

R which is I take the sum over all points in the Curve um of um the the order of r at P times some formal instance of the point P it's like this bracket P you can think about it as as like a variable X or XP maybe um it means that I I only want to record to I mean for every point on the curve

what is the the multiplicity of this point with respect to R and obviously like if K is algebraically closed this this was another small proposition for for K algebraically closed e e of K is infinite H so if I want to to to cover also the the case of K algebraically closed I need to know that this sum is is always finite but it's

okay because we we know that a rational function can only have finitely many zeros and poles over over e and and so the sum has finite many nonzero elements and the rest I mean whenever I have zero times P I just I just ignore it I mean I just treat it at zero um and of course if K is algebraically closed that then we have so if if K is algebraically

closed then we know that the the um the sum of all um multiplicities of zeros and poles in P is going to be zero but K not is not necessarily algebraically closed so we we would like to to extend the the the view of what is a divisor to also these cases in which K is not algebraically closed we don't have this symmetry um and and therefore it makes sense

to to abstract a little bit and and talk about divisors as a as a formal artifact so um a DE or over e is a um a formal sum D is equal to sum NP bracket P where P runs over all the points

points on the curve um and I have two conditions one only finitely many um NPS are different than zero and um each NP for for each P n p is an integer and you see the collection of all divisors is denoted as div or d e over

K all divisors and and within this collection of all divisors I have um a sub okay so before I get to that this collection of all divisors is naturally an a billion group why because if I have a formal sum ER the is a su NP bracket

p and I have D Prime which is sum NP Prime P then I I can sum them I can sum the coefficients D plus d pre is the sum of NP plus NP Prime Prim I mean at at the formal instance of P bracket p i do NP plus NP Prime and I run over all all P this is again a finite sum because each of these guys is is is a

finite some I mean so for example the could be um let's say 2 * o - 3 * P and D Prime could be um - 3 o + 4 q just formal sums that has have each of them has only finitely many Expressions that are non zero and if you want to to add them together H well you know what

to do you you get so D D+ D Prime is a minus O A - 3 p + 4 q okay so it's important to to distinguish the um I mean a divisor is like a is a formal expression that is an obstruction of what what we eventually want to capture I mean eventually we want to capture zeros and poles of rational functions but we we extend the view to

to all formal sums because one of the main reasons is that this is an a bilan group I mean if I don't extend it to to this H to this generality I don't necessarily have inverses here the inverses are very easy I mean what is the in what is minus D minus D is is some you just do minus over the coefficients but if you only considered a um de resource that come from rational functions you don't we don't know at least a priori that we have the the inverse of of a given divisor like

this so within the the set the the group of all divisors of e there is the so-called principal divisors principal divisors are um um divisors of the form d r for some R maybe you know a prior again there could be more than one rational function that um that have

a um the same divisor so if you give me a specific divisor like like this a prior there could be many rational functions that have this divisor in this case this divisor cannot possibly be a a principal divisor why anybody this divisor D I'm saying this divisor D cannot come from a rational function because this sum of its

coefficients is not zero is not zero so in order to correct it I would need I mean if I want a a d to be a principal divisor to have a chance to be a principal divisor I would need a a let's say the double Prime I could do 3 minus 3 p and this could be a a potentially a principal divisor but there is a second condition that we will talk about soon that really is necessary for a divisor to be principal divisor namely that if you

remove the brackets and you do the you do the sum in the elliptic cve that is you do 3 * o minus 3 p you get O Okay so and this I will come come back to it soon but this is the second condition of what what does it mean to be a a principal divisor I mean like an equival a characterization of principal divisors um and ER so so here in order for D Prime to be a principal divisor P needs to be

a point of order three that is 3 * P needs to be O Okay so that's that's kind of a side remark we soon get back to it and um we would like to characterize so this we would like to characterize is this a um subset of the set of all divisors um that we care mostly about I mean this is a a divisors of rational functions H so one one part as as we said before we

know that principal divisors have to have degree zero I mean degree is the the sum of all the integer coefficients in a deor so if you want you can think about it like this we have an ailan group divisors over e h and we have a map that I call degree um that goes from this ailan group into the the a bilan group of integers that takes a divisor d which is like sum n PP and sends it to the sum of

coefficients and and this is actually a group homomorphism group map H in fact I have two group two group Maps one is from the the the divisors of e over K into the integers and two from the the divisors of e over K into the billion the the elliptic curve group and this I call sum sum is I have D is some

NP bracket p and I said it to some NP P this this is Meaningful in the eliptic curve right because you can always do integer times a point in the elliptic curve and you can sum points in the elliptic curve okay and and this brings us to a a a the the main theorem that we H we worked quite a lot uh to prove which is called abob and abob says a

divisor D is principle if and only if two conditions need to be met one is that the degree of D is zero and two that the sum of the is zero is O and the the important part here there is if and only if it means that if I I can now cook a

rational function now moreover any two rational functions say R and R Prime um such that the divisor of R is equal to the divisor of R Prime satisfy r = c * R Prime for

some C in the field k okay so this this is a this is a very strong result and this summarizes a a a big portion of what we did on elliptic curves um and is is then becomes very useful in later in the the construction of the whale pairing and and so on but um but it's important that you you you you understand the formulation okay it

really it's it's not hard to remember I think and uh it it really plays a pivotal role any questions about uh about this okay just a this is jumping jumping a little bit back I mean there is this issue of of a kind of calculus of divisors that is I mean if I have um

it's an effect of of principal divisors I mean if I have two rational functions RN R Prime I can look at them in um a a as two functions from E of K to the projective line and in the projective line I can um I

mean the projective line is just a K plus some additional Point Infinity so let's say if um p is on E of K such that um R of P and R Prime of P um are not zero or Infinity are not well I can there is a slightly more precise formulation but let's say this will definitely be true then I can

do um I mean r and RP Prime are like algebraic objects right it's like they are quotients of polinomial so I can always multiply R and R Prime um this this is an element in the field of rational functions but a a a I can also take R of P this is an element in the field K and I take R Prime of p and I multiply them and I I claim that this is equal to R of a sorry to to the

this this algebraic object R * R Prime and then evaluated at P okay and and this like a recurring theme in in all this course is that you can multiply Infinity with infinity you can multiply zero with zero but we don't have a consistent way of multiplying zero with infinity so in fact RP and RP Prime P could be zero or Infinity but I cannot I cannot have this result this equality if one of them is zero if like let's say RP is zero and RP

Prime p is is infinity then it's going to be a an issue but using this this uh this result we can um we can deduce a a kind of a calculus of of divisors namely that um the divisor of a product is the sum of divisors and the divisor of a quotient if if this quotient is is is well defined I mean if R Prime is not the the zero rational function this is the

difference of the of the divisors and divisor divisor times divisor of constant time R is just divisor R okay so um this kind of a results allow us to manipulate and we use it in the I think in the maybe even in the construction of the wh pairing but definitely in the in the proof of properties um okay so this this was

ABI and then the next the next big thing that we did was um a identifying torsion points so but before I before I move to this any questions about is this covering a um the the the issues you had with the divisors yeah it's very very nice thank you very much okay and er okay and so we we

arrived to this issue of the last ingredient basically in order to do the whale pairing is to identify the the N torsion points so I remind you e of n e of n is the collection of all points on the curve such that n p is equal to O So it's like it's not exactly points of order n because order is like order of a point in an a bilan group is the minimal number such that if you add this point to itself you get zero but [Music] here it could be that

that that let's say let's say n is equal to 2K so it could be that um n is equal to 2K and um p 2 p is already zero which means of course that um NP is going to be zero and this means that that um NP sorry that P is a point of order of is an N torsion

point but the order of p is not n the order of p in the group E of K is going to be two so it's like a generalization of order um it's like all the points that get nullified by multiplication by n and we we already had I mean when we discussed a a points of order two I mean when n is Prime n torsion points and points of order n are the same the the the difference be could

could be when n is not a prime so when we talked about points of order two It's also two torsion points but it's not um but it's because two is a prime um but more more generally we want to understand e of N and H ER we already know that e of two this is always in fact you it's very easy to check that this is a subgroup of e of K Bar um I mean because if you you take two points of two n end torsion points p

and Q and you add them just like you add in the elliptic curve and and then you multiply by n then you get NP plus NQ and this is O so so also p+ Q is going to be an in torsion point and and similarly minus p and minus Q are going to be inion points so this is in fact a subgroup of the the elliptic curve I mean the k bar rational points of E um and and as a sub I mean since it is a subgroup is is a group in its own right

we know the number of points the number of points of order two or or two torsion points we know them to be four right this is like a um the point infinity plus three others let's say P1 P2 P3 um and we know that E2 has a group structure so if I do P1 plus P2 I get something either I get o or i get P3 or or something I mean something within this

group um and so I would like to to to to say what what is the isomorphism type of this of this group E E2 and it was not hard to see that E2 has to be isomorphic as a as a group to Z mod 2 direct some Z mod 2 this is basically because there are only two two groups of order I mean of size four ER there there is a a a integers module four and um and this guy this is called

the client group and in integers modu four I I I have an element of order four namely one I mean if you do 1 + 1+ 1 four times you get zero and in E2 I never have I don't have an element of order four I all the elements are of order two so the only way I mean E2 has to be a isomorphic to a group that has a a that all its elements are of of order two and and that must be decline group it cannot be integers modul okay but that that's like a baby

case that's easy and we want I mean the big Theory here is that um e of n is isomorphic to integers modu n direct some integers modu n i I forgot to mention again this issue with algebraic closure um I mean e of n is intentionally defined even if k even even if K is not equal to I mean if K is not

algebraically closed the the N torsion points are always defined as [Music] um the points in in the I mean the k bar rational points okay and and then this this theorem is always true um but in order to in order to prove this theorem we we need to to analyze well first so so um Stepping Stones one is to

analyze a so I have the multiplication by n map this goes from Ek to EK and this I mean p of course goes to NP and we basically want the the kernel I mean we e of n is going to be the kernel of this map right it's like the point the the points in EK such that NP is equal to zero but um this this is a new creature it's like

mapping from the the points of the from the K rational points to the K rational points and in a polinomial way because we know that the addition I mean the group law the on the elliptic curve is is constructed by by polinomial or quotients of pols so we needed first we needed a notion of a a rational rational map um rational map it means that H let's say raw from e to e it doesn't

have to be from E to e it could be from E to e Prime but in our case we only care about um um rational Maps between the curve to itself H let's say the curve has a um the equation y^2 = x Cub + a X plus b then raw is is like a pair of rational rational functions so R comma s where R and S are rational functions on E so

RNs in particular can be regarded or can be viewed as functions from E of K to the projective line so k k Union Infinity H and and what I want from from raw is that it for so is defined over over some some field K and I want um for any

extension let's say l of k um and any point on the curve let's say this point is a b and it is on the curve over L it's like an L rational Point H then I I can plug in P into R and I can plug in P into s right because R and S they each of them has two

variables so I have r a i just write RP this is a and I have a s of each of these guys is is a um is an element in the in the projective line um and this I think of it as like the x coordinate and this is the y coordinate and in order I want that RP SP

will belong to e of L so I I want to I want raw to to map points in E of L to points in E of L and the only way to guarantee it I mean what what does it mean to be a a point in E of L it means that the if I take the y coordinate and square it I get a a a the same thing as if I I Cube the x coordinate and I add a times the x coordinate and I add the constant B so I want that s

p squar would be equal to r p Cub + a r p + b okay and then I I am guaranteed that raw induces a a function from E of L to e of L for any extension L and again this any extension is important because I might not start with an algebraically closed field like this this K could be FP and I I I would like

to to consider the algebraic closure of of the field I started so whatever equations I want to be satisfied I want it to be satisfied over all extensions and and there is a main example of this this is I mean a multiplication by N is a rational map this is this I I can think of it going from E to e so for any any field that you you put I mean that contains the field in which e is defined any extension of it I can do

multiplication by n and this is so EG multiplication by n is rational H any questions Nicholas or anybody else okay and so so then multiplication by n in particular it has two coordinates this we denote them GN and H and we went about analyzing the the so each of these guys GN for example is going to be a rational function on on e

as a rational function it has zeros and poles right so we went about analyzing the zeros and the poles of of these coordinate functions of multiplication by n in order to eventually understand multiplication by n and er this became a an interesting mix of techniques because we we had a um what was it how's it called recurrence relations so we analyzed GN and hn and eventually we had a theorem that

identifies a um the the devisor of GM minus GN this was um in terms of um in terms of um e m and and e n and I think well e m minus n and so on um but this whole thing well again I forgot us another detail um this theorem

is only true if um if the gcd of of p andn is one I mean if if P does not divide n basically and if P divides in then it becomes something else it's it's a mess we didn't touch it at all because we don't need it I mean in the construction of the whale pairing we always assume that that the the prime does not divide the the the N the n in the end torsion points and P here is the the characteristic of the the ground field okay so and and this of course

this propagates into into this theorem so so P does not divide n otherwise we don't know um but using this we we recovered another stepping stone that um the number of points on e n is n squ and then using the we had an argument using the the classification of of finite a billion groups that says that en en

is is isomorphic to ZN some ZN ER and okay and that that led us to H to the well pairing but I think ER I want to leave a a little bit of room to kind of open discussion or questions so I will not H review The Well pairing so one question this last thing is the same result that we get we got like some minutes ago when you were talking about the end option points right uh can you ask say again yeah so

this last thing about like the the group of the ENT point of the little Cur being isomorphic to to ZN plus ZN it's like we got this result here but like few minutes ago you also got the same result uh when you were uh doing the endorsment point to stuff yeah I mean I I I guess I I wrote it in the beginning just to indicate what is where are we going this was like a bit this was like this was set as a as a big goal okay okay okay in some point and then we and then we had Stepping Stones to to it okay I see okay thanks yeah so I mean the way here here if you

remember H to to to um to prove that the number of points of n is n Square this was recurrence relations and and after we knew that this is n Square then we we use the classification of a billion groups n some some basic properties of En because we know we know that for example it cannot be z modulo n Square because in Zed modulo n Square we have an element

of order n square right and in N all the elements have order at most n ER other questions if we have an offline meeting in uh in death con maybe we could review than the well parents there and uh the Miller algorithm because I I missed last session because we had an adting and I looked through the Miller algorithm on my own and I kind of like I technically understand what it does but I didn't get any idea why it

looks like okay yeah sure we can we can talk about it I tried to Google something like I don't know um video like YouTube video or articles but no one explains uh no one explains it yeah so I will explain it's not a big deal H yeah but I guess I mean it's not going to be today we ran out of time other requests I mean for Devon or or otherwise then H thank you all for uh

staying this this uh this long with the course and uh yeah see you around thank you so much it was incredibly cool it was incredibly cool thank you yeah thanks a lot see you in person then yeah yeah see you Inon bye bye thank you again for