>>: It is my pleasure to announce the next speaker, who is Melanie Matchett Wood.
Who is going to speak about composition laws.
>>: Melanie Matchett Wood: Thank you. So my talk is going to be pretty light on technical
details. Which I think is probably okay. We've heard a lot of talks this week. But I'm more than
happy to talk about technical details later if anyone's interested, but I'll mostly leave them out of
the talk.
So my goals in this talk, I'm going try to do a few things for us. I want to review two examples of
explicit group laws which I think are probably familiar to almost everyone in this room. And I want
to review those to put them in the context of a much larger story.
So it turns out that these two examples are just the first stepping stones in a much larger story.
And this larger story is relatively recent. A lot of the results of the larger story are in the past few
years, and already it's having a lot of really exciting theoretical applications. But computationally
not much is understood about it. So I want to suggest just a huge range, actually, of open
problems in computational number theory and algebraic geometry that stem from this larger story
that we’re just beginning to understand.
So the most classical example of an explicit group law is this bijection, probably, I think, best
attributed to the Dedekind-Dirichlet, between gl2 (z)-classes of primitive binary quadratic forms
over the integers, and isomorphism classes of quadratic rings and elements of their class group.
So don't worry too much about if you think sl2 instead of gl2 , or what I mean by, by twisted. You
probably know some version that classes of binary quadratic forms, exactly tell you elements of
the class group of quadratic orders and number fields at least.
So you're probably familiar with some version of this bijection. And one nice thing about it is that
these objects on the left are very concrete. Just ax2 + bxy+cy2, where a, b and c are integers. So
you simply have to write out three integers, not so bad. And you get an element of a class group
of a quadratic ring.
And one nice thing about this picture is there is a group law on the right-hand side. So if we fix
this c, the elements of the ideal class group, while it's a group, so they have a composition law.
They have a group law, and therefore okay given this bijection, there is a group law on the left
hand side. And of course Juss understood this group law before we really had the ideas of ideals
and ideal classes and class groups. And so this group law could be understood in another way.
But now this is probably the best modern viewpoint on the group law on binary quadratic forms, or
more precisely classes of binary quadratic forms.
So one nice thing is it's not just theoretically that there exists a group law. In fact, the group law
in binary quadratic forms over the integers, can be given explicitly in terms of those coefficients a,
b, c, bi-polynomial formulas and some gcd operations. So it’s a very straightforward thing to
compute. There is one trick, I said of course, the group law was not actually on binary quadratic
forms, but on gl2 (z) classes of binary quadratic forms. And so if you really wanted to compute
explicitly, you also need a reduction theory to find a unique reduced representative in each gl2 (z)
class. But of course, this, Juss knew how to do, and we certainly can do pretty easily today. So
that, that lets us and then, explicitly, by finding reduced representatives for each class, compute
the group law.
And I'll just point out that the discriminant b2 – 4ac of the form on the left hand side, is the
discriminant of the corresponding quadratic ring on the right hand side. Which is kind of
important because I said to get the group law we needed to fix the quadratic ring. And then we
have a group law on the objects. So you need to be able to fix the quadratic ring in terms of the
data of the binary quadratic form which we can do in this simple way.
So after that example, another, I think, very familiar example to a lot of folks, and there's lots of
ways you can say this. I'm just going to try to say this in one way. Is if we take a q to be the
power of a prime, we have a bijection between, again, gl2 classes of primitive binary quadratic
forms over fq[t] and isomorphism classes of cd, where say c is a quadratic extension of fq[t] and
[d] is an element of the class group of c. So what changed here, this used to be z and I just
replaced z with fq[t], and here I've replaced z with fq[t]. I changed this letter to d instead of I
because in this context most of us think of these as divisor classes instead of ideal classes. But
that was purely for psychology, and so it looks like I've just taken the theorem and replaced z with
the fq[t], which is essentially what I've done. And again we have that these classes of primitive
binary quadratic forms over fq[t] are just given by the simple polynomials. You just need to write
down three elements in fq[t] and that’s what it means to give a primitive binary quadratic form.
And again, we have a group law on the right-hand side. If we fix c and thus on the left-hand side.
So we’ll move that bijection up to the top and make a few remarks about it. In case it doesn't look
completely familiar to you yet, a quadratic extension of fq[t], I haven't been very precise about
what I mean by that. It's just a double cover of the affine line over fqt, so double cover of the line
is just a hyper elliptic curve. And the class group at least in the case when a hyper elliptic curve
is smooth, the class group of a hyper elliptic curve it's Jacobian. And so here I'm talking about
elements of the Jacobian of hyper elliptic curves, and those are given by these binary quadratic
terms over fqt. And just like before where b squared -4ac gave you the discriminate of the
quadratic ring, here b squared minus 4ac, gives you the branch locus of the map from c to a1.
Which in characteristic, not equal to 2 is enough to determine the curve the hyper elliptic curve c.
So there are lots of analogies between z and fqt. And so no one is, perhaps, surprised when I
can take a theorem I can pull z out of it and replace it with the fqt I. I think a lot of you are familiar
with this idea that the computations of the group law on Jacobian's of hyper elliptic curves are
essentially the same as Juss’ composition on binary quadratic forms over the integers.
Yes, so there are lots of things that are very special about Z and FQT, but often things that are
true for one part are true for the other. But this is, in fact not one of the those things that, that
relies on the special properties of the integers and FQ Brackett T. So in fact, if we have any ring
at all, so I guess I mean, for me rings are communitive, so maybe I should have said that, but any
communitive ring, or if you like to think more generally, we could be working not just over a ring
that somehow the affine case we could be working over for any variety, or any scheme, if you
want something more complicated than that, you can probably put it in there.
So in amazingly general context, we have a similar bijection which of course there are lots of
rings and people have thought about a lot of examples of this over the years and there are far too
many people who have thought about this to list them all, but I think only recently, maybe, have
we understood the whole story. So there is a bijection between equivalence classes of primitive
binary quadratic forms over a ring r, and isomorphism classes of cd, where c is a quadratic r of
algebra and d is an element of the class group of c.
So essentially, this is taking the theorem that we have seen two examples of with z and fq.
brackett t, and saying that it is completely in general, with no using nothing at all about, about z or
fz brackett t. And the only thing is that I have to tell you a little more of what I mean by all of
these things, and the theorem. But already the take away should be that there is something that
all of these words mean. That this is true. So there's just that same theorem. I'll tell you before I
set up a quadratic ring over z and then I set up a quadratic extension, so maybe I'll be a little
more precise now. I'll say c is the quadratic r algebra. So quadratic r algebra is an r algebra that
is locally free ranked to, as an r module.
So if we think for example just, say if r equals z, well over z locally the three modules are free.
And so I just mean so quadratic z algebra would just be a z algebra, meaning a ring that's free
ranked two as the z module. So for example, that includes orders in quadratic number fields,
includes a few more examples, like z ajoin x along x squared is also a quadratic zalgebra. But
this is precisely what I mean by these quadratic extensions. And if we'd like to think of r
geometrically, so for example, if we take the geometric space spec r, or more concretely, if
instead of fq of t, we’re thinking about working over the affine line, over fq, so if you weren't really
thinking about rings, but you were thinking of curves with maps to the line, then a quadratic
algebra over a our base, is just a double cover of the geometric space.
So just like hyper elliptic curves are double covers of the line, if we, if r here we think of
geometrically, quadratic r algebra’s exactly correspond to double covers of, of the geometric
space that we're working over. Here the class group so, is precisely the group of invertible r
modules or geometrically, you could think of as the group of line bundles on your space r, when
the quadratic cover is smooth, for example, if we’re working over the line, and we have a smooth
hyper elliptic curve, then it's the class group is just the Jacobian group of divisors, mod principal
of divisors.
Though in some sense from the point of view of competition laws, maybe you don't even care
exactly what the class group is, as long as I could tell you if it's a group. Okay so this is just that
same theorem, and now I've said a lot about the right-hand side and what those objects are. So
the left-hand side, I should warn you, binary quadratic forms over r, can be a little more subtle.
Yeah, uh huh?
[Inaudible question]
>> Melanie Wood: Yeah, absolutely. There's no irreducibility condition. In fact, I only put the
primitive condition up here for simplicity. So that I didn't have to say some technical things. In
fact this, you can state this theorem where you use any binary quadratic form, including the one
in which a, b and c are all zero.
[Inaudible question]
>> Melanie Wood: Yeah so if you, well, you can have, so primitive just means that the
coefficients don't have a gcd, but if you have forms that are reducible over q, that factor, say like
over the integers. If you have a binary quadratic form, that factors into two linear factors, that'll
correspond to an ideal class in z plus z.
[Inaudible question]
>> Melanie Wood: Well, the things, so primitive actually corresponds to the fact that they'll, like
ideal classes are invertible. If you take non-primitive things over here, you get non-invertible ideal
classes. Your reducibility of the form, corresponds to the quadratic algebra being a domain. So if
you take reducible forms, then you get quadratic algebras that are not domains and you'll get,
say, r plus r. So yeah, in fact, you know this, this question illustrates one of the nice things about
this bijection, is that any question you might want to ask about the objects on one side, can be
easily answered in terms of properties of objects on the other side.
Yeah, so irreducibility gives you c being a domain, primitive, gives you, the ideal classes are
actually invertible ideal classes. If you take away primitive, you can state a more complete
theorem that involves non-invertible ideal classes. But then it wouldn't be a group. Since I'm
trying to talk about group composition, I stuck to the primitive things here. Is that, okay great.
[Inaudible question]
>> Melanie Wood: Good question. So yeah, I was going to warn you about binary quadratic
forms over R. They're not in general given by ax squared plus bxy plus cy squared with letter a,
b, c and r. So things get a little more complicated than in the two cases that we've seen. This,
this meaning that binary quadratic forms are given in the simple way is only the case when all
locally free modules over r are free. So for example, over z or over fq bracket t, all locally free
modules are free, and that's why we just have to write down these three numbers a,b and c, to
get a binary quadratic form over those rings.
So for example, all locally free modules over r are free if you're working in a Dedekind-Domain of
class number one. Or in a polynomial ring, so this is really a deep theorem that all locally free
modules over r are free in this example. But so if you're working over a polynomial ring which is
just like working geometrically over affine space, than in fact, you get this simple representation of
binary quadratic forms over r, just three numbers. And I will, in fact, tell you later what in general
binary quadratic forms over r are.
Okay. So I just want to say so from this last example here, where I said okay in this case where
we have a polynomial ring over a field, all locally free modules are free. You know, here’s a case
of this theorem, which I don't know, my guess is, no I think hasn't been particularly thought about
very much. So there's a bijection between gl2z classes of primitive binary quadratic forms over
fq, now joining two variables t1 and t2, with isomorphism classes, of quadratic fq bracket t1, t2
algebras, and elements of the class group of, of c. Those [inaudible] algebras. And here again,
now to write down these primitive binary quadratic forms, you just need to write down three
elements in this ring.
The c on the right hand side, these quadratic algebras, they correspond geometrically to surfaces
with agreed to maps to the plane. So if you like to, you know, think of this as how this is, I mean
a very straightforward generalization of a hyper elliptic curve, which is a geometric space with a
double map, or degree to map to the line, now here we’re talking about surfaces with degree to
maps to the plane. And what I'm telling you, is that there are class groups that can be given
explicitly in terms of binary quadratic forms whose coefficients are just polynomials in two
variables over the field that we’re working over here.
I just said fq for concreteness. So now that we I've talked about this general theorem, I want to
talk about how the forms correspond to elements of the class group. So I think in these the first
two examples I talked about, I think you're probably familiar with the fact of the correspondence
can be written down very concretely. And so, in fact, here I want to answer this question a little
more conceptually. Why is it that binary quadratic form should have anything to do with class
groups of double covers? And I want to sort of convince you that it's just not some cosmic
accident. And so I'm going to actually illustrate over fq bracket t, since it seems like maybe the
best example for this, for this conference. But the idea that I'm going to explain to you here works
over any ring or variety or scheme, it, over whatever base you want, with appropriate technical
details filled in, this idea tells you why binary code quadratic forms, parameterize class groups of
double covers.
So in this example, or now I'm thinking over a specific base, I'm going to consider the affine line
across the projective line. So in this case, for concreteness, the affine line is going to have
coordinate t, and the projective line is going to have two coordinates x and y. Those will be my
projective coordinates. Of course I have a map from the affine line across the projective line, to
the affine line just by getting my projective coordinates. And here's the only construction, in some
sense the form, so if I have a binary quadratic form over fqt. I’m now writing this aftof x squared
plus b of t of xy plus c of ty squared, just to remind you that these a, b and c are polynomials in t.
And that form cuts out a curve in a1 cross p1. So a1 cross p1, that's a two-dimensional space. I
put one relation on it here. And we see that the form is homogeneous in x and y, so it actually
defines a legitimate kind of form on p1. Of course and the degree doesn't matter for t, and so this
form, this quadratic form, cuts out a curve in this a1 cross p1. So I'm just moving that fact up to
the top here.
So this curve in a1 cross p1, has a degree to map to a1, i.e. it’s a double cover of a1. Well why is
that? I just take the map from a1 plus p1 down to a1, and why is it a double cover that just says if
I fixed the t, if I plug in the number and give an a, b and c here, there should be two solutions in
p1. Well, there are because this is just a quadratic form on p1. So that's what makes it a double
cover of a1. And or a quadratic fqt algebra equivalently. And this is the double cover. So the
form itself in a1 cross p1, cuts out the double cover, that is the curve. In this case it's a curve, but
in general it's the quadratic algebra for the double cover in this bijection.
Which is think is very simple and probably, I mean, seems, I think, often to be overlooked. Okay
so I need to give your curve and an element of the class group, so there's the curve, and if I just
say intersect c with the line y equals zero, so that's a line in the space, I get a divisor on c. In this
case I get a bunch of points. And that gives me the element of the class group. So very simple
to get this divisor, in general we have to sort of pull back o of one from the p1. Okay I said I
wouldn't get too many technical details. Maybe that's one little one. Of course I took y equals
zero but that was just to do something simple. I could've taken x equals zero, or their similar
lines. And I would've obtained equivalent divisor, so the same element of the class group.
So the construction of a quadratic cover and an element of the class group from a binary
quadratic form, I just want to convince you conceptually is very simple. The form itself cuts out
the double cover. I just take basically any line coming from a point on the p1 and that gives me
the divisor. So this description as I've given it agrees with the classical correspondence between
binary quadratic forms and ideal classes of quadratic rings over z. So it's giving you, if you write
it down into formulas, it gives you the same formulas as Dedekind-Dirichlet gave you. And if we
take just the, you know we forget about c, and we take the ab over fqt, this gives the mantra
representation of points on the Jacobian of a hyper elliptic curve. The reason in this case people
traditionally forget about c, I guess, is because you can recover it from b squared -4ac if you were
remembering that instead.
So this very simple conceptual description gives us concretely the two explicit constructions that
we know about. I'll just tell you how this last point here works. So remember that b squared -4ac
is the discriminate of the quadratic algebra, or the branch locus of the quadratic cover. So over at
qt, let me call that b squared -4acf. So in characteristic not 2, of course, if I give you that a, b and
c, you expect that to get a divisor on the curve Z squared equals F of W. Where F is this
discriminate here in a 2. Is how you probably usually write that hyper elliptic curve. So let me for
preciseness, let's see prime be the curve that's defined by this equation. Then I can explicitly, I
give you an isomorphism. I'll just write out here between c, c for me was the curve cut out by my
by binary quadratic form, and c prime is the more traditional definition of a hyper elliptic curve in
a2. And here is just an explicit isomorphism from c to c prime. And it turns out that if you look at
this isomorphism, you'll see that y is zero on c. That was the divisor I picked. Exactly when a of
z equals zero and z equals b of z on c prime which are the conditions you expect in the Mumford
representation of a divisor on, in a class group of a hyper elliptic curve or a Jacobian of the hyper
elliptic curve.
And so this gives that usual representation. I should say that we had to say characteristics not
equal to 2 to write, of course the hyper elliptic curve in this way, the theorem that I stated has
absolutely no conditions on the characteristic of the ring, anything about 2. It's completely
general, and it's totally well behaved, with respect to 2 and taking and working mod 2. And
doesn't have any, yes, there's no sort of special cases at 2 there. But of course, if you want to
represent your hyper elliptic curve in this way, instead of representing it as aax word plus bxy plus
cy squared, then you have to make cases. You have to be careful about 2.
I want to say that so in general, just like in the two cases that we've seen, the beginning in fact, in
general, the composition law in this theorem, can be given uniformly in terms of polynomial
formulas and some gcd operations. And that's actually true in a precise sense. The composition
law, it pulls back from composition on the universal primitive form. That's a precise abstract
sense that says, there is a universal composition law that works in all rings. But of course for
each r that you might be interested in, the actual method, if you were going to implement it, of a
computation of this composition might differ, or might actually be reasonable.
And for each r, the reduction theory to find a unique representative in equivalence classes of
forms. Remember we had not just binary quadratic forms, but we had gl2z classes of binary
quadratic forms. And we needed a reduction theory to find a unique element in each of those
classes that we can say this is our favorite element. We’re going to use this to represent that
class. And each r, this is potentially a new problem. You know as far as I know in almost any
case, both theoretically to find a reduction theory and algorithmically to actually be able to
implement say, reducing an element.
So other examples that I think would be interesting to study. Already from just the description I've
given you, would be orders and numbers fields with, with trivial class groups. Then the reason I
put that trivial class group condition is because I haven't even yet told you really what a binary
quadratic form is when we don't have a trivial class group. And this example that I just talked
about, the affine plane over fq. So in both of these cases, the main interesting aspects would be
understanding a reduction theory of binary quadratic forms, and then actually being able to
realistically implement the composition law and the reduction theory.
So those were some cases where locally free, r modules are always free. And I promised that I
would tell you more or I would tell you what this theorem actually says when locally free r
modules are not necessarily free. Okay. So I'm going to give you the definition of a binary
quadratic form over r. And it starts with a locally free rank 2 r module z. So a binary quadratic
form has several parts. It has this locally free rank to our module c. It has a locally free rank 1,
our module l. And it has an element of the second symmetric power of the rank 2 module
[inaudible] with the rank one module.
So geometrically, if you prefer to think geometrically, this is a rank 2 factor bundle on the ring. A
rank one factor bundle on the ring and then a global section of this ring, three vector bundle
constructed out of the ring two and ring one factor bundles. Okay. So that's what a binary
quadratic form is completely abstractly, but I'm going to give more and more concrete. But first
let me just do a reality check. An example, when v and l are free, so let's just say v is the free
ring to our module created by x and y, and l, since it's supposed to be just ranked 1 if it's free we’ll
just call it r. So x and y here are just formal to remind me that I have two copies of r here.
Then an element of the second symmetric power of rx plus ry tensored with R, the tensory with R
doesn't do anything. r just the forms x squared plus bxy plus cy squared. So when v and l are
free in this definition, then elements of this objects into [inaudible] are just those binary quadratic
forms with three elements of r to write them down here, x squared plus bxy plus 2y squared.
But even if we're in rings where locally free r modules are not always free, things aren't totally
helpless. So for example, if r is a Dedekind domain, say like a maximal order in a number field,
or a smooth affine curve, then, essentially then we know what all the locally free reign 2r modules
are in the locally free rank 1r modules are. And essentially, we get a type of binary quadratic
form for each element of the class group of r. So before we had just sort of one type. We just
had to write down a, b c. We always get this type because of course there's always a free ring 2r
module and a free ring 1r module. But in general the class group is not trivial, will now have
several types of binary quadratic forms. A type for each element of the class group, roughly. But
each of these types then is completely explicit, and I'm going to give them more concrete
examples on the next slide.
But I want to say that morally you would imagine here if you're working over some r, you,
computationally, you would compute this class group once. This class group of r, and then you'd
be computing, say, lots of class groups of quadratic extensions of r. So there might be some
work here to compute this class group once and, you'd do it once and for all. But then you'd
compute class groups of lots of quadratic extensions. Okay. So I promised I'd give a concrete
example of these different, wild types of binary quadratic forms, and I have to convince you that
they are not actually that scary. So let's say, okay, be a maximal order in a number field, and I'd
be in non-principle ideal. A more concretely I'll take everyone's favorite example of this, the
adjoining square root of -5 and here is the non-principle ideal in that ring.
Okay. So now I'm going to take v. I need it to be a locally free rank to okay module, but not free.
So I'm just, take it is ok, x plus, here is the ideal lie, and I'll just let l be ok. So instead of having
okx plus oky, I just have okx plus the ideal time of y. So now this gives me, you know, what I
want to show you is that this element of the class group gives you a new type of binary quadratic
form. So elements now sent to v tensor l, are still given by ax squared plus bxy plus cy squared.
But no longer are a, b c allowed to be any elements of the ring. a has to be an ok. b has to be in
the ideal. And c has to be in the square of the ideal.
So that's a little more complicated than requiring a, b and c to all be in the ring, but I hope that
you feel that this is a pretty simple thing too. And now, you have to write down three numbers,
but they're not all in ok. They might be required to be in some ideal. And the other kind of
change here is that instead of gl2, before, when v was just free rank 2 module, and we had gl2
acting on it. Now the group GLV that acts on the form to give us the equivalence classes.
Replacing the gl2’s the equivalence class, is a group of major cs, where instead of having
elements of ok in each place, I have elements of ok up here. And there still just 2 x 2 major cs.
These are elements of I, elements of I inverse and elements of ok. But still something fairly
concrete. Of course here there's only one non-principle ideal class. And so for this specific
maximal order, we can just take usual binary quadratic forms with a, b, and c on the ring, and we
can also take this type of binary quadratic forms, and that'll give us all the binary quadratic forms
over this ring. In general, if we had a bigger class group of our maximal order, then we'd have to
do this kind of construction for an ideal in each class. And we've had some list of types of binary
quadratic forms.
And in fact the types of binary quadratic forms, I think of appeared, in this conference already
there's morally, with a few sort of transformations, when we're on Jacobian's of hyper elliptic
curves, this distinction between degenerate and not degenerate divisors that you have to use two
points to represent a minus twice infinity, what it or just one point to represent. If you untangle
that in this language, in a certain context, it would essentially be coming down to two types of
binary quadratic forms. So this, so this is something that in some example, we've already seen
with these different types of divisors on Jacobian's hyper elliptic curves. And if you moved into
higher and higher genus, and you're trying to compute on those Jacobians which you know you
may not be interested for cryptographic reasons, but for other reasons if you wanted to compute
on those, you would essentially see that more types appear for reasons that are coming from
something like this.
So in this case, say over a max [inaudible] number field, the question of reduction theory is
almost entirely open. There's some very recent work with corona on it. But it actually would have
lots of interesting applications not just in this context. But for some other number theoretic
applications, and so it's a pretty simple case, pretty close to, to the integers that we're used to
working with. But the reduction theory here, I think is a great open problem. And the composition
in a practical sense is an open problem. Theoretically, the composition is given. I describe to
you the very geometrically, the correspondence and you know on the right-hand side that we
have the class group. You just multiply elements over there and that gives you the composition
on the binary forms. But I think to understand this practically, for example, to implement it, is
quite a bit trickier. One of the reasons why, is that in this case we have different types of binary
quadratic forms. Composition is locally given by universal formulas, in the sense that these
modules v and l, I said were locally free rank 2 and rank 1r modules, and so locally where they
are free, the composition is given by the global formula.
But patching together those formulas that you have locally, into a global answer, I think is a
nontrivial problem. So understanding the composition law explicitly, where the v and l are
nontrivial, I think would be really great, and it would have a lot of applications. So, for example, in
some examples where ok has nontrivial class group. Even in the example where r is p1, okay so
now we have, in this example, r was a ring. Well, okay well, now it's p1, whatever all of that
meant, but I still essentially defined for you what a binary quadratic form would be. So this is a
parameterized Jacobian's hyper elliptic curves, but now I'm thinking of them as double covers of
the projective line instead of double covers of affine line. Theoretically, this makes almost no
difference whatsoever. But practically, this kind of coordination of Jacobian hyper elliptic curves,
is different.
p1 has nontrivial rank 2 modules and rank 1 modules. And so you get actually, you’re writing
down different things, and this case would be interesting because somehow we know that it can't
be that far off, from working over a1, which is already well understood. So we sort of know what
the answer has to look like, but maybe it would help us get a handle on these subtleties that
come from having, that come from having non-free rank 2 and rank 1 modules. But lots of other
examples, for example, instead of p1, if you worked with r in elliptic curve, you'd parameterize in
Jacobian's of what are called bi-elliptic curves. Bi-elliptic curves just means curves with a degree
to cover to an elliptic curve. And so this is a very, would be very new territory. And perhaps
instead of doing this, you might first work in the affine case. So work with an affine elliptic curve
[inaudible] so working with actually a ring here, a maximal order in a function field of an elliptic
curve. And this would still, you know, approximately give you Jacobian’s of bi-elliptic curves and
the affine case is probably easier than starting with the projective case.
So, I mean, these are just sort of the first examples beyond what we already know and
understand. And I think there are already a lot of interesting open, and I think reasonably
accessible things to understand. Okay so, so far I've told you this very long story about forms
that parameterize class groups of quadratic algebras or quadratic extensions or double covers.
Those are three ways of saying the same thing. But this, again, is actually one step in a much
larger story. And this larger story has only, I think become apparent, recently. So, for example,
I'm going to give a theorem and it was first shown over z and I'm just using the function field case
here because maybe it's a more familiar. So here's a bijection between so it's supposed to be a
similar kind of thing. On the one hand we have gl2z cross gl3z cross gl3z Classes of primitive
pairs of 3 x 3 matrices over our ring. And then on the right-hand side we have isomorphism
classes of triganol curves over our field and elements of their class group. So I'm going to
unwind what all of these things mean, but I hoped to convince you that it's a pretty strong analogy
to the theorems we've seen before. So that's the same theorem. We just moved it up. I said
pairs ab of 3 x 3 matrices. Really you should view it as a three-dimensional 2 x 3 x 3 matrix.
Which you could also think of as a tri-linear form. And that tells you how gl2, gl3 and gl3 act
because it's a 2 x 3 x 3 matrix. It's a trilinear form. So they act, it acts on those three sides of the
three-dimensional matrix.
So, again, it's very concrete writing down a, b down to giving three elements is giving eighteen
elements of this base ring. Which is certainly more than three, but on the other hand, is very
concrete. The reduction theory you might think, oh, gl2, gl3 and gl3, and this is some huge group
you have to do reduction theory for. But the reduction theory can be done for each of these
groups separately. So before we had to know how to gl2 reduction theory. That in some sense
can be carried over, and now the question is doing a gl3. And the only really new ingredient
reduction theory-wise, would be doing the gl3 reduction theory. So that's what this stuff on the
left is. Certainly it's a bigger case, but I think morally a pretty similar to the forms we had before.
And then triganal curves. So trigonal curves are just curves with degree 3 maps to the line. So
we've been talking a lot about hyper elliptic curves. Hyper elliptic curves are curves with degree
2 maps to the line. And trigonal curves are just curves with degree three maps to the line. So
this is telling you about the class groups of triple covers of whatever base you're working over.
Here I'm working over that affine line over fq. And this class group is just the same for smooth
curves the class group is just the Jacobian. So this gives the parameterization of Jacobian's of
say, for example, of triganal curves, over fq. But this story, I mean you know, now that I told you
that there's a, an N equals three case, you probably won't be too surprised that the story doesn't
stop with these cubic extensions or triple covers. So in fact, we have a bijection and now the
great thing is that it doesn't, the complications that came from the N equals 2 to the N equals 3
case, you might think it's just going to keep getting worse and bigger and bigger. But it's actually
fairly well behaved.
So now on the left-hand side I have something that looks very similar to what I had before. I've
just replaced 3 by n. So gl2Z cross glnz cross glnz classes of pairs of n by n matrices, or really
you should think of this as a 2 by n by n, and so it’s three-dimensional matrix. Say over your
favorite ring like fqt. And on the right-hand side, I now have isomorphism classes of c and d,
where c is a certain kind of engonal curve over fq. And d is an element of the class group of c.
So it's another story for another day. One can actually say exactly which engonal curves appear
here. But I'm not going to talk about that today. I'm just going to tell you that some of them
appear. But what I'd like to make clear today is that the curves that appear, even though it's not
all engonal curves, you get their entire class group. So while you don't get all the curves you get
their entire class group. So you have a complete composition law on the left-hand side. Any two
elements you can compose them, and their product will in fact be another 2 by n by n matrix.
So what are engonal curves? Well as you might guess, engonal curves are curves with degree
N covers to the line. So here, you know, the line that I'm talking about is a1, because I'm talking
about this specific example, fq bracket t. And so as with binary quadratic forms, there is a version
of this theorem over any ring or variety or scheme or anything. I've given you one example
working over fq bracket t which tells you about the Jacobian's of certain engonal curves, but you
could take the integers of course and you could, and here, we've got class groups of certain
orders in rank and number fields, and you could take, you know again, the plane elliptic curves,
and get, and get a lot of different, and get a lot of different examples. This is just one example.
So this is the same theorem, just moved up. And I think there are a lot of problems here. So for
n, even greater than or equal to 3, even equal to 3, to implement these composition laws
explicitly, like I said, we know theoretically that they exist, but implementing them explicitly is an
open problem. And I think would be very useful. And to understand the reduction theory, so
here, even with just one n and your favorite base ring, I'm thinking that with the integers, it might
not be too hard, and then you could go from there to try to consider other cases. But here, so of
course, I would suggest starting with an equals 3 case, and I think that these, that these
computational problems are, are completely open and very accessible, and a lot, like I said, these
results already have lots of theoretical applications, and I think it would be great if we could also
use them computationally.
I think that's all I have to say. Thanks.
[Applause]
>> Thank you for the talk. Are there questions?
[Inaudible question]
>> Melanie Wood: Yes, so, she asked, how do you go from your 2 by n by n array and get the
curve. And I will tell you, it's very easy to say. I'll try to tell you on the top of this board over here.
So a and b are in by-in matrices. Okay. So if I take the determinant, so ax plus by, so here x and
y are formal, and I'm just formally making this one by-in matrice and I’m taking this determinate
so this then is a binary, binary just means two variables, x and y, binary form of degree n. Okay.
And remember before I had this example where ax squared plus bxy plus cy squared cutout a
curve in a1 cross p1. Well the fact that this was a degree 2 in axom Y, didn't matter. Similarly, if I
have a binary form, you know a0x to the na1x to the -1y plus any to the n, this also cuts out a
curve in a1 cross p1, now with a degree in map to a1 instead of the degree 2 map. So it’s
completely analogous to the binary quadratic forms situation. So yeah, you just take this
determinant, and that gives you the degree n cover.
Any further questions?
[Inaudible question]
>> Melanie Wood: Well, she said that, that this description was reminiscent of the [inaudible]
group, and was this related or did it have some…This, does this theory somehow relate to that.
Well the best answer that I can give you is, so not these examples that I'm talking about, but
other examples of parameterization of these algebraic objects by forms, are related to, in Carl
Reuben’s talk, he mentioned this theorem of [inaudible] where they’d gotten that,[inaudible] where
they had gotten information about average to [inaudible] so those kinds of counts come from
similar parameterizations that are part of this story. I don't know that it ever really gets the hands
on the [inaudible] group in exactly the way you probably would want to. But yea, that is related to
at least getting, working explicitly with these [inaudible] groups.
>> Any further questions?
If none, then please join me in thanking the speaker.
[Applause]