Computing abelian varieties with complex multiplication
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Computing abelian varieties with complex multiplication

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CC Attribution  NoDerivatives 3.0 Germany:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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2020

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English

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Abstract 
Elliptic curves with special complex multiplications in their endomorphism rings can be used for constructing class fields and cryptographic systems. I will give an introduction to this theory and then move from elliptic curves to more general curves and abelian varieties. Several difficulties occur, and I will demonstrate my SageMath package for overcoming these difficulties, which has been successfully used for constructing complete lists of CM curves of low genus.

Keywords  complex multiplication abelian varieties computation SageMath endomorphisms Shimura's reciprocity law algebraic curves 
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00:00
my name is more coastline and i would like to tell you about the way that i use mathematical softer and i will also use this presentation to attract the us my own the cold the subject is complex multiplication of a billion for eighty's and i will explain this by starting with the most basic age which is the case of complex multiplication of. look to curse for simplicity l a shame that kids a field of courtesy different from two and three and then an olympic career he overcame his youth project of curved given by an equation waste radicals execute plus a expletive me for a and b. in case one nice thing about these curves as a sewage of two points you.
00:41
in a new point which local people are skew this edition makes the the set of pointless curve into a billion group and i will be looking mostly at the and more fun ring structure of the serbian group will take the reins of all of the trade group more physicians for me to. the this is a ring with a point by sedition and with composition as are remote location. for example for every integer and you can add a point the to itself and times and this gives you are an amorphous movie he which will also call an which is the multiplication by an end. for most of the curves over the of shills of directors and jerome this is everything for no other and were thousands and the an abortion ring is just as more for to the rings up but sometimes happen to be extra an amorphous firms so for example if we take it easy to be the curve why squared equals x. two plus extra. this as a poster is a scourge of minus one inside the base for alqaeda let's call it i then we get a more prison which also call i used by sending extra why to minus extra mile i why and if you look at what i swear it is indian workers rings that means that you're applying this site twice. so the point x. y. is meant to the point extra minus way where i squared and that is exactly minus the point exwife for the group lol on the undertaker so this means of the animal humoring contains the square root of minus one as well and in this case we say that yes complex multiplication but. during of goshen integers the eye. so this complex multiplication the is rare for look the curves over the complex numbers and we like to find like to stress fi which looked occurs have this complex multiplication do this will introduce the village close to normal let's first have a look at the j. invariance have lived tucker's.
02:48
j. with the finest seventeen twenty eight times for a cute divided by for a cute plus twenty seven be squared terrain be ordered confusions of the alert to curve and the nice thing about this gene variants that to live to curse or a some more fick over heels of the closure of the big issue if only if there is the same day. in the area with this ginger and can be used for specifying a specific alert to curve and then will use the who gets close to a meal to specify old the clip the curse of complex multiplication we should fix an imaginary could redick number fuels capital k. then who would cost full meal of that field. is the products for all of the curves over the complex numbers who is an amorphous ring as i swore to the natural order of k. of extra minus j. of the so this turns out to be a fine and products it also turns out that dish wholemeal this you don't get doesn't just have complex. both issues that has integer go fish and so this is really a finite the object a discreet objects that you could write down. and the roots of this film you are exactly jane variants of the undertakers complex will the location by dimensional order of cape and mobile only doesn't specify exactly which that the curves over see have complex multiplication it can also be use them to nice applications so the first. the petition is competition will cost field theory the roots of this follow me oh generate exactly the who were close chilled of case which is important to gloucester theory and the second application is that you can use this to create look to cursed with straight order and order for straight properties over find it feels. if you reduce just call me or more below pride and then you take a route their get a gene variant of analytic earth over finance field that's also has a specified ring inside the end of morphine read this gives you a lot of control over for been is an amorphous and and because of that also over the number of point. so this curve that it's ordinary are super singular what kind of georgian it has this has a lot of different applications in case another that we know that we want to computers poll no real question is how do we computer it will they have been many methods. post the over the past two decades but the method i want to present is a complex and political a third are very classical method which works as follows the lift to curse of the complex numbers they are known by the various was maps to course formed the two lettuces in the.
05:41
the complex numbers and because this is an equivalent of categories it also tells us how the and working evenings correspond to each other surrey and working memory of the look bigger. his is the same as during a complex numbers that multiplied the letters into itself and this is another reason to call this complex multiplication it's the complex multiplication is not let us do so. from this can be deduced that the and more simmering is this ring of integers of the field k. if and only if the lettuce years for his after scaling an ideal of history of injures and then if you look at the ability to curse up to a summer from that. response to these ideals up to skillings which is exactly the clause group of this of this feels. so that is how to compute all the critics love to curse for the last full meal and then if you want to know these jim variants if you read the letters thousand preserved which can always do a scaling then this journey can be viewed as a function of how and that function is a question of merger forms it's a merger. function and their very first place of evaluating this j.. a sense that we can compute issue but last fall a meal that is all very standard for example if she starts seage a tight hilbert close to a meal of minus four hundred four so that's the field of discriminate minus four hundred four then starting sage stage more time than computing this will last longer.
07:25
will this column use computers almost instantly. and which you can see year this number fourteen is that clause number of that field and then the school fissures tend to be pretty big you see the biggest school fish and is over year it's the last one and this really huge. so in practice for these competitions especially if you want to make this number four hundred four even bigger they are need to have smaller coefficients and in order to do this the function jay is in practice replaced by a smaller function but another more.
08:08
to function in order to make the call fission smaller and then we call this value for style for that smaller function as we call it costs in various we want a smaller clawson variant. if you do this world a smaller functions they tend to have higher level so it's not automatic that this is still have the same degree in order to prevent a degree from blowing up to have to go fish comes from all over the degree becomes bigger than still doesn't help you need to recognise swear that this effort. so is still in the same tools and for that we can use schumer a source of property law. so if you have a goal was all the more physical. official the field in which a fertile lives this corresponds to claw superior to do some kind of ideal in an ideal group. and generals are she crossed the law says that in order to to get a full trial and then to get to get what sigma effort so is you can act on the function as if in a certain way through a certain matrix depending on this ideal that score sponsors. sigma and the easily way that intel so in order to know whether your efforts toll is suitable only need to know is what are you at play this matrix on it is still the sabres. and. this is all known there is well into the curve case and i would like to know go to the higher genus case of the case of higher dimensional been of radiation your genus kers. the picture there is like this given a curve over a field k. you can take its jacoby in which is a bowler a civilian variety over case the dimension of the jacoby as the same as the genius of the curve and she looked over of creating clothes feels.
10:18
and the genus arabia to g. to be the most three then this man has almost are generated there are also some special classes will be under eighteen's which you also get a surge it to the and those of the cases where we can actually work with curves and motorways we have to work with the beating her eighty's and these are being directed. it is as repeal of the curve case when you take the pill complex numbers to correspond to complex tour a complex gene an angel space which low some kind of lettuce so in theory you can do very similar things are. your field k. and serve measure could read it here to work with sea and fuels which are fields of the form take a totally real number filled cage zero a joint square root of the totally negative element to it so first example if you take kids are to be q you get exactly the imaginary good ridicule. will switch it feels that you have four she and over to curse as well. so here's another example if you take a zero to the queues for to that is totally real and then creative to minus three doesn't totally negative elements that so that gives you a gentle. and this has the same occasions you can generate fields from close to period comes close fills n you can create curse which especially for gene is to win three can also be used in cryptography however there are few technical difficulties first of all i said college story with her. human for this also means that you need to work with the right kind of daisies of these lattices lander another thing is that if you want to generate cost feels oversee until k. they need to be alert to only immediately of or eighty two have called the actual cation but he read. flex fuel of k. which is another field which you can construct from k. but it's always the same escape for example if k.. is no god our of degree for over to you and your to normal closure and then the diagram of shock schools and looks like this picture here and the reflex shoe is another fuel it's here it's not a nation where fifty k. in fact when you go to genus higher than two years. and sometimes happens that katie and the reflex chill don't even have the same degree of the queue. another problem should just do now easily copy the the source formula to curse they are just generate a certain some fuel of the old class children some of the old class children self one reason for this is that you need to connect this case with this case are through some kind of. have more and it's called the tide norm and that's some kind of map that connects k k are in it a computer at this as well. finally once you've created juror complex multiplication in variants of the curve of gene is bigger than one. it's only to create the curve itself from those variants and the methods for doing that often introduce use go fish and for which amended algorithms to make the school fish smaller so those a song of the difficulties that you encounter with this and the results. so for constructing clause fills the a complex will be cation that his work in progress of a ph d. students jurors asuncion room and the supervising together with less energy in bordeaux.
14:06
some of the things that we do already have has a complete list of sea and the curse of genius to over rational numbers and also over some critics feels this is a joint work with her thomas only for him but we're was a massive students in work and we construct a piece. curse. we created the list of careers and hundred culture her ph d. thesis was about. proving that this least that we create the four was actually completed i mentioned that all these methods for numerical so a priori they only give approximate approximations of results and they and it's something like that you. some say sling talks about that this confirms to actually proof that they have sheehan but in this case we were able to bounce denominators of the variants and then if you computer them with interval arithmetic to sufficiently the person than you actually prove that the results are correct. prettiness three and so special cases of genes hired and three we also got similar lists also completely so over to you and finally and i managed to find the appropriate general aviation of humor she was. city law that is explicitly love for competition and takes care of all these difficulties from the previous slide and was able to find smaller last year and using that. so now i'd like to show you this this computer code which you can also get from get to the core jewish and strangeness received imported into this said session and then i can create the sea and field this is young to old jets that also allows us to create from.
16:04
sample see and guides to complete school education things like that so let's take one c. and eight i can ask for the reflex field corresponding to this specific see him play it also creates that. then i can ask for one period matrix of the usual lettuce so here's a period mater's to let this is generated by this cool inverted this call investors and to standard visitors. here's a from childers creates the cost for meals actually in a proven way if the this. it takes a bit longer because actually computer to nominate a formula us and they are not very short so that means of the interval arithmetic his to be do them to a pretty high precision but there it is turns out that this is a minimal colonial of the first in various. the modular spaces higher dimensional so you need multiple invariance now and the other two functions than interrelate the other injury uncertain terms of the first service and when you interpret this appropriately looking at the documentation of this function specifies four points in this more july. space corresponding to the genus to curse that have was jacoby and have an amorphous a brain that much more her case. and you can also use all their modular functions so for example the rose in any variants are what you have. functions i can compute them so here's one that sets sold sort of feta function square stearns another say to function square divided but it could have two others were third of functions and i can usually wait this in one of the spirit major she's for example. to russian two hundred than your it is now i'd like to compute some gall country it's so here's an example to the dollar could use a computer that by acting through ideals from certain clause groups that we take one ideal been primed factor. seven in this reflex feels close real function. and the function of. you want to know how does the golf more person course point to peacefully artema acts on the value of death observed. well then this function give output to matrices and and you you have to let you actual f. and that the inverse inverse of an exit and the gives you. the appropriate go action in a way that you can actually compute. so here those major cities and is this for before made to say you is over over the threshold embers and use this for reform it trips over his at malls aids and we can let you act on the road in an inherent will we get another rosen a new variant so different. when it's also a product of two squares of status divided by a producer is of feta us and we can that the matrix an inverse act on the. on the period majors said the new period matrix year it is and we can evaluate. and then we get the appropriate go work when you go it's also computed in america. ok so those are some of the things that you can do with this code issue any questions please awesome.