17143
>> Kristin Lauter: Okay. So today we're very pleased to have Tomoyoshi Ibukiyama, visiting us
from Osaka University, where he's a professor. Professor Ibukiyama was a student of Professor
Ihara, and he's an expert on Siegel modular form and super spatial abelian varieties, among
other things. Today he'll speak to us on supersingular modular forms. Thank you.
>> Tomoyoshi Ibukiyama: Thank you very much. First of all, I'd like to thank Kristin for inviting
me and giving me a chance to talk here. And I'm very enjoying it. So thank you very much for
that I'd like to talk on the Siegel modular forms. But of course there are a lot of stories about
these kinds of things and I would like to apologize that I will only talk about the works around me
so I might not hit many things, maybe, many important things but I just want to talk directly related
to my work.
So the aims of my talk here are the following, how one can generate results on supersingular
forms and the arithmetical code on algebra to higher dimensional supersingular varieties. I give
all the definitions and also in particular I would like to give a survey on some old results, many
obtained in the 1980s, and also I got some new results recently, and also some projects are still
going on. So I'd like to talk something about it. So it's something about supersingular germ three
and isometrical coded algebra or related [inaudible] forms. And also I'd like to talk some
background about the programs, if I have time.
So we start from classical results. So this is very classical. Mostly done around the 1930s and to
1950s. The classical setting. The supersingular elliptic curve E is an elliptical curve, defined over
a period of [inaudible] satisfying the whole combined condition one or two so in general elliptic
curve, any torsion points, there are the number of N torsion points is in general N squared. So N
squared, N torsion points.
But when N is P, then and also if N is defined over classical P field then something else happens.
So in general E has P, P torsion points over only one P torsion points, which is zero.
So among two of them, supersingular means that E has no P torsion point except for zero. So
this is a definition of a supersingular. And also differently we can say that N over E is maximum
order O over the definite coded algebra E.
E lands on infinity and lambda is all other primes. Lambda means if you B and QPs and it
becomes division. And if you take B tense torsion it becomes so-called [inaudible] and O as a
place tense QQ, where Q is the prime different from P becomes just a matrix algebra over size
two over QQ.
So these two are different conditions, and this kind of elliptic curve is called supersingular. So we
would first like to see what kind of things is known for these cases. This is very classical.
So mainly this kind of mathematics was done by Deuring and Eichler. I don't know the history so
well, but as far as I know these are the main persons who did this.
So any supersingular curve E has a model over this scale. As a model just means a direct,
another supersingular curve E which is asymptotic over E over algebraic [inaudible]. And also
another elliptic curve is defined over FP squared. I mean extend over order 2 over P.
And also all supersingular elliptic curve isogenous. These two are rather classical. But maybe
it's defined over the scale is due to Deuring. And secondly, the number of isomorphism classes
over algebraic algebraic code P is finite. So compared with the general elliptic curve, of course
there are infinitely many elliptic curves, and for each J invariant you have isomerism cross elliptic
curve over to zero infinitely many elliptic curve even over characteristic P view. But if you take
supersingular ones there are only finitely remaining.
And the number of such isomerism crosses equal to the so-called cross number of B. So the
cross number of B is defined as usual as [inaudible] over F maximum order B.
And so and also we can say a little more about the field of automation. So to explain that we
need to explain about type number. So the type number of B is defined as follows: So B has
maximal orders but not right in isomeric field since this is coded algebra, so there are many
maximal orders which are not isomeric with each other. And also the number of such maximal
orders are finite. And that number is called type number T over B.
And actually T is less than or equal to the class number, and also it's the same to say that B
cross could also conjugate C cross, because if it is isomorphic then it's B cross conjugate.
Conjugate C cross and the isomerist is the same as the isomers.
Anyway that number of crosses is called type number T. And the result related to it,
supersingular elliptic curve, is number of isomerism crosses super elliptic curve which have a
model over FP which is given by 2T minus H. This suggests that 2T minus H is non negative and
actually H is in between T and 2T.
So these are qualitative results in a sense but we can give also the quantity of -- sorry, quantity of
all this. So in 1938, Eichler gave a cross order number for the coded algebra. So actually he
gave a code -- general coded algebra, cross number for general coded algebra, and his proof is
rather tricky in his first paper. He used structure of unit group or maximal orders and then he
used so-called mass formula for the coded algebra, and then you get cross number. But later on
around 1950s he re-proved this theorem by using trace formula.
Then another thing is that Deuring, in 1950, re-proved by Eichler, the type number was given. So
that Deuring's method is very direct and he categorized something very directly and Eichler's
method is to use so-called trace formula and in both cases of course the result coincides and 2T
minus H is given like this.
Here H minus P or H minus 4P is a cross number of order, which is quadratic order which is given
minus B or minus four B. So relatively, speaking, this is a cross number of Q square root minus
P. So cross number imagine here.
But imagine a quadratic field have maximal orders and no maximal orders. So minus P and
minus 4P sometimes means that cross number of some order which might not be maximal. For
example, when P is 3 module or H minus P just means a cross number [inaudible], imagine
quadratic field and H minus P means a cross number of order inside the Q square root minus P
which has a conductor too. So it's not maximal order.
When P is over O, then minus P cannot be the scheme so the H minus V just means zero and H
minus 4P means cross number of Q square root minus P.
So this is a result by Deuring and Eichler. And then just another very, very interesting relation
between modular form. We have some integration with modular form. So here by A 2 gamma
node TI denotes space of modular forms of weight 2, where gamma node P is a congress
subgroup defined like this. This is where known group used by many things, for example modular
regression is based on gamma node P and so on, so this is a very famous group, so in other
groups this group can be called the holy subgroup in the case of [inaudible]. So the reduction
modular P over this group becomes so-called the Polaris subgroup and the Polaris subgroup
means the maximal server group inside the group.
So apart from the matrix. So that modular homes of gamma node P has some integration with
the cross number.
And maybe this was noticed by Eichler first. And then Ogus claimed that this is somewhat related
with modular equation. For example, if you have a modular equation, a modular equation is an
occasion which gives a model of gamma node P. If you reduce that, then all singular points come
from supersingular points. That's a kind of connection.
And also you have if you take similar model of gamma node P, modular P then it is crossing of
two cards which crosses only at supersingular points. To say that, I do not know if this is a
proved point of this fact or not. Anyway, they gave some geometric meaning of this.
>>: Modular equation, the classical.
>> Tomoyoshi Ibukiyama: Classical modulation.
>>: So JF tau and JF tau.
>> Tomoyoshi Ibukiyama: Yes.
>>: It would be the modular form of rate 2.
>> Tomoyoshi Ibukiyama: Modular form of weight two. They're not class form, so they're class
form plus one.
OK, so I would like to try something higher dimensional case. So my strong belief is that anything
related about supersingular things is controlled by isometrics. Every supersingular geometry is
reduced to geosymetric coded algebra. That's my belief. F or example, by strong
approximations the cross number of MJB, MJB is across matrix over P is one, bigger than Y
equal to 2. This is very much the result of Eichler or Kinesia something like that. So this is
conclusion, strong approximation server.
And this fact is reflected in the foreign theorem. So I don't know about the history, but this
theorem was obtained by maybe Deligne in the past. She would ask this question to several
people and the third answer that can be proved by this, but also Deligne had some paper and
other people like Ogus use that, I think. But I don't know the precise history. But usually people
quote other people, I think.
And so then if you take any supersingular elliptic curve, EI and EI prime where I runs over 1 to Z
then you take any product. Then any product of these are all isomorphic to the things. Since the
cross number one isomer this one is result. So this is very -- in a sense this is a stupid result and
it's not so interesting in a sense. So this is the original class number one. If you think this is
important you can go farther. Oh, sorry.
Then I can generalize to some other direction. But for that I must explain some of the geometry
definitions. So I need some geometric object. You might notice this very well, but I would like to
explain it.
So the [inaudible] is set to be supersingular if it's isogenous to this power of G. This power over
E. This is a supersingular elliptic curve so E can be supersingular elliptic curve because they are
all the same. So this condition is stronger than the condition that there's no P torsion. So even if
abelian variety has a mean T torsion, no T torsion then you cannot blame the data if it's Z. It's
just a different thing.
So you can control supersingular abelian variety but no P torsion abelian variety. It's quite
different and also I must explain what is prioritization. Maybe you know very well but I would like
to repeat. So by ATI write zero pick zero A which means algebraic zero, ATI brand zero divides
it, divided by the active device.
And isolambda A over T is set to be prioritization if lambda comes from some effective device or
[inaudible] and the S is defined cross over D and D is translation D by S and D is the devisor but
this is algebraic equivalent to zero the cross means being active to the why S is some point in
pick zero A. So the ample divisor you can define this is isogenous and if and if the map is
obtained like this and you call this prioritization.
We say that it is principally prioritization if lambda is isomorphic. So we can consider the role of
supersingular A lambda, where A is supersingular and lambda. I forgot to say when A is
isomorphic to E to the Z, then A is called super spatial. But supersingular is very different from
that. So super spatial abelian variety, there are infinitely many even if you consider the principal
polarization. So the low cost of supersingular principal supersingular abelian varieties have some
dimension in the modular of principle abelian varieties. So it is not finite number of points.
There are infinitely many points. And also it has big dimensions when the dimension grows up.
For example, when, for example, G is 2 dimension is very small. Dimension is just one. But in
general it becomes very big.
Now I explained almost everything about isogeometry but I must leave you some isogeometric
code and I think that maybe this is not popular, I think. This is less popular than the abelian
varieties. I'd like to explain -- actually, I don't know how to explain this. This is a bit of news. But
maybe I should explain that. Maybe I should explain that.
So that B is as before coding algebra which [inaudible] and O is maximal order. So that I said
that there are many maximal orders but I fix one. It doesn't depend so much. I fix maximal order
O and we take a group of similar code homogeneous form of B. The code homogeneous form of
B is B to Z. So the code homogeneous form B to Z is just you can divide by use your inner
product X 1, Y 1, bar, plus XY 2-bar plus something. So if you consider the synergies of group of
similar of that metric then you get this G.
And you can define authorization of G local components GB, et cetera, as usual. But so that this
might not be very popular. But if you take G by complex number then you'll get GSBGC. And
that fact it's said that the G is a Q form of GSBQ. So that somehow this is related with similar
groups. So this is another Q form with sympatric group. And the sympatric group is of course
connected with Z modular form so it's very close to consider Z modular form if you consider
something about G.
Now, I must tell you something about radius. So prefix O. So the left O modular L inside the B to
Z is called lattice when it is left O and it's also lattice in the usual sense.
On the norm error of norm is two-sided [inaudible] or respond by X transport Y, where XY runs
over elements of L. And L is called maximal if it's called the same norm. This is local property.
Local property means that if it is maximal at all local places, then L is maximal, globally. So this is
just local.
So if you would like to check which kind of orders or lattices become maximal, then you must
check local things. First you must check which kind of lattice exists as a maximal lattices locally.
So this is well known by Siegel. So here are the paper run in 1963, I think. So the heuristic L and
the computation at B. So essentially there are only one or two isomeric processes on the lattices
so at place, which is different from P at V. Finite plus V. Then L is L norm compression O at B.
So actually by assuming you can change normally a little, but doesn't matter essentially it's norm
B. And at P there are two lattice L P, norm is OP and the other one is norm is A. Where pay
two-sided prime meter P over P. The OP has only one two-sided L over P, so that's I denote by
T. So these two cases are different.
And also that genus of L is global lattices inside B to Z such that local behaviors is the same as
error. So which means that there exists all local raise V and V becomes LVH for some HGV,
NGV. So locally they're isomorphic, and the set of lattices is called the genus. So the locally
there are only two different lattices at P so there are only two genera of maximal lattices globally.
So let's call the first case, normally it's O case, principal, genus and norm is take, let's call it the
nonprincipal genus.
So we would like to denote that genus by LPG or L and PG. So the principal genus principal
genus and non-principal genus. Both are very important.
Then I got to talk something more to the -- we'd like to define some open subgroup over there.
UL is group which picks L out there, where LG is defined like that. So that some people might
ask how they act on error so I wrote the definition. So that error G is also global lattice.
So then if L is L kappa, L kappa stands for principal genus or nonprincipal genus so we call U
kappa for this group. L kappa. So actually the U kappa is open subgroup of GA. And since there
are two cases, principal genus and nonprincipal genus, so what's the local behavior? For
example, when V is not P then the local is just only one. So that's -- this is just a maximal
compact subgroup of SP2ZP actually. This is just so-called hyperspatial maximal component
subgroup. But when the press is P then we have two maximal compact subgroup.
Principal genus P and U nonprincipal genus P. They're different. And they represent a different
maximal compact subgroup of their local group. There are other maximal subgroups, but these
are very typical subgroups.
So the definition of cross number is given as follows. So that the definition of genus L kappa is L
kappa is a set of global lattices. So the G axioms are right. And so the G orbit is called the class.
And so the L kappa over G is a finite set. And that finite -- the number of that finite set is called
class number. Otherwise, if you use this group U kappa, then this class number becomes just
you saying G over G over UK. So this is just [inaudible] and this is just usual thing to define class
number so this can be called a class number UK. But I prefer to call this a cross number of L
kappa.
So I used 30 minutes. So the type number. Let's think of type number. If you take maximal
number order G they're all isomorphic nothing to say because class number of 1 G also type is
also one. So instead of that we must define G type number, type number related to Z. So the
whole fixed principal genus, or nonprincipal genus, we can take representative of lattice over G
over K. That's denoted from L one to L H. Then write order of right J to LJ writes it's just error
MJB which picks LJ. And this is order. This is also maximal order. And all these maximal orders
RJ are conjugate with each other inside the LGB because they are all isomorphic. But if you ask
the conjugate C classes of G, with respect to G, then it's different. So the J might be conjugate
over GRJB but J might not be conjugate over G. This number over G congested class is G type
number. I like to call this G type number, so that this is not common since I defined it. But
anyway ->>: Isn't G the terminate is in Q?
>> Tomoyoshi Ibukiyama: Yeah, asymmetric groups. So let's start from application to
supersingular geometry. So we pick supersingular key find by T minus O. There are many we
pick. So that first relation was obtained around 19 -- actually '82 but published in 1986. The
number of isomeric classes of principal production P to Z is class number principal genus if G is
bigger than one. If G is one, just only one principal genus, only one principal possession, so it is
not interesting. But when G is bigger than one then the principal process from E to Z is equal to
the cross number. So here I'm taking this so this is super spatial, not supersingular. And this is a
sketch of a proof. And then we have another object so that we have locus of principally pride
super abelian variety. Super-single abelian variety. Super single abelian soft varieties. So this
locus, has some dimension and actually this is connected, geometrically connected, but not in
principal in general at all.
In most cases it is not irreducible. The number of that irreducible components of S G1
supersingular locus I call it, is equal to the class number of principal genus and if G is odd, and if
G is even then it's a class number of nonprincipal genus. So we have two kinds of results related
with the class number.
Then I must talk about here are the definitions. So I would like to define what is a real definition,
let's assume A is defined over some period K. We say that abelian variety lambda defined over K
isogeometry is defined over K. Using these two very natural dimensions and also we say A
lambda over any field K has a model over k, small k, if there exists abelian variety over K,
[inaudible], which provides abelian variety over field.
>>: Is that already on important on not -- they're giving K over K isomorphically don't really care.
>> Tomoyoshi Ibukiyama: I don't really care. In between these two -- it might not be defined
small K.
>>: I know. But ->> Tomoyoshi Ibukiyama: So isomerism is any field, yeah.
>>: Not given K.
>> Tomoyoshi Ibukiyama: It might be become bigger but it doesn't matter. Yeah, it doesn't
matter. Any field. Some algebra, field you have some isomerism anyway. It doesn't matter.
This dimension is slightly different than the other but I will not explain it.
So now we would like to talk about principal polarization, division of period of principal
polarization so any principal E to the Z is defined over P scale. This is very well known. So I
wrote this in the introduction of the paper, this is a class I should cut it off.
So we did. And then another theorem. So the number of classes prioritizations of A to Z which
have a model over FP is equal to twice type numbers minus class number.
So this is exact order of Deuring's result. 2 times T minus H gives a number principal
formalization [inaudible] and also in particular we can consider this we have C over genus defined
over JC is congruent isomeric to E scale.
I think, yeah, I said that since in this case you can really count the number and you can give the
formula for the number. So, for example, even for the genus 3 case, every abelian varieties,
principal abelian varieties are Jacobean you can do that but I didn't know the numbers so I did not
write it.
But I have another result further along this line. So for any odd prime P there exists a similar
project genus defined over P such that rational, number of rational number of points over P scale
is given by one P scale plus 6P. Here six is two plus three and three is the genus. And so that
this is Bayes maximum. It's bound over Bayes maximum bound.
And this kind of thing is actually very, very subtle. So the proof of this factor is very difficult. And
the proof is given by the whole idea. So if you take E 3, I mean essentially the zeta functions of
Jacobean are equal to zeta functions over the could be. So if the Jacobean has a very good
property whose data function is easy to calculate then we are done in a sense.
If you take supersingular elliptic curve E then the [inaudible] over FP scale is just minus P. So it
gives correct number of rational points.
So if you find the curve C such that Jacobean is isomeric to E 3 then you are done. So you might
think that since we have a program for T minus A so 2 minus A can be variegated so we can
show the existence, but that's completely false. Because even if JC is isomorphic to E 3 it
doesn't mean it's isomorphic to P scale. The data function depends on the definition. So if JC
and E 3 are isomorphic to P scale we can get anything.
So another order Bayesian is the as sent is a program. So E 3 is purely certain Jacobean over
curve of genus 3 but we need some dissent property. But unfortunately dissent doesn't work so
much. Because usually they say if Jacobeans are isomorphic then curves are isomorphic.
But that claim is true only over isomorphic cross field so we must descend the data. The descend
data doesn't work so well because automorphism is Jacobean over plus minus one so there's a
difference when there's no public.
Long time ago [inaudible] asked me if there exists a [inaudible] curve of this sort and actually it is
not true. I mean in the category of this we cannot say that. So and also even if it's E 3 over FP
scale, for example, then -- no, sorry. Even if -- how to say it?
Even E 3 lambda has a model over FP, it doesn't help us so much, because we must check
whether it is isomorphic over FP scale. With that property we have to have several isometrics
and we need some subtle twists for servicing of that kind of existence, existence over M in the
[inaudible] group and so on. So I characterize some and I prove this. I think Christian did some
related work along this line related to that known scale PKs.
Now, I will talk about locus. So any irreducible components have model of P scale. So this is
rather well known, I think. So the next theorem is rather new, because I did it last year.
So the number of used components over S G1 which is a model over P is again equal to twice
type number minus class number. So it's principal genus when G is odd and it's the number of
genus when G is even.
And this number is also equal to trace R pi. I don't explain what R pi but R pi about type of
[inaudible] G over grade zero. So that type inversion is just no matter gamma node P, for
example. And in this case, too, there's some numbers over U kappa P so that this is obtained by
that numerator. So this is the genus type version. And trace is just equal to 2 T minus H.
So this is the new theorem. And then I'd like to talk something about explicit formulas, because
there was a formula by Eichler, so in case of degree one there was a formula like this. That is a
formula. Eichler's formula of class number.
But in our case -- so this is the formula for the principal genus. So it's much more difficult to
assess.
So you might ask why there are many such many times because it seems 1 minus 3 over P, there
are, RP is in many places, but actually this is just a contribution of some elements in the
[inaudible] group in the trace formula. So I just wrote them separately so the same thing appears
in many other places. And here, for example, minus 3 over P is just original symbol. Original
symbol and so on.
So this was by Hashimoto and myself. So the nonprincipal genus itself we have a formula, and I
did not give it here. It's in my paper as well.
And also we have -- we can determine the automorphism of LI. LI is a representative of lattices
of principal genus and number of principal genus. So the principal genus case automorphism
arrives, by automorphism by LI, I means is automorphism of lattice LI which resolves a metric.
So it's something like finite group in the autonomic group or finite group in G.
Actually this finite group is something like S five or S four or something like that. This is cyclic or
something. You can give all the groups here. And you can actually count how many lattices
have the same autonomic group or if you give a concrete group, finite group, you can count how
many lattices gives that.
And so, for example, for principal genus this automorphism group is nothing but automorphism
group of cap of genus 2. When genus is two. So I'm here talking when the case G is two. So
this is two dimensional.
So R known only for saying G is 2 and G is bigger than 2 I don't know. But then principal genus it
is automorphism group over cap. The thing for nonprincipal genus this automorphism group is
automorphism of family which gives us these components. So there are different objects.
But cap case you can just give a model of cap and count automorphism. And these components
you can do it in that way so that I calculate this number by using trace formula. So N is kind of a
trace formula I use.
Then I'd like to talk something about type number. So you can characterize type number two. So
if you divide the G-1 means that G transfers G as 1. So it's essentially sympathetic group of rank
2 but this case I'm calling compact form I can't tell you exactly. But if you take G1 infinity then it is
just a compact sympathetic group of rank two. So here SB 2 means the compact sympathetic
group of rank two and SB2 minus 1 is SL5. SL5 is just automorphism group of definite [inaudible]
size five. So this is class automorphism and also the type number is given by conjugation by SB
2. And if you take a conjugation by SB 2 then [inaudible] disappears. So essentially it gives you
the action of SL5. So the type number is essentially the class number with SL5. It should be the
class number of SL5.
So usually be able to do that. But actually if you fix some lattice, some genus inside the form,
and if you want to show that the certain type number is equal to the class number of quaternary
lattice you must say which kind of quaternary lattice should appear there. That's a different
problem.
So that detail is not so easy. But Eichler describes that in this case. So lattice seems P is -- and
the degree G is 2. Then the type number, both type numbers are equal to the class number of
genus of quaternary lattice is zero. Here genus means set of quaternary lattices which are all
isomorphic locally. So if you fix this, if you take this equivalent 2P scale or 2P then it consists of
only one genus.
And for principal genus we take quaternary lattice over discriminate 2P square. For nonprincipal
genus we take lattices of the schema for 2P.
And we need some condition, too. Some local conditions. So because at two isomer class is not
fixed by the discriminate, so that we need some condition two. And we assume that at two lattice
is something like this. So on the right-hand side I'm expecting lattice as a symmetric matrix so
that we just take 200100 so it's 5 by 5 symmetric matrix and that matrix gives quaternary L lattice
at 2. So the class number of this quaternary lattices is equal to the type number.
But of course this kind of regulation you can give it as G is 1 case in that case the type number is
exactly the class number of quaternary quadratic forms. In that case SB 1 over 1 minus one is S
3 so it's a normal form it's maximal order in that case but it's exactly the same as a class number
of quaternary lattices with some fixed discriminant.
Now on the other hand [inaudible] gave a class number formula for such quaternary lattices a
long time ago, around 1977. So it's a very, very complicated formula. But anyway he did it. So
now we know the class number is XP 3 and also the type number XP 3 so we can calculate the
2T minus H. So this is a result.
So the number where of irreducible components of S 2, S 2 means genus 2 to everyone, principal
supersingular locus defined over FP is given by these numbers. So this is the result here. B 2 is
generalizably new number. It corresponds to the Q square root P, exactly corresponds to Q
square root P and B is the abelian number P scale number.
And here H scale minus something is just a class number of imaginary quadratic view. So it
contains class number so that sense it's a complicated formula. But you can see that in the
one-dimensional case that the formula is just contains class number. So this kind of thing should
contain class number, so it's not surprising. So now I'd like to talk another subject. Sorry I
change topics so often.
So let's talk about geometric forms. So e can talk about geometric forms supersingular locus. I
don't explain what is geometric forms that it is an [inaudible] function invariant by certain XA
groups and we take a so-called paramodular group, KP over P, and also [inaudible] P. They are
inside the simplex group of size four.
Some people write mainly American people write SP 4 Q instead of S P2 Q. I'm usually right. It's
by two. So allow me. It's size four. So we have a very interesting relation here. This is a
theorem. When genus is 2, then 1 plus dimension of cusp form of wave three belonging to the
parametric group is equal to the irreducible component of S 21 and also if this P is bigger than
five than BP 3 means BP zero subgroup. Three means weight three and new means new form
inside here. I don't give the dimension, but we can give the precise dimension for that. But
anyway we can characterize that dimension.
And that dimension is equal to isometric genus over S 21. Here S 21 is not irreducible and S 21
has many similarities but still we can define isometric genus. So that isometric genus is equal to
S 3.
>>: The 3 form.
>> Tomoyoshi Ibukiyama: No, S 21 is the supersingular locus.
>>: So it's a curve.
>> Tomoyoshi Ibukiyama: This is actually a curve. Supersingular locus is actually crossing many
cobs. Crossing many cobs. The components are cobs. I don't know the reason for this. I think I
can get both sides to compare. I should say something about ground about this.
So there are the [inaudible] conjecture of homogeneous for G homogeneous forms case I have
very explicit conjecture since 1980s about these. So one side it's KPNPP and other side is
nonprincipal genus and minimal sub-parametric group inside. I have very, very precise
conjecture about this and also I have a dimensional formula of both sides so which support this
conjecture. So both sides we can decide and both dimensions are the same. So that if you can
characterize all the trace of [inaudible] you can prove it. But I don't know that part. But very
strong support from dimensional formula. Dimensional is the same so maybe they should be the
same.
Now I could characterize this case bigger than five but in case three or four I could not. But I did
it two years ago. So then I found that there's a relationship between supersingular things and this
one.
Also since a class number is essentially the corresponds to homogeneous always, if you believe
this conjecture, but the geometric meaning is not clear. So if you take a small model of
paramodular group or [inaudible] group and reduce it modular P, then maybe you can explain
why you can give [inaudible] proof of this. My proof is I just calculate both sides independently
and then coincide so they are the same. That's my proof. But that's stupid proof ->>: What do you mean by [inaudible] what you're trying to say is you have more front on the G
group, such that principal -- not principal genus UP you can produce which [inaudible] form of the
KP is that what you want to say?
>> Tomoyoshi Ibukiyama: No. No. I'm talking about geometry. If you take a similar model and
take the reaction modular P, for example, then maybe some geometric object inside may be
expressed by the supersingular things.
>>: It's the geometry of S 21 that you're talking about?
>> Tomoyoshi Ibukiyama: No, no. Yeah, yeah, maybe inside the similar variety corresponded to
the geometric form of modular P contains S 21 or something like that.
>>: Do you expect that you don't know the challenge, the explanation, but would you expect an
arithmetic characteristic generalization of this you encounter higher dimensional?
>> Tomoyoshi Ibukiyama: Yes, I think so. But I have no results for higher dimensional case. I'm
just aiming that's future work. So I don't know at the moment.
So and also I should say that I characterized these dimensions around the 1970s when SB is
equal to Y 5. Why don't I do that with weight three. And the weight three is more difficult than the
general case?
The reason is that the, first of all, [inaudible] doesn't [inaudible] and secondly if you want to use
geometric wave three in something like that then such kind of method depends heavy on the
[inaudible] but if weight is very small then there's no punishing theorem. Actually it's not true. It
doesn't punish at all in general.
So that my remedy is to prove this kind of banishing directory for these special [inaudible] groups.
So I do not know the general theory, but I can prove that this action banishes only for these
groups. That's my solution.
>>: Actually, weight three is also banishing, actually not smaller.
>> Tomoyoshi Ibukiyama: Not smaller. Wave three is ->>: It should be maximal.
>> Tomoyoshi Ibukiyama: 3 is the bottom. Wave two, I don't know anything.
>>: Yeah.
>> Tomoyoshi Ibukiyama: It's very mysterious. And I don't have any conjecture, anything I don't
know. [Inaudible] but that's the only result, I think. And also this kind of banishing may be where
we two for vector case so I gave this comment in the [inaudible] because I think this kind of
banishing is also bad for the vector case. I did it over the [inaudible] case [inaudible] case. And
the vector Barrett case I have a similar type of conjecture in terms of weight [inaudible] so this
conjecture this kind of banishing is barred in the vector Barrett case. But this is different. So I
don't talk about it.
So I am almost time. What's next? So it's not the end of the story, when these two. So maybe
you notice there are strange things. One side there is locus supersingular locus of supersingular
abelian varieties. On the other hand, there is a super spatial abelian varieties, principal
[inaudible] so the supersingular locus of course contains super spatial abelian points as a point.
So the super spatial abelian varieties are the finite many points inside the locus S 21. So, for
example, you can ask to which irreducible components that super spatial abelian variety is on, or
not, or something. It's kind of an isomer if I can question. So the components belongs to the
[inaudible] U energy PG and super spatial elements corresponds to the WP and G and what's the
relation between this when P is on W.
So the isomeric if I can answer is given right away if it's W. If W contains common W [inaudible]
heuristic minimal [inaudible] and this is not trivial theorem at all. You need some isomer if I can
about it. It's an easy, easy theorem. And so we would like to generalize this to the general
computation. So, for example, when G is 3 then we have some results or already with [inaudible]
aiming to generate these kinds of things to understand all the calculations of structure of
supersingular locus.
But, true, it's unfinished project I'm aiming to do that. Thank you very much.
[applause]
>> Kristin Lauter: Questions?
>>: When you said [inaudible] what's a super spatial really is in more your objective, and here
you talk about supersingular meaning it's E squared, for example. And with the principal
[inaudible] what's a super spatial mean?
>> Tomoyoshi Ibukiyama: So super spatial means E scale, E to the Z.
>>: Super ->> Tomoyoshi Ibukiyama: Super spatial. Supersingular means isogenous to E to the E. And
super spatial means ice more if I can E to the G.
>>: What about the polarization.
>> Tomoyoshi Ibukiyama: Polarization is completely different.
>>: Then [inaudible] meaning isomorphic, isomorphic to this with the parameterization.
>> Tomoyoshi Ibukiyama: No, no, no, just abelian variety. If abelian variety is isomorphic to E to
the G then it is called supersingular. There are many different [inaudible].
>>: So there finite many of them.
>> Tomoyoshi Ibukiyama: Finite, many of them. I said it's equal to the class number of principal
genus. And also -- yeah. Supersingular abelian variety, which are not necessarily super spatial.
We can count also the [inaudible] to some extent when genus is 2, for example. But I don't know
the general formula for that.
>>: Counting on the supersingular [inaudible] name again to 1, we get to 1 again?
>> Tomoyoshi Ibukiyama: [Inaudible] I think.
>>: [Inaudible].
>> Tomoyoshi Ibukiyama: Sorry. Sorry. Right. They are all isogenous to E to the G. So the
class is only one, sorry.
>>: Principal, you could [inaudible].
>> Tomoyoshi Ibukiyama: No, [inaudible] but anyway if you fix abelian variety, then the principal
polarization on that is finite on that.
>>: Still finite.
>> Tomoyoshi Ibukiyama: Yeah.
>>: I see. But isomer if I can is abelian varieties are infinite.
>>: One and two.
>>: I wondered if you could say something more about, so your theorem you proved the
existence of a curve with, like, P squared plus one plus 6 P points? On the bottom when you say
what the obstruction is, and you can't necessarily find a hyperelliptic curve with this property, so
I'm wondering what the technique then is to show that you can actually get the model of P
squared.
>> Tomoyoshi Ibukiyama: So, first of all, we know when E 3 principal polarization E 3 has model
over FP. We know that. And here the problem is that we cannot descend it to the -- for example,
if we descend this to the AP scale, then JC and E 3, there exists isomorphism over quadratic
extension over the base period. But not the base period itself.
So if you start at P scale, then this isomorphism might be over FP force. If you start from FP then
quadratic extension is P scale so you can do it. But the theory is not enough.
Because you can count -- when this has a model over FP, I mean I explained to you that there's
some way to characterize the model over FP if only T minus H is nonzero. So you might think of
this -- this is the end of the story but that's not true again. Because model means that you might
change E 3 itself to another abelian variety which is isomorphic to E 3 but not necessary isometric
over FP scale. And that A might not be E 3 itself. Of course it's isomorphic but the model, there
are many models so the model might be different from E 3.
And if it is not isomorphic over FP scale, then still we have a problem. So the ideal problem is
that we try to obtain the principal polarization over E 3 defined over FP. E 3 itself, not the model.
But E 3 itself defined over FP. Just another condition. So we can write down that condition
carefully by the isometric code in algebra and we can show it existence of that kind of
polarization.
>>: What's it mean from the trace of R pi very nice but complex, formula. I wonder if we can look
at it FP change [inaudible].
>> Tomoyoshi Ibukiyama: What?
>>: The P. The trace of R pi become [inaudible] alpha another formula. You look at that
[inaudible] generating, is that part of a homogenous form?
>> Tomoyoshi Ibukiyama: P change. If P change then that means [inaudible] changes.
>>: Generating formula. The trace of R pi if P to the -- Q to the P.
>> Tomoyoshi Ibukiyama: You mean something like a hypo polynomial?
>>: Yes, for example.
>> Tomoyoshi Ibukiyama: But that can't be done because we don't know the trace of TP scale or
something like that. That's the only case.
>>: The classical, if you know [inaudible] have the class member, quadratic fields. [Inaudible]
clearest class number. Generative function. The model form.
>> Tomoyoshi Ibukiyama: Yeah, yeah.
>>: You wonder ->> Tomoyoshi Ibukiyama: That I don't know.
>>: So ->> Tomoyoshi Ibukiyama: Like a [inaudible] or something.
>>: Or something.
>> Tomoyoshi Ibukiyama: I never thought of it. So I don't know.
>>: Okay. You want to have P parameter, what do we mean, you know what I mean.
>> Tomoyoshi Ibukiyama: Might be simple, but I don't know.
>>: Okay.
>> Kristin Lauter: Any more questions? Thank you.
[applause]