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>> Kristin Lauter: Hi. So today we're very pleased to have Everett Howe visiting us from CCR,
Center for Communications Research. And Everett and I have a long history together. I actually
went to the University of Michigan to be a post-doc there so that I could have the opportunity to
work with Everett, who was an assistant professor there at the time, only just in time for him to
leave before I arrived. But, luckily, we've caught up with each other later and collaborated on
various projects. Everett is an expert on a billion varieties and polarizations, among other things,
and today he will speak to us on Genus-2 curves with a given number of points.
>> Everett Howe: Thank you, Kristin, and thanks for the invitation for coming here. It's a real
pleasure to visit Microsoft. And I'm even happier to get some snow. It's been a long time, I think
Michigan was the last time I had some snow. This is essentially the second part of Peter's talk.
And he had the harder task of trying to explain the nonexistent result, the lower bound result.
And I have the easier task, I think, of explaining how we can solve an easier problem.
So, again, the title is Genus-2 Curves, with a given number of points. And the genesis of this
question, as Peter explained, is an easier question, given some integer N, construct an elliptic
curve, some E over some finite field, with N points.
And this was solved in Reinier's Thesis. There's this efficient algorithm that heuristically one
expects will produce such an elliptic curve over a finite field in time over polynomial and log N.
So that's good.
But you can think of ways of generalizing this question. And what Peter talked about was
something hard. So given N, construct some curve C over some finite field, say of Genus-2, with
the number of points on the Jacobian of C equals N.
And as it turns out, the only ideas we had for doing this, kind of the obvious generalizations of
Reinier's thought were too hard. So this was too hard to get a nice polynomial time solution.
And, in fact, I think the most efficient method we know of for solving this is to pick a prime Q of
the right size and pick a Genus-2 curve over that field and hope that the Jacobian is the right
number of points, which is not a very satisfying algorithm.
>>: What about the discriminative ->> Everett Howe: I think that's worse, I think.
>>: Can you even count the number of polynomial Jacobian efficiently?
>> Everett Howe: That may be hard but you can check to see whether there are N. So if N is, for
instance, square-free, then you could take a random point on the Jacobian and multiply it by N
and see if the point is killed by that.
So you actually could do this fairly efficiently. The testing the Jacobian part, but the whole
algorithm and picking things at random is not advisable.
>>: Are you sure we have factorization?
>> Everett Howe: Assuming you have factorization of N maybe that N is square-free, there are
some assumptions.
But there is a third question. There's another way of generalizing the first question, which is to
construct some C of Genus-2, and this time you don't ask for properties of the Jacobian, you just
ask for properties of the curve. You want the curve to have N points.
And this is what I'm going to talk about. So the idea that we initially had, I should say this is joint
work with Peter and with Kristin. The idea that we initially started up with was, as in Peter's talk,
we want to see if we can find some degree 4 field. So this is a totally imaginary quadratic
extension of a totally real quadratic extension of Q. And we want to see whether we can find
some element in this field that will be the Frobenius for our curve.
So we would like K to be small in some sense so that we could compute the goose across
polynomials for it. We would like pi times pi bar to be some, probably a prime if it's a prime power
that will work. But it's never going to happen in practice.
And we would like the number of points on C, that's going to be Q plus 1 minus the trace of pi.
So that should be the number N that we are given.
So you could write down what all these conditions mean and then you could try to think of ways of
constructing small fields with these properties. And we did that and we had no good ideas.
Everything is just too hard to control all of the things that we had to control.
So we're kind of stuck. And that was very disheartening, because we thought this would be an
easier problem, and then we were stuck. Until we had a clever idea that we should cheat.
The whole problem was dealing with these quartic fields. They're kind of hard to deal with.
Saying what it means to be small for these fields is, you know, not trivial.
And instead we had the idea of, you know, instead of having the characteristic polynomial
Frobenius being irreducible polynomial, defines a quartic field, we thought what if we had the
characteristic polynomial Frobenius factoring?
What if it factored as the product of two characteristic polynomials of elliptic curves over FQ, what
would happen then? If we could construct a curve whose Jacobian had this characteristic
polynomial, then the number of the points on the curve, which we would like to be N, would
actually be equal to Q plus 1 minus the trace of this factor, minus the trace of that factor.
So our idea now is going to be to find two elliptic curves over FQ so their traces satisfy this
condition. And essentially what we want to do is to take two elliptic curves and glue them
together.
So take two elliptic curves and glue them to get a Genus-2 curve. So later on in the talk I'll
expand this outline. I'll tell you how we can choose these curves over some finite field so that the
traces add up to the right number. But before I get to that point, I want to talk a little more about
this general idea. In fact, I think in the abstract that I sent out, I say that the talk would be kind of
about gluing together curves and various applications. So one application is going to be solving
this problem. Almost solving this problem. Solving this problem five-sixths of the time.
So the reason I want to talk about this is it's a very old idea. Although maybe it's a little bit of a lie
to say that it's a very old idea. I would like to say that people have been doing this for 180 years,
but that's not exactly true.
What people have been doing for 180 years have been the opposite. Breaking Genus-2 curves
into a product of elliptic curves. So let me review some of the history of this idea of finding a
Genus-2 curve whose Jacobian has two factors, has elliptic factors.
So I'll start with the slide. So here is -- well, the author is missing. Here is -- the author is still
missing. This is Legendre's. Here it is. Legendre's, this is the third supplement to his treatise on
elliptic functions, which was printed in 1828.
So that's the 180-year thing. And at the very end of his treatise on elliptic functions, he talks
about ultra elliptic functions. And so what he's concerned about here is, as people were, is
computing the values of integrals. So what he notes -- maybe I'll write this on the board as well -what he notes is that if you're trying to compute the integral of DX over the square root of X 1
minus X squared, one minus K squared, X squared, and he has limits of integration but let's not
worry about them, he says what you should do is make a substitution.
And actually you'll notice -- well, let me write the substitution first. He says take P to be that. But,
in fact, you notice he says something else. He forgets the K there. Something must have
happened to the tech file on the way to the printer.
But if you do this, if you make the substitution, then this integral that has a degree 5 polynomial
on the bottom, underneath the square root sign, turns into something involving these elliptic
integrals. I don't know how clearly you can see this. But now there's a linear term in P and a
quadratic term in P. He shows that this ultra-elliptical integral can be written as the sum of two
multiples of two elliptic integrals.
So this was interesting. This was 1828. Legendre. But in 1832 Jacobi was writing a review of
Legendre's book, and the review was in Crow's Journal, and it was actually kind of hard to find
this article, because it's a book review. People were referred in the literature to this work of
Jacobi. And so you look in the index for this volume of Crow and you find that there are a couple
of papers by Jacobi in Latin. And so you look through them and remember your high school Latin
and try to see anything involving split Jacobians and you don't find anything, it's very frustrating.
In the index of the volume they don't mention he also wrote the book review. So you finally find
the book review. And it's really interesting to read the introduction to the book review. It's very
purple prose. It's about the old Legendre, 80 years old writing this treatise on elliptic functions,
and then Oble comes in because oble has invented this whole new generation of elliptical
functions, and it's very romantic.
But in a postscript to his book review of Legendre's paper, Jacobi says, ah, actually this family of
integrals that was under considered can be generalized. Maybe I'll turn this off for a minute. And
he says look instead at integrals of the form DX over the square root of X. Let me see if I can get
the notation correct. It's, I have to say, awful notation. X times 1 minus X times 1 plus a lower
case kiX, 1 plus lambda X. One minus lambda chi, X. And not surprisingly, there are misprints in
this paper involving these things.
There's sometimes where the Xs are gone and the Chis are gone. It's a mess, and have you to
know what the answer is when you read this to figure out what he's saying.
So what he says is, let's see, well the way we would look at what they're doing is we would say
consider the Genus-2 curve given by this polynomial. Maybe I should say if you take lambda
equal to 1 and said chi equal to K you'll get this special case.
Well, what we would say is that this Genus-2 curve has an involution. You send X to lambda chi
over X and Y goes to some kind of messy thing. But the point is you could just see that there's
an involution just by looking at this transformation. You see that the pole of this function at infinity
gets sent to the pole at 0 and 0 goes to infinity -- I'm sorry. The Weierstrass point at X equals 0
goes to 1 finite infinity, vice versa. The one at X equals 1 goes to over here and vice versa and
these two get swapped.
So this Mobius function, Mobius transformation, takes the zeros and poles of this right-hand side
to the zeros and poles of the right-hand side and therefore it turns into an automorphism of the
curve, an involution.
And if you take this Genus-2 curve and take the quotient of it by this involution, it turns out that
the quotient is a Genus-1 curve. So this gives you a degree 2 map from C down to an elliptic
curve E. And so that's how we would understand Y it is that an integral involving this degree 5
polynomial can be expressed in terms of sums of integrals involving degree 3 polynomials.
That's where that comes from. So this family of integrals that Jacobi wrote about has this nice
property that via quadratic transformations you can write them in terms of elliptic intervals. And in
1867 -- so in 1867 Koningsberg -- I'm sorry Koningsberger [phonetic], in 1883, independently,
communication was not as good back then as it might have been, Picard, they showed that
Jacoby's family is complete, in the sense that if you have any Genus-2 curve that has a degree 2
map to an elliptic curve, it comes -- it lies in this family.
So any hyperelliptic integral that you can convert into a sum of two elliptic intervals essentially
comes from this family. Okay. So this is very old stuff having to do with polynomial substitutions,
and I want to somehow get to Jacobians. And since between the 1820s and now there are the
late 1800s, it seems natural to go via period matrices. Previous so if you want to get to a billion
varieties, you have to step through period matrices.
And in 1884, another kind of round-about story, Covaleski published a paper in 1884 that she
actually wrote in 1874, but it didn't get published for 10 years. And she cites unpublished work of
Weierstrass, who describes the period matrices of any abelian -- the period matrix of a curve -what did I say? A curve that gives rise to an abelian integral that can be reduced to an elliptic
integral.
So he says if a curve gives rise to some abelian integral that you can write in terms of an elliptic
integral, that his period matrix has a special form.
So, of course, you can write different equivalent period matrices, but there is a period matrix of
the form, can be written -- let's see. So you have -- write the full matrix. You have the identity
matrix on the left-hand side and then you have a Tau in the upper left-hand corner. You have
stuff below and to the right of that Tau.
You have a rational number, A over N, and symmetric, in the second spot over, and zeros
everywhere else. And this N is related to the degree of the transformation which takes the
abelian integral to an elliptic integral.
So Covaleski cited this result in this paper that was in limbo for 10 years but didn't prove it. So it
was in 1884 again that Ponkerey [phonetic] proved it. And this special case where the genus of
the curve is 2, and this was also Ponkerey [phonetic], I think. No, this was Picard. Ponkerey
[phonetic] proved this and then I believe it was Picard that proved in the Genus-2 case, if you
write the period matrix in a slightly different way, so you, instead of having identity you do this
anti-identity, you get Tau 1, Tau 2 and a 1 over N and a 1 over N.
And in 1884 so Picard was not aware of this virus Strauss result until 1984 when this paper of
Ponkerey [phonetic] came out. Actually, I think Ponkerey [phonetic] saw Picard's paper and said,
hey, this has already been done. It's just that the news didn't get spread very well.
But in 1884 Picard was able to show that this result follows easily from Weierstrass', even though
Weierstrass has something which is not necessarily a 1 in the numerator. He shows you can get
a 1 in the numerator. So I might show something about that. Yes, it was Picard.
So right down there you see the critical period matrix. Okay. That's enough of that for now.
What does it all mean in terms of elliptic curves and abelian surfaces, how would we understand
it more than 100 years later?
So I'll write over here for a while. If we have this period matrix with a Tau 1 and 2 and a 1 over N
and a 1 over N, let's look at the elliptic curve corresponding to this Tau 1. So this is some lattice
sitting inside of the complex numbers, and let's look at the other elliptic curve that corresponds to
the lattice 1 Tau 2 sitting inside the complex numbers. And let's look at some N-torsion points on
these curves.
So I'm going to let P1 be the N-torsion point given by the class of 1 over N model lambda. And
I'm going to let Q1 be the end torsion point corresponding to Tau 1 over N model lambda. And
over here we'll have a P2 and a Q2, and I need to get the [inaudible] right. I'm going to use the
same idea, 1 over N and Tau 2 over N.
So then you could write a map from E1 times E2 to the Jacobian of this curve C whose kernel is
the group G that's generated by P1 Q2 and Q1 P2.
>>: How was it defined?
>> Everett Howe: C is going to be the curve with this period matrix and you can kind of visibly
see that in here there is a USB lattice with Tau 1 and Tau 2 as the sublattice that's the product of
two one-dimensional lattices.
So if you're going to translate the stuff involving lattices and period matrixes into abelian surfaces,
this is what you get.
And the swapping of the P1, Ps and Qs here, is related to the fact that you've reversed the
identity matrix here. And to make this even more modern, what we're seeing is that given an
isomorphism from the N-torsion of elliptic curve E1 toward the N-torsion of an elliptic curve E2,
that anti-respects, the vay pairing [phonetic], it reduces the vay pairing. So anti-respects the vay
pairing, so I guess that means that if you take the vay pairing, which I'll denote by little e, E2 of
CiP, ciQ should be E1 of PQ inverse. Given such a thing you get a diagram. So here we have
multiplication by N and here we have the graph of Ci, which is the kernel of the vertical map.
So if you take this diagram and turn it on its side, you'll get this left-hand column, and if you take
the dual of this map you'll get this map. And Jacobians have these things called principal
polarizations that will in this case map the Jacobian to itself leading towards dual.
It will fit in a diagram like this where this is multiplication by N. So in some sense this diagram
and the words that are said are kind of the modern-day translation of what this period matrix
means, the fact that can you write the Jacobian with a period matrix of this form. Okay.
So what all this is talking about in some sense, then, is how, you know, for a given integer N there
is some way, kind of some abstract way that you could get two elliptic curves and produce a
Jacobian.
So the question is, the question we face now, since what we want to do now to solve our original
problem is we want to find two elliptic curves over a finite field and I still have to say how we're
going to find those curves but once we've found them we want to find a Jacobian that's isogenous
to the product.
And this family of constructions, depending on N, is going to give us a family of ways of doing
that. And we'll only be able to use these constructions if they're actually constructions, if they
actually can use, write down equations to solve these problems.
So let me turn now to the question of given an a value of N can you actually write down equations
of curves that will fit in a diagram like this? And the answer is: Yes, you can do that in certain
cases, and I guess we've already seen one of those cases.
So now we're moving from observing that some Jacobians can be broken apart to the problem of
given two elliptic curves how do you glue them together. So how explicit is this? So it depends
on the value of N. So if we look at the case N equals 2 first, I've already told you. I've written
down this thing that Jacobi wrote down. But I would rather show you a different way as well. So
this is I think simpler than Jacobi. It's at least something that I can remember and that's saying
something.
Suppose you start with an elliptic curve E1 given by Y squared equals X times X minus 1 times X
minus lambda. And for now I'm going to work over an algebraically closed field. I don't want to
worry about any arithmetic, I just want geometric stuff.
And for E2 I'll take Y squared equals X, X minus 1 X minus mu. You can always write elliptic
curves in this Legendre form.
Suppose you let P1 be the point X equals 0, Y equals 0. I'm sorry, X equals 1, Y equals 0. And
Q1 be the point lambda 0 and here I want to actually reverse the notations. So Q2 will be 1-0 and
P2 will be mu 0. So how do you take this data and produce the curve C that's supposed to exist?
It's very easy. So I propose that that is easy enough to remember. And you actually see that
there are going to be some exceptional cases from this. So let me show you the exceptional
case and then say what happens when you do care about arithmetic and not just geometry. So
the exceptional case is going to happen when this right-hand side has multiple roots. That would
be a bad thing because you wouldn't get a Genus-2 curve. And so this will give you multiple
roots if actually, if and only if lambda equals mu. Is that right? I believe that should be correct if
and only if lambda is mu. Because this would be 1 and you'll get multiple roots there and also
here. And if these two things are equal then you cross multiply and you find again that lambda
equals mu.
So what happens in that case, what happens in this diagram, the thing is that you get a principle
abelian polarized surface down here, but it's not a Jacobian. And what happens is that left side
becomes -- fits in a diagram where the right with the -- is the Jacobian there is actually E1 times
E2 again, and the map is this map whose kernel has size 4.
Instead of getting a simple principally polarized surface, which is a Jacobian, you get a reducible
principal polarized surface, which is a product of elliptic curves. And again if lambda equals mu,
then in fact E1 and E2 are the same elliptic curve. The problem only comes up when you're
trying to take an elliptic curve and glue it to itself by identifying the two torsion subgroup with
itself. And that does seem like an exceptional case.
Okay. So that's kind of what is going on geometrically. If you care about the arithmetic, if you
want to have elliptic curves that are defined over some base field, then you can work out what
happens. And geometrically you're still doing this very simple thing, writing down an equation
kind of like this.
But it is a little more painful. And I'll just show you what you get. So this is from a paper that I
wrote with Bjorn Poonen and Franck Leprévost. So here it's not even particularly worth going
through aside from the fact that you see that this thing which is going to be the right-hand side of
a equation for Genus-2 curve has kind of complicated factors coming in that deal with your
arithmetic, but it's still X squared plus something X squared plus something, X squared plus
something. So you can still see the traces of the geometry occurring even though the arithmetic
makes it more complicated.
But the point is that it's very explicit. So you can, given these two elliptic curves and identification
of the two torsion groups, you can easily just write down the Genus-2 curve that you get. So
that's the case N equals 2. The case N equals 3 is also fairly explicit. At least geometrically.
So N equals 3. So in 1876 Arete [phonetic] wrote down a family of examples. He translated into
modern terminology, he looked at cubed this family of Genus-2 curves and he observed that if
you set U equal to 4 X cubed minus 3 X, you could express this right-hand side as a cubic
function in U so that this degree five polynomial and X, this Y squared degree 5 turns into a Y
squared equals a degree 3 rational function in X which gives you an elliptic curve.
And, likewise, you could also make a different substitution. 2 X cubed minus B over 3 X squared
minus 1, and again you could write this right-hand side as a cubic expression in V, which gives
you, again, a different elliptic curve that this Genus-2 curve maps to.
So there's a one parameter family which is not enough parameters. And it was only in 1885 that
Grassaex [phonetic] wrote the final answer. He said look at this family of Genus-2 curves where
there's a condition, you need that this Q should be 4 times B plus 4/3s AP.
So it looks like there are four parameters but really there's only three, and really there's only two,
because you scale X and these parameters can be scaled as well. It's really only a two
parameter family. Which is good. That's the right number. And now the substitutions that you
use are even more related to the equations before the curve.
So this is kind of the geometric picture again. It's not quite as simple as the geometric picture for
the degree 2 examples, the N equals 2 examples, but it's still not that hard to write down. But as
far as I know no one has gone through the kind of painful details of working out the arithmetic that
will make everything work over some fixed base field.
But that actually won't matter for our application. So I will get to our application. I might as well
show you the paper of Gorsaw [phonetic]. So this is an announcement of it at least from Comp
Predeux [phonetic] in 1885. And, again, they're always talking about integrals. But you get this
polynomial I guess using T instead of X, but he gives the substitutions that you need to reduce
that to elliptic integrals. So there's one of the substitutions I gave.
And I think that even might be all for the pictures. Okay. So then you can look at higher ends.
Bolsa [phonetic] did some work for N equals 4 but I've been unable to decipher it. I've looked at
the paper not terribly carefully, but I haven't been able to kind of extract nice parameterized
families for degree 4 expressions in them. But it should be possible.
For N equals 5, there's much more recent work. I actually haven't seen this, but from what I
understand Maka, Jesca and Vokline [phonetic] have a paper where they explicitly write down
parameterized families of Genus-2 curve that have degree 5 map elliptic curves and I believe
that's as far as anyone has gotten.
Okay. So that was the historical digression. Let's go back to our original problem. So what we
wanted to do was to construct a curve with a given number of points, and our idea was to write
our N as Q plus 1 minus S and T where S and T were traces of elliptic curves. And our algorithm
for doing this maybe now you'll understand why I waited so long to say it, because not only were
we cheating by looking at split Jacobians we're going to cheat even more.
So step one is to find an elliptic curve over FQ, with number of points equal to N. So we use
Reinier's thesis to produce a Q and an E. And whose trace is T and that satisfies that relation.
So we want to find a Genus-2 curve that has end points. Now we have this elliptic curve with T
points. We have the right T here. The only way we can find another curve that will help us if we
find a curve with trace 0.
So we want to find -- now FQ is fixed. So we want to find some second elliptic curve E prime
over FQ, with trace 0, which is to say that the number of points on the curve should be Q plus 1.
And you might recall from Peters talk that this was supposed to be a hard problem, given a finite
field, a particular finite field, finding an elliptic curve with a particular trace is typically a tough thing
to do.
Unless the trace you're looking for is trace 0. And in that case it turns out to be a very easy thing
to do. If you write down a supersingular elliptic curve over this field it will have trace 0, assuming
Q is a prime, which it most likely is.
So it turns out to be very easy to write down curves of trace 0. So essentially what happens is
that there is a particular elliptic curve in characteristic 0 over Q that will be supersingular for half
of all primes. And if that first one doesn't work, then there's another elliptic curve over Q that's
supersingular for all fault primes, and if that doesn't work you try another one. And there's maybe
seven, I forget how many. It's the number of class number one fields.
>>: Seven [inaudible].
>> Everett Howe: 11 fields or 11 discriminates.
>>: Nine.
>> Everett Howe: Nine. So we have nine choices already of elliptic curves defined over Q that
are likely to give you a supersingular curve. And if they don't, there are other things you could do.
So it's not at all hard to find a super singular elliptic curve. And once we do that, now we have an
E and an E prime with the right number of points with the right traces. If we can glue them
together, then we will have solved this problem of constructing a Genus-2 curve.
So step two is going to be to glue. And here there are actually going to be some problems.
Because if we use the N equals 2 gluing, then what we need is an isomorphism between the two
torsion of E and the two torsion of E prime. An isomorphism as groups with an action group of
the GAWA [phonetic] of the base field.
In particular, this will mean that the trace of each one, the two traces will be congruent to one
another, modular 2. And this S we already know is 0. So T is going to be an even number. And
therefore N, which is Q plus 1 minus S minus T is going to be congruent to Q plus 1 mod 2, and
Q, the way we made it, it's going to be a large odd prime. So this will be 0 mod 2.
But it turns out that's the only restriction, given any even number N, if we can apply Reinier's
method to produce this e over FQ with end points, then we can find a super singular elliptic curve,
glue the two curves together along the 2 torsion and construct a Genus-2 curve with the right
number of points.
So that is nice except for this one annoying problem about the thing being even. So we can hope
that maybe if we look at N equals 3, which is a little less explicit but we can deal with it. We're
only working over finite fields. It's not that hard a problem. Anytime we run into a problem, we
can just reapply the elliptic curve construction to get a different prime and try again.
And, here, if we're going to be gluing along three torsions. So now we need an isomorphism of
these groups. So now it turns out that we'll get a congruence modular 3 between these traces of
Frobenius.
So now N, which is Q plus 1 minus S minus T will be congruent -- and this is 0 -- to Q plus 1.
Mod 3. And the one thing you could say about a prime plus 1 is that it's not going to be 1 mod 3
unless Q is 3, which it isn't. Because Q is going to be big. So now if N is not equal to 1 mod 3,
we have a hope of doing this. And it turns out with a little bit of fiddling you can write down a
supersingular elliptic curve for which you will be able to tie these things together along the 3
torsion.
So with N equals 2 and N equals 3, we can explicitly solve the problem for N not 1 mod 6. And
factored. We need N to be factored so we can apply Reinier's method. And if we had explicit
constructions for N equals 4 and N equals 5, then we can again do things for N not congruent to 1
modular some numbers.
At least this gives us the solution for five-sixths of the numbers which is not so bad. So before I
give an example, I will give you two more applications of these ideas of gluing. What can it be
used for.
There's a nonconstructive question. Given Q, can you give a list of the possible characteristic
polynomials of Frobenius of all Genus-2 curves over FQ. These are called vay polynomials of
Genus-2 curves over FQ.
And this question is answered now, this is work -- this has been a work in progress for a long
time. But the final answer is given in a paper that I wrote with Enric Nart and Christophe
Ritzenthaler, it just appeared.
And one of the critical cases is figuring out when is there a Genus-2 curve with this as its
characteristic polynomial of Frobenius. And you can see where this is leading. If you can tell
whether or not it's possible to glue two curves together, then you can determine whether or not
such a product of vay polynomials is itself a vay polynomial. And so being able to look at this
glue construction in some detail, to figure out when it works, when it gives you a Genus-2 curve
and when it gives you a split Jacobian is something that we had to look at. So there was one
other application of gluing. This is what the paper with Poonen and Leprevost was concerned
with was constructing Genus-2 and Genus-3 curves over the rational numbers that had
interesting parties. To give you an example of what I mean by an interesting property, we were
able to write down a particular curve, which you can kind of tell from the exponents is obtained by
gluing together two elliptic curves.
So if you look at this curve if you look at this Jacobian over Q, this is a group, the moreday group
of the curve. And it has a torsion part of order 63. And then it has some free part. I think the
rank might be 3, but I'm not sure. I couldn't remember off the top of my head.
But so using these gluing techniques we're able to take elliptic curves that have interesting
subgroups and tie them together to get Genus-2 curves whose Jacobians have surprising
subgroups. You may not think it would be possible to get torsion of such a large order on a
Genus-2 curve, but it is.
Okay. So I think what I'll do to end with, I'm going to switch over to the computer example now.
So I have MAGMA loaded up here with a program that will do the computations to glue together
curves with elliptic curve with the right order and a super singular elliptic curve to produce a
Genus-2 curve with a given order.
This is not an efficient program. It was just kind of a proof of concept. So I can't do large
numbers. But I can say let C be the Genus-2 curve gluing of curves that will give us that many
points.
>>: We can't see ->> Everett Howe: That is a good point. Thank you. Thank you for pointing that out.
All right. So initially yesterday we were hoping that maybe getting an example with 10 to the
2009 points, well, I have code that works for smaller numbers.
And actually it would be nice to verify this. Right? So the thing is that computing the number of
points on a curve, well, let's see what C is, first. Oh, my screen is too wide. There we go.
So that's over a reasonably big field. Now we know this must be a prime. And given by explicit
equations. We can see how many points does this have. And it takes a while to count points on
a Genus-2 curve on MAGMA because it doesn't know it has a split Jacobian and I can just
calculate the number of points on an elliptic curve. But on the other hand this should convince
you that I didn't get this curve and this program just by looping over curves until I found the right
number of points, because that would take too long.
And, hey, it's right. And if we want something that isn't even, you could say G2 what did I call
this? Triple? Yes. 200903, we want one that's not 1 mod 3. That's yesterday. That was fast.
So D is that and actually you can see that while this one had the suspicious form of only even
exponents so it's a double cover of elliptic curves this one doesn't have that suspicious form
because it was constructed as a triple cover of elliptic curves.
And great. It does actually work, and it was faster to create these things than it was to verify
them. So that's good. And I believe that is it.
[applause]
>> Kristin Lauter: Questions?
>>: I have a question. The 63-torsion there, can you see that 63, how it related to the gluing that
you were doing? Why were you gluing stuff in order to get 63-torsion?
>>: We were just gluing together just to try to get large torsion. So we took an elliptic curve that
had 7-torsion and an elliptic curve that had 9-torsion, and we found examples where the two
torsion subgroups were isomorphic as gama models. So we were able to glue them together
using this explicit gluing and get something with 63-torsion.
>>: How big a torsion could you get there?
>> Everett Howe: The largest Genus-2 thing, I think we had a torsion of order of 512, maybe.
Could that be true? 16 squared. No, that can't be true.
>>: [inaudible].
>> Everett Howe: It might have been 128. It might have been the largest example we had. So I
have to find the paper. Actually I can go to the introduction to this.
>>: Is it known what the largest torsion ->> Everett Howe: No. For Genus-2 curve? No. There we go. So Genus-2 curves, we got
various torsion groups and various parameterized families. So I guess the largest torsion group
was this one, and it was parameterized by a positive loop service. We can get quite a few
Genus-2 curves with that kind of torsion with Jacobian.
But this is a little bit harder to explain because you start with elliptic curves with 2-torsion on them
and we glue them together, but kind of naively you would expect to lose 2-torsion by the isogeny
that goes from the product of elliptic curves to the Jacobian because there's 2-torsion in the
kernel. We had to do some extra work to make sure we got torsion back in the end to make up
for that.
So explaining the 63 case is a little bit easier.
>>: [inaudible] so theoretically possible?
>> Everett Howe: This is what we've achieved. And what's theoretically possible I don't think
anyone knows an upper bound for the torsion on Genus-2 curves.
>>: What are the largest torsion structures known for simple Jacobians?
>> Everett Howe: So Leprevost has a number of papers where he looks at prime orders and
other people have looked at it as well. I'm not sure what the current state of the art is. But
people are getting prime orders up to 50s and 60s, I think. But I don't know what the largest
group structure is, the largest torsion subgroup is.
>>: Is it upper bound?
>> Everett Howe: Not that I know of. There might be.
>>: I think sthoh had a posting in the last six months or so of a new record. I'm thinking four
digits but I can't remember exactly.
>> Everett Howe: Okay.
>>: So [inaudible] conjecture measures [inaudible] for the subject curves, a lot of modular
functions, right?
>> Everett Howe: Yes.
>>: So there isn't any accompanying conjecture being built?
>> Everett Howe: There might be, but I'm not up to date on what's been done on this.
>>: This is something simple. If the Jacobians is a list of unaltered points of the curve how come
there isn't a one-to-one mapping of the Jacobian to make the problem equally hard or equally
easy?
>> Everett Howe: So for Genus-2 curve essentially a point on the Jacobian is a pair of points on
the curve. But that's only essential. There are exceptions. There are pairs that have become
equal to other pairs and there's a single point will give you a point on the Jacobian as well.
So counting that discrepancy between just square the number of points on the curve and divide
by 2, I guess, is exactly the difficulty in figuring out how many points are on the Jacobian.
>>: I was going to ask a related question. So can you run this kind of a program for the
symmetric square of the curve instead of the curve with the Jacobian.
>> Everett Howe: Say that again, so what do you mean?
>>: In other words, let's say question number four, add to your list given an N, et cetera, can you
find a curve, Genus-2 curve, let's say, such that the symmetric square over cubed is N points?
>> Everett Howe: I think we can do something. The details are going to determine whether it's
actually feasible or not. I think you could probably do something.
>>: You would need to know the number of points over quadratic.
>> Everett Howe: Yes, as well as the number of points.
>>: It's closer to the Jacobian than the curve, perhaps?
>> Everett Howe: Yeah, I don't know. I don't know. That's a good question.
>>: If you knew that, potentially you could use the Jacobian map and have a number of points on
the Jacobian there?
>> Everett Howe: Yeah, that's something to consider.
[applause]