Let me welcome you to the next session. It is a great pleasure for me to introduce the next speaker, Kazuyo Kato of Kyoto University. Kato is one of the world leading researchers in algebraic geometry, especially arithmetic, algebraic geometry and number theory.

His work is noted for its brilliance and originality and he has research in many different topics which include higher dimensional class field theory, Piatik Hodge theory, log geometry and especially the special values of Haseve L functions and Iwasawa theory.

In particular, his work on the last topic gives some of the most striking known cases of the famous Birch and Swinnerton-Dyer conjectures, one of the great conjectures in modern number theory. In fact, his lecture today is on Iwasawa theory.

This theory has proven to be one of the most powerful tools in modern number theory, perhaps a little surprisingly but really powerful as all experts know. Kato likes metaphors and artistic impressions as you'll see in his lecture. In fact, he's written a poem about prime numbers

which can be found in the 2003 volume of Documenta Mathematica which is dedicated to Kato's deep influence on the subject of arithmetic geometry through his profound research and through his many students. He has won many prizes. I won't go through them but I'll just mention

the Imperial Prize of the Japanese Academy. Professor Kato. Thank you very much for the nice introduction and thank you very much for the invitation.

So, Iwasawa theory is a deep theory which relates theta values to arithmetic. And the mysterious point is that theta values

are analytic objects and the analytic objects and the arithmetic objects are connected in spite of the vast distance between their innate natures. There are, as you see, there are many questions around there.

In this talk, I first read you some story and then describe some recent developments in Iwasawa theory. So, sorry. Yeah, yeah, this one.

Oh, here it is. So, the study of theta values was started by Oida. So, this is the given zeta function, the standard zeta function. And Oida discovered that zeta two is a pi square over six in 1735.

And Oida was very happy that he solved the difficult question, what is the sum of the inverses of all the squares? And also he was happy that the answer he obtained, the appearance of pi was surprising.

And he proved that zeta r, or r, positively even integer is a rational multiple of the r power of pi. So, and then I.

Yeah, so this is Oida, yeah. And then next, yeah, so next is this, yeah.

So, Oida also considered the value of zeta function at negative integers. Then he sees the formula he obtained.

And for example, even negative integer, the value is zero, and for all the negative integer, the value is expressed by the value of a positive integer, the m is here, one minus r.

So, one minus r. So, then zeta minus one is related to zeta two. And zeta two, so it is one minus one over 12. And so, all negative zeta values are rational numbers. And in fact, these negative integers are outside

the area of the convergence of Riemann zeta function. And so, for example, this zeta minus one is, in fact, diverges as an infinite sum. But today, we can, we define these values

by the theory of analytical continuation. Oida did not know the theory of analytic continuation, but he had a method to compute such divergent series and defined the correct values. So, here is a list of some values.

And then the denominators are, in fact, simple things. There is some simple rule which prime number appears in the denominator with pitch multiplicity.

There is such a simple rule. But for the numerator is really a delicate object. And here, the 691, this is the prime, appears suddenly here. So, now, next, Kuma appeared, Kuma comes.

In the middle of 19th century, Kuma discovered that the special values of Riemann zeta function have the two remarkable points,

two remarkable arithmetic properties. Why is it written here? These two discoveries were the starting point of the Iwasawa theory. So, here, Kuma's theory means that if you pin

R is prime, and if R and R prime are negative integers, and R is congruent to R prime of modulo p minus one, then zeta values are congruent to modulo p. And this theorem of Kuma was generalized later to congruence modulo p to the n for higher power p.

And then, in 20th century, it was this such periodic properties discovered by Kuma was summarized by Kuma and Leopold.

This was in 1964, as the existence of the periodic Riemann zeta function, which periodically interpolates these zeta values.

And the next theorem of Kuma was the Kuma's criterion. So, this Kuma's criterion relates zeta value and ideal class group.

So, if prime p divides this zeta value at negative integer, and for some odd integer R, if and only if the ideal class group of the q zeta p contains a group of mod order p. So, here, zeta n, in general,

means the primitive n-th root of one. And so, this is just the class number, or the order of the ideal class group is divided by p, and that's the same thing, yeah. And so, for example, as we see, the zeta minus 11 is divided by 691.

So, this means that the p divides the class number of q zeta p if p is 691. So, these are Kuma's criterion, and also Kuma's criterion was Kuma's. These two arithmetic properties of zeta values

are very mysterious. These cannot be imagined from the analytic definition of the zeta function. I hope to talk about the importance of the ideal class group, because that was the subject of,

oh, sorry, I lost some fear there. Yeah, yeah, yeah, yeah, sorry. Yeah, yeah, sorry, I lost something there. Anyway, Kuma proved that if p doesn't divide the zeta value

or p doesn't divide the zeta value for all the integers, negative integers, or p doesn't divide this class number, then the film has lost the equation x to the p plus y to the p equals z to the p has no solution.

And this was, how this was proved is that, sorry, I lost the, sorry. Yeah, yeah, yeah, sorry, yeah. So, the film has equation x to the p

plus y to the p equals zeta to the p is denoting the multiplicative form but using the primitive piece root of one. And then, in the multiplicative form, he tried to decompose,

consider the prime decomposition of both sides. But in this field, the prime decomposition doesn't work in general. And so, but if the ideal class group is, the ideal class group is not divisible by p,

then the arithmetic of this field becomes very simple and so he could prove some strong result on film as last segment, which was the strongest result before was proved completely by a different method.

So, ideal class group vanishes if and only if the prime decomposition, unique prime decomposition in this field holds. And then, if the ideal class group becomes big, then the arithmetic of this field becomes complicated.

So, in other words, the, so, yeah. The ideal class group is a beta group which relates number theory harder. If the ideal class group is big, then the number theory of that field becomes complicated.

But Kuma discovered that the ideal class group is a sweet group which has sweet relations with zeta values.

Next is the Uesawa theory. Yeah, in the late half of 19th, I don't know, 20th century now, from 1950s, Uesawa started Uesawa theory. So, in Uesawa theory, compared with Kuma's study,

Uesawa theory is finer in the point that it considers not only the order of the ideal class group but also the action of the Garawa group on this ideal class group.

And for this, here, we can take many cyclotomic fields. And then, not individual zeta values, the unifying object, the periodic Riemann zeta function

is here. And so, this theory is the relation between zeta side, periodic Riemann zeta function, the arithmetic side, ideal class group.

Uesawa was a very nice person, and he was always smiling. You cannot see the smile because it's a bit too big. But yes, when we study Uesawa theory,

we are very much surprised by the mysterious properties of zeta values, and by the shock, we naturally started to smile as him. And I think this may be the reason why he was always smiling.

So, how the Garawa action appears is explained by here, the theorem of Elbrun-Ribbet.

This is now regarded as a part of Uesawa main conjecture formulated by Uesawa. So, for R, negative integer and odd, and then the P, zeta R, if and only if this class group has some P part

with some special Garawa action. So, as we see already, zeta minus 11 is divided by 691. And we already know that this implies that the class group contains the group of order,

some group of order P. But what is the meaning of this level? So, this explains, this is for each R statement, for each R, so if you put minus 11, then this means that the class group

has some order P part on which the Garawa group acts in a very special way with here, minus 11. So, for example, yeah, this is, so if P is 691, then this minus 11 appears here.

There is some special part on which the Garawa group acts in some shape of minus 11 twist, yeah, yeah. And the Iwasawa main conjecture formulated by Iwasawa and proved by Meza-Wiles

has this form. Sorry, sorry, sorry, sorry. Yeah, yes, yes, yes.

So, Iwasawa's main conjecture proved by Meza-Wiles is equality of this form. So, I don't explain some details, but here you have the prerequisite function on the left-hand side. And here, you have some object,

arithmetic object, which is defined by using real class groups of cyclotomic fields of roots of one of the higher power of P. And so, in the theorem of Liebherr and Elbron and Liebherr,

in Kuma, they considered only the field of this root of one, but here we can consider higher power of P, the bigger cyclotomic fields.

And then, each zeta values are unified in periodic zeta function. And so, then this is a stronger form of the theorem of Elbron, Liebherr, and the result of Kuma. So, here the relation between zeta side

and arithmetic side are connected by a beautiful equation. Many people tried to generalize this Iwasawa theory to Iwasawa theory of algebraic varieties,

and we now hope to explain the Iwasawa theory of elliptic curves. Oh, sorry, sorry, sorry, yeah.

So, let there be an elliptic curve over rational field of rational numbers,

and for simplicity, I assume that it has good ordinary detection at P. Good means that we can take the modulo P over the elliptic curve is still an elliptic curve. And the modulary means that if we did that such elliptic curve, modulo P elliptic curve

in characteristic P, the Pthosian point of the valued point is asymptotic to z, modulo P over z, or zero. And if it is non-trivial, then we call the elliptic curve ordinary, and if it is zero, if it is supersingular.

And then the ordinary means that the modulo P is ordinary in this sense. And this supersingular case, Iwasawa theory of elliptic curve exists, but the shape is rather different and a little more complicated, so I assume for simplicity that it is a good reduction.

And we take the cyclotomic ZP extension of Q, which is the unique extension of Q, unique Galois extension of Q whose Galois group is asymptotic to ZP. And then I also take the imaginary quadratic field K

in which P splits, and take K infinity, which is the unique Galois extension of K whose Galois group is asymptotic to ZP times ZP. ZP squared means ZP times ZP. And then we form the Iwasawa algebra.

It is the inverse limit of the usual group algebra of the Galois group. Here, Q in and K in are just the intermediate finite extensions of the base field, and then the union of those, these fields. And so we take, this is called the completed group ring

of these profinite Galois groups. And the first one, this Iwasawa algebra is asymptotic to this ring of one. Formal path is in one variable over ZP, and the second one is two variable.

Then for, so the original Iwasawa algebra is such a complete group, commutative periodic group algebra, yeah. And the periodic zeta function is essentially living in such algebra.

And so now, and then, in the case of relative curves for zeta side, instead of the periodic Riemann zeta

function, we have the periodic zeta function over the elliptic curves. So the first one is cyclotomic periodic zeta function. Second one is the imaginary quadratic version of that. And these periodic zeta function periodically interpolate

the complex zeta values, these complex zeta values. So this is a, so before I explain this, E has a complex zeta function, which is known to be a homomorphic null, and then it is defined

as some Dirichlet series. And we can twist the zeta function in this way by Dirichlet character, and then that Dirichlet twist is defined in this way by just putting the kain on this, that place of the Dirichlet series.

And then, those values are complex numbers, but if you divide by some invariant called period, then the ratio is a rational algebraic number. And so then we can regard those numbers periodically.

And so here, chi is a Dirichlet character, oh sorry, for this Dirichlet character, but for this version, chi is a quadratic, imaginary quadratic version of the Dirichlet character.

And this is a zeta side, so instead of the periodic zeta function in classical Iwasa theory, the periodic zeta functions of the, periodic functions of the elliptic curves are periodic. And then, this is arithmetic side. So instead of the ideal class group,

we take the Thelema group. Thelema group is some, it sits here in some exact sequence. Here, you have the group of, group of rational points here, and the, but tensor with Q over ZP.

And then, you have again, also some group called Thete-Sharovich group. And this group is conjunctually always finite. And so, Thelema group is almost this group. And so, but in Iwasa theory, Thelema group is better,

behave better than group of rational points. And so, in Iwasa theory, this group appears. But in Burt-Sina-Tondaya conjecture, those two groups appear, and in Iwasa theory, elliptic curves is closely related to the Burt-Sina-Tondaya conjecture. But I am sure that in this talk,

I don't talk about Burt-Sina-Tondaya conjecture. On the date, after tomorrow, Darmon will give a talk on Burt-Sina-Tondaya conjecture and on his excellent results. So if you are interested in Burt-Sina-Tondaya conjecture,

please, please attend to his lecture. And so, on arithmetic side, we need some definition, characteristic idea. That is, for a module who are this type of algebra, and for M are finite region-related

two-genre module, then M is almost isomorphic to that direct sum. And then the characteristic, this is pseudo-isomorphism to this module of this form, precisely. And then, the characteristic idea of M

is defined to the product, to the idea of bar generated by this product. So this is a non-zero principle idea of bar. And this is a module-theoretic version of the order of the finite abelian group.

The order of the abelian group is that if M is the finite abelian group, then M is isomorphic to such direct sum and the order is the idea of Z generated by, the order is equal to the idea of generated by this product.

And so, and then the order of the finite abelian group is the most important invariant of the finite abelian group. And here, we are considering the, such similar group with Galois action. And then, the lambda module structure, lambda cyclotomic lambda module structure,

these measure structure of the US algebra is described as the Galois action. Because the Galois group acts on the, on the, on the cinema group, the group ring can act on the, on the, on these real cinema group. And so, so the, this lambda module structure,

this describes the Galois module structure with those cinema groups. So the most important invariant for these Galois module structures is indeed the characteristic idea. And now, now listen to progress the following.

So first, because the characteristic area is defined only for Tojo modules,

as order is defined for finite abelian groups. So, so we should know that these, these modules are Tojo. The arithmetic side is a Tojo group. And then, this is, I proved that X cyclotomic is a Tojo module.

And then, from this, we can deduce that this X is a Tojo lambda module. And this into work of Skinner-Uban gives a theorem now, a big theorem now, that under some mild condition, we have the, the analog of US main conjecture

for elliptic curves. The idea generated by periodic zeta function is equal to the characteristic media of X real cinema group for this cyclotomic case and for this quadratic imaginary case.

And, and so this is a analog of US main conjecture. And this means that 20 years later, we proceeded from classical US theory to US main conjecture of US theory of elliptic curves.

And this is a consequence of two divisibilities. And I proved the one divisibility for cyclotomic case under mild condition. And this is arithmetic side divides the periodic zeta side,

and recently Skinner-Uban proved under mild condition that for quadratic, imaginary quadratic situation, they proved that the periodic side, the zeta side divides the arithmetic side. And from this, we can deduce the above theorem.

But our difficulty as speaker, my difficulty as a speaker is that the paper of Skinner-Uban is not yet completed, but they wrote a shorter version of the papers

and they gave many lectures. So I think I can put their work as theorem here. But of course we have to wait that they complete the paper.

In the case, he has complex multiplication, but by that imaginary quadratic is K,

the theorem was approved by Rubin long ago. And also there is another version of Iwasawa theory, another world of Iwasawa theory. For the Iwasawa theory of E, but over this abelian extension,

so-called anti-cyclotomic extension, then there is a theory, anti-cyclotomic Iwasawa theory of E. And so there is some special extension called anti-cyclotomic extension, which is also one of the cyclotomic extension, obtained by cyclotomic extension.

And then there are some beautiful theories on this extension. And Bertolini and Darmon proved also the arithmetic side divides zeta side, just like this for this anti-cyclotomic theory. And they and other people, Colibaldi and Howard,

and obtained strong results on this theory. I am sorry that I am not introducing many important works of many peoples in this talk. There are many, many important talks which I don't introduce in this lecture.

And for this subject, Darmon will talk, I think, on the day after tomorrow. And for concerning such theories,

there are two methods in Iwasawa theory. Two methods in Iwasawa theory. One is Euler system method to prove that the arithmetic side is a tojon, and also the arithmetic side divides zeta side. And another method is modular form method

to prove that the zeta side divides the arithmetic side. For the classical Iwasawa theory, the first proof given by Meza Maris was modular form method. And Reita Rubin gave the second proof

by Euler system method. And so for this elliptic curve version, so for example, this result is by Euler system method, and this shows that the arithmetic side is small enough. And also for this also,

this is proved by Euler system method to prove that the arithmetic side is small enough. And the theory of result of Skinner-Uban was proved by modular form method similar to Meza virus,

and they showed that the arithmetic side is big enough. And so I hope to explain how these methods. First, I hope to explain the Euler system method. This was discovered by Colibargan, other people like Ting or such people

also had similar ideas. But yeah, Colibargan is the most important part for this method. And so the method is that the data function and the arithmetic objects are too far. These data functions are analytic

and the arithmetic objects are arithmetic, so the natures are pretty much different. But the data function has some arithmetic inclination called Euler system, which is arithmetic family. And so Euler system is arithmetic, so it is very near to arithmetic objects.

So by studying this factor, we can relate the data function to arithmetic object in spite of the vast distance between them by using the Euler system, which is arithmetic object.

So for example, in the case of classical Uesawa theory, the arithmetic inclination of Riemann's data function and also the Dirichlet series is a psychotomic unit. It is arithmetic object, in fact,

one minus just one minus root of unity. And but this arithmetic object is related to data values, and so I am sure that this is an incarnation of data values, though they don't say that they themselves are incarnations. They are silent, but we can see that,

for example, in R and C, by taking logarithm of psychotomic unit, then we obtain the data value in this way, just to receive the expansion of log. And also this is related to the following factor also.

Again, we take the logarithm and then take some twist by Dirichlet character, then we have the derivative of the Dirichlet character to Dirichlet function to sequence zero. And also, not only R and C, in QP, by the theory of Kuma and Uesawa and Kurzweil and Kurma,

this family of, this is, in fact, an inverse system of, for normal system, so this family gives a periodic Riemann Zeta function. This arithmetic object produces a periodic Riemann Zeta

function by some homomorphism. And then the periodic Riemann Zeta functions are related to Zeta values for negative integers. So this element is periodically related to those negative Zeta values. This is really a mysterious thing.

So I wonder, I wonder, I wonder much, yeah, yeah. So Zeta function enters the arithmetic world, transforming themselves into, oh, sorry, sorry,

into Euler systems. And, yeah, I don't know, and they produce the formula, arithmetic side divides Zeta side. This is a spirit of Euler system method. And so this is too mysterious, so I become a little crazy.

So Zeta function transforms to Euler system, but maybe there is some area called Zeta land or something like that, from which a Zeta function comes to the CR and the R as a Zeta function, analytic Zeta function. And then they come also to the periodic world

as a periodic Zeta function, and then come to the arithmetic world as Euler systems. And so maybe no one knows that such land, but there may be some land called Zeta land or something like that. Is it like Disney land or something like that? I don't know, but maybe this is like

a neighbor land in the story of Peter Pan. Yeah, something like that, I think, yeah, yeah, yeah. So if we study Zeta values, then they are too mysterious, and so we sometimes become crazy, yeah, yeah.

So there are many Euler systems, and not so many, so sorry. The problem is that we find only a few Euler systems, so psychotomic units are one example. So psychotomic units are in combination

of Riemann Zeta function, and it lives in the psychotomic field. And then, but this group is understood as K1 of that field. And as I explained, they are related to this Zeta value in R and C and in QP, the periodic Riemann Zeta function.

And also for the Zeta function of elliptic curves, we have also an incarnation Euler system called Berenson elements, and also Hygna points.

Berenson elements live in K2 of the Mujra curve. And in R and C, it is related to the derivative of the Zeta function, complex L function, of elliptic curve tested by chi at zero, S equals zero. And also, the Berenson elements are related

to PRDK function in psychotomic theory of elliptic curve. And Hygna points appear in anti-psychotomic theory. They live in Jacobian Mujra curve, which is regarded as K0, a part of K0 Mujra curve,

and related to these derivatives. And periodically, again, related to the periodic function of the elliptic curve for anti-psychotomic extensions. And so probably, always for any kind of complex Zeta function, there is some arithmetic reincarnation

which live in similar varieties, in K0 similar varieties. So those are K groups, and Mujra curve is a similar variety so we expect that always some such Euler system appear in K groups of similar varieties,

but the problem is that only few Euler systems are known.

So for the explanation of the split of Euler system method, I already used Peter Pan, a story for children in England. I now hope to use the story in Japan for children. I am very sorry that I do not talk about

a story in Spain. The title of this story is the Crane return to Sanxu. So a man saved the life of a crane, which was shot down by another person, so here. And then later, crane came about transforming herself

into a girl, and then she produced, in his home, she produced beautiful clothes for him.

He wished to return to express her thanks to him. Yeah, this is a story, yeah. I don't know, I'm so sorry. Ah, yeah, yeah, okay, okay, yeah, yeah, yeah. So anyway, yeah.

So I think in this story, Zeta function, the Zeta function is presented as a crane. Yeah, for a crane, it is difficult to enter a house of a man. So since the Zeta function is too much analytic, it was, it is difficult to enter the arithmetic world.

So she transformed herself into a Euler system, psychotomic unit. And then after, psychotomic unit, psychotomic unit is, I'm so sorry. So this is psychotomic unit, and this is the aircraft loop. So they are human beings, so now, so very near.

So, and after she entered the arithmetic world, then she produced beautiful clothes. Here are the beautiful visibility in main conjecture of the U.S.A. theory. Oh, thank you, thank you.

Thank you very much, thank you very much. Thank you very much. Yeah, I worried if, yeah, but still this explanation may not work because this is mathematical, or mathematics, so, yeah. So I think I have to give more mathematical explanation,

but the mathematical explanation is, in fact, there's a complicated factor. So I just explain roughly. So how we can obtain the divisibility in the case of psychoclassical U.S.A. theory. So then the system of psychotomic units, the older system, produces by a procedure

discovered by Kolibergian many, many principal ideas, as many as expected. And showing that the ideal class group, which is a quotient of the group of fraction leaders by principal ideas, this one is as small as expected. So there's some method of Kolibergian to produce

many, many principal ideas using the system of psychotomic units. So then this group of principal ideas become as big as expected. And that means that the quotient group becomes as small as expected. And this shows that the quotient group,

the ideal class group, is small enough. So then ideal class group divides the set size. This is a rough story of the Euler system method. Now I hope to explain the modular form method

used this time by Skinner and Uma and before by Meza-Weis. So the modular form method is the following. So the Riemann zeta function and the Dirichlet function

is a zeta function of modular form of GL1. In fact, the Dirichlet character is regarded as a modular form of GL1. And this zeta function of the elliptic curve

is a zeta function of the modular form of GL2, thanks to the solution of similar tangential conjecture. The zeta function of elliptic curve is now a zeta function of the modular form of GL2.

And then the split is that the theory of elliptics can be studied by using the theory of modular homes of bigger algebraic group. In the case of Meza-Weis, it was GL2. So to study the usual classical USR theory,

which is the USR theory of GL1, Meza and Weis used GL2 modular forms. And this time Skinner and Uma uses the modular forms of U2 to study the USR theory of modular forms of GL2. The full description of the theory method is rather hard,

so I just explained the speed using the case of Debe's theorem. So how Debe's theorem was proved by the modular form method and Meza-Weis, in fact, extended the method of Debe.

Debe maybe gave first the modular form method and Meza-Weis extended it to prove some conjecture. So Debe's theorem is this one. If P divides the value, then the ideal class group contains a Galois module with order P with R twist.

And how is the proof is the following. But this is a part of the zeta side of the device, arithmetic side. This shows that we have not very, very part of the ideal class group. So this means that arithmetic side is big enough. So this is showing that arithmetic side is enough.

There is some special non-trivial part. So this is a part of zeta side device, arithmetic side, compared to the Euler system method. So there are three points, key points for this method. The first one is that the Riemann zeta form values appear as constant terms of Wiesenstern series of GL2.

And so in this way. And the second, yeah, here, this is the Euler series and the Riemann zeta value appears as a constant term.

And then the second point is that the Euler form of GL2 produces a Galois representation into GL2. This is a part of Langlands correspondence,

the correspondence between modular forms and Galois representation. And the third point is that the ideal class group is regarded as a group of extension classes of finite Galois representations. For example, this condition, ideal class group, contains such special Galois part

is equivalent to the existence of the extension of this form. This is a representation of the modular PZ with some condition here. And then how the method works, how proof works,

is happening. So if P divides zeta R, then the Euler series, modular P has no constant term because constant term was divided by P. So then, because constant term is a zeta value.

So then, roughly speaking, the modular form without constant term is a cusp form, so, roughly speaking. So then, the Euler series is equivalent to a cusp form modular P. And then, our example is that, so, example,

691 divides zeta minus 11, then in this case, as instances of wave 12 is coming to the cusp form delta, eigen cusp form delta modular 691. This is Raman-Jan's Congress.

This cusp form is the famous cusp form delta. And then, we take the Galois representation associated to this cusp form, yeah. Then, by two, yeah.

Then, the composition is, we take the composition to take the, we take the modular P of the Galois representation. Then, by the fact that F is going to the Eigenstern series modular P, this modular P Galois representation becomes some simple property that it is not reducible,

but it is an extension of one-dimensional representations of that form, and if it satisfies that condition. And for example, the non-splitting of this extension comes from that F was a cusp form. And so then, so anyway, we have this extension

from the cusp form, and then, then we have, by three, we have the consequence. This was the method of giving. And now, so the, now given the method of Skinner-Uba

is an extension of this, yes. Yeah, there are three, again, three points.

So, first point is that the special value of elliptic curve, the data function of elliptic curve is, appears as a constant table of the Eigenstern series of the algebraic group U2, and here, U2 is defined by fixing some imaginary quadratic field,

so that quadratic field appears there, E-K, yeah. And the second point is that the Eigen modular form of U2 to produce is a gallery presentation. This is, but not yet written, but Skinner and Uma, Uma tells that this part

follows from the work of Ramon and Ngo. Ramon will give a talk in this Congress. That was, I think, the day after tomorrow, yeah. Yeah, I think so, yeah. And the third part is that the Selma group

is also understood as an extension group of gallery representations, finite gallery representations. So, the class group is replaced by Selma group here. And then, by using the similar method as before, they could obtain the elliptic curve version of the modular form method.

And in the, yeah, so this is a simple, this is an explanation of the method. And in the remaining time, I hope to explain the non-commutative USR theory.

I have still five minutes, yeah. So, this, yeah, so USR theory considers the gallery action arithmetic objects, but usually, gallery actions are non-commutative. But so far, I only talked about the abelian gallery extensions, and so abelian gallery actions.

And so, it is natural to think about non-commutative theory. Yeah, and elliptic curve over Q, again, do the reduction at P, and then we have natural non-commutative gallery extension that is the field gallery extension

obtained by adding the P and S root to the points of B to Q. And so, then the group here is non-commutative here, and then the USR algebra which is obtained is again non-commutative. In fact, in this non-commutative USR theory, we can take any, essentially any gallery extensions,

but to fix the idea, I take here that there's a special natural gallery extension here. Already in the previous ICM, Huber-Kings formulated a conjecture of non-commutative USR theory,

but they did not present how to formulate periodic functions. So, in their formulation, periodic functions did not appear. And after that, it is now found that how to formulate periodic functions. So, this is an interesting point. So, I hope to explain it.

So, in this non-commutative theory, in the arithmetic side, we can have a module again, just in a similar way. So, I put here arithmetic side, yeah, and then also the zeta side.

Sorry, zeta side disappeared, sorry. Ah, yeah, no, no, yeah, zeta side is here, yeah. Sorry, both sides are here, yes, also, sorry. The arithmetic side is that we again take the, in the same way, we take the Thelema group here, and take the limit, and take the dual Thelema group.

Then the non-commutative Thelema group is acting on the Thelema group, Thelema group, so then the grouping works on the dual Thelema group. So, we have a ZPG module. Always, we can show that this is finite regenerative.

It is an easy fact. But, tojon is very difficult. We cannot prove that this is a tojon module. And so, but anyway, to formulate the conjecture, we have arithmetic side. And, but the big problem is that on zeta side,

what really does the p-addicted L function do is a really difficult question in this, for non-commutative theory. This is because non-commutative rings are not good places. Non-commutative rings are not good places to live for zeta function.

For complex zeta L function, which has a home of real product, like factor at two times factor at three times factor at five. If this lives in the commutative ring, then what is the order of the product is not clear. If you put the product in this order,

then probably the people who like the prime number 11, for example, will be angry, saying that five is our lovely prime 11, not at the top. So, this is really hard situation.

So, the commutative rings are not, does not match real zeta function. A periodic zeta function do not have the real product, but still, if it is hard for complex zeta function, then it will be hard for periodic zeta function to live in commutative ring.

But there is some way to limit this difficulty that the non-commutativity of the ring A, this vanishes if you go from the multiplicative group it equals to the K one, which is the abelian group defined in that way.

So, there is a canonical one from, it equals to the K one. And then we, and then the idea is that, the idea is that zeta function can live in K one.

a non-commutative ring, non-commutative ring, yeah. So, for example, we can think about Selberg's data function. Selberg's data function is also some kind of Euler product defined for discrete compact subgroup of SL2R. And then this is a product, but here, in that form,

so I don't explain much. But then, but then, it's also gamma ranges of prime elements in gamma. But we can define also some final version of that, putting just here gamma here. But the product is in the, in fact, in the L1 space,

or L1 ring, L1 gamma, it is a space of L1 functions on this discrete group, this group set. And but it has a, it is a ring for the convolution. And then this product is in this L1 cross,

but the product, the order of the product is not clear because this is non-commutative. But if you go to K1, then our product can be defined. So, of course, because we are taking infinite sum, we have to maybe divide some closure of zero or something like that, or yeah. Some modification may be important.

But this gives some impression that the data function, complex data functions, can live also in K1, or all our products can live in K1 non-commutative ring. So, but still, no one considered this refinement of the Selbog Zeta function.

But if you, but here, Hainan-Bass, Bass already considered something like this in the case SL2, SL2-R was replaced by SL2-Q. Yeah, in that case, the Selbog Zeta function is called the Ihara-Selbog-Zeta, Selbog-Ihara-Zeta function.

Now, conjecture for the existence of the non-commutative version of the PRDPL function is the following. So there is, there exists a PRDPL function

for this non-commutative value extension in this K1 with some localization of the, of the, oh, yeah, so O is the completion of the maximal under-mified extension of ZP.

And if this is some denominator set to which we can take the localization, and then there should be a PRDPZeta-L function which periodically interpolates these L functions, complex L functions, L values with L-chain row test.

This is row with some, called L-chain character, representation, finite image representation of the group G. So, yeah, so yeah, in this Selbog version, if you push this gamma to one,

then you get the original Selbog Zeta function from here. If you take the representation of the group, then this Selbog Zeta function comes, it goes to the Selbog Zeta function twisted by the representation. So similarly, we hope that this is some PRDPZeta function

which goes to the twisted version of the complex Zeta function. Ah, sorry, I already finished my time, sorry. So I have to finish, yeah. Yeah, then just I need a few words here. So the main conjecture is, sorry, yeah, yeah.

I find my time is already finished, yeah, yeah, sorry. So the main conjecture is committed by using that form. The PRDPZeta function is related to the Selma group,

real Selma group. So this is the expression of that idea, yeah. And this is compatible with the conjecture of working formulately in the previous ICM.

There are theories, deep basis on the theory of bands and flux. And so then we have, yeah, I finished that time. So there are still, but still we don't,

we cannot construct this non-commutative version of the PRDPZeta function, and we cannot prove the main conjecture. But still we have some partial results on the total ideal class group version of this. So we are now considering Selma group, but in original universal theory,

we are considered ideal class group, and then we have some results on ideal class group version, but I have no time to explain, so I don't explain it, so I have to finish my talk. This is the last seat, yeah. Yeah, so recall that these negative integer,

negative Zeta values are negative integers each. So L-brand and Lebesgue's theorem was for each value, but that became stronger by unifying these to classical universal theory. Those individual values are connected and unified

to appear in the Zeta function, and the classical universal theory became stronger than the theorem of Lebesgue and L-brand Lebesgue. And so the dream is that commutative universal theories

of various Zeta functions can be unified to the unified non-commutative universal theory. This, the results on this direction started to grow now. So I hope that this goal is such a unified non-commutative

universal theory is not at the infinite distance. Thank you very much.