So this is a joint work with Farid Mokhan and Jack Tillween.

So the aim is to construct sheaves of locally analytic, overconvergent, siegel modular forms

and to relate them with classical forms. So this has already been done by Andre Atta, Jovita Piloni. And we try to use an alternative approach, somewhat more naive, based on the construction of overconvergent Iguza towers.

So I start by fixing some notation. So p is another prime integer. g and n are integers such that p does not divide n. I will denote by w the ring Zp of zeta n,

zeta n being a primitive nth root of unity, and by k the fraction field of w. Next, I will consider x, which is the periodic formal completion

of the zegel variety of genus g at level n.

So this is a formal scheme over w. I will also denote by a the universal Abelian scheme.

And by capital X, the generic fiber of x in the sense of Reynolds. So this is a rigid space over k. So in this context, we have various objects.

So first one is the Hodge eigth function. So this is a map from x to the interval 0, 1.

That sends a point x to the truncated valuation of the periodic valuation of the AASO invariant. I write it like this. So here, more precisely, if L is a finite extension of k

and x an L point of x, so that corresponds to an Abelian scheme, A x over the integers of L. So the AASO invariant is just a determinant of the Frobenius acting on the Lie algebra of A x

mod p. And with this function, we can define subsets of x. So if v is between 0 and 1, I will denote by x of v

the inverse image by this map of the interval 0, v. So in particular, if v equals 0, so we can get the ordinary locus, generic fiber

of the ordinary locus. Also, if v is non-zero, this provides a strict neighborhood of the ordinary locus. I should write ordinary locus.

So we will use a normal formal model for x of v that I will denote by purely x of v, which is obtained by considering the periodic completion

of the normalization of the maximal open of the blowup of x of x here along the ideal generated by AASO and any element of valuation v. So the maximal open

on which this ideal is generated by the AASO invariant. So there are other objects that I will also consider. So we have a tower T over x.

This is the torso of basis of the conormal sheaf. So I denote it by omega, the conormal sheaf of the universal abelian scheme at the unit section. So this provides Zariski GLG torso.

And we can define classical Zegel modular forms as functions on T that are invariant under the unipotent

lower triangular matrices. There is also another object that lives above the ordinary locus. This is the Iguza tower.

So I will denote it by T infinity. So this is a map PR. And this is the torso of basis of the periodic take module of the etal part of the universal abelian scheme

above the ordinary locus. So this is an etal GLG of ZP torso.

And similarly, we can define the periodic modular forms by considering functions in the elements in the completions of the functions T infinity that are invariant under the unipotent upper triangular

matrices in GLG of ZP. So those objects are related thanks to the architect map.

So for every n integer, we have the architect map associated to the finite etal group scheme, which

is the etal part of the P to the n-stortion in the universal abelian scheme, to the conormal shift like this. So this is a map.

I will call it n between shifts on the finite etal site of the ordinary locus. And we can pass to the limit and obtain a map of state

from the periodic take module of the etal part of a order to omega hat. So this is morphism of sheaves on a site.

So I x-ord profet. So a site which is similar to that defined by Schulze. So with objects are projective systems of finite etal maps from some TM to U, a Zariski open in x-ord.

And whose coverings are given by families like this, GIM, such that for all n, the union of GIN of TIM is TM.

So this is a simplified version of the architect map. And this, in turn, allows to define the architect ecosystem

map. Do you see something on the UIs? Do they cover U or? I think they do.

This is finite, yes. So it is obtained. So it's a map between the Igusa tower and the restriction of T to the ordinary locus, which

is simply defined as follows. So HDI of some basis of the periodic take module. So we just apply the architect map to the dual basis.

And so this allows to relate classical modular forms to periodic modular forms. And so the idea is that one should be able to construct such objects

on some x of v for a small v. More precisely, this is the idea. There should exist some v0, a rational between 0 and 1, such that for all v between 0 and v0,

we have the following diagram. So we have the classical Igusa tower. Here we have some GLG of zp, netal GLG of zp torso,

that extends the Igusa tower above x of v, and also a map, HDI v, extending the architect Igusa

map above x of v. Right. So before I give some ideas on the construction of such objects, I would like to make a comment about canonical subgroups.

So recall that if n is an integer, then when v is less than 1 over 2 times p to the n minus 1, so here the 2 should be replaced by a 3 when p equals 3,

we have a canonical subgroup of level n over to, I call it hn, over x of v. Right.

So in particular, using this description, this provides also a finite et al covering of x of v. And indeed, one expects that the nth floor of the overconvergent Igusa tower, assuming of course that v is less than v0,

identifies with the torso of basis of this canonical subgroup. So to the G, hn. Right. But of course, as n goes to infinity,

that's this bound here shrinks. So one cannot construct the tower, so yes, T infinity v by using PD-visible groups. So we have to use some runabout way. So I will explain the strategy.

So it comes from a sort of simple observation that one can construct the periodic representation attached to the et al part of the universal ordinary abelian

scheme in some unusual and a bit complicated way. But that might be generalized outside the ordinary locus. So this works as follows.

So one starts with this PD-visible group. So one can attach to it, it's a duodony module. So this provides an f crystal with a filtration.

This f crystal plus filtration. Such that, so the Frobenius maps the filtration inside PM

and such that the resulting phi module M modulo fill M tensor f P. So this is et al. Having this, there is a construction by Dwork

that associate to this a unit root subcrystal. There's a Frobenius too.

So that has the property that M is a direct sum of U and the filtration. So how should we think about this f crystal? Should we think in the sense of crystalline side? Or you are in the rigid and the rigid?

No, here this is on the special fiber. And the last step is a CATS correspondence.

So which is a kind of period correspondence in the simplest case. Because it's in the unrammified case. And one gets the periodic state module we are looking for. In fact, we get it's dual. And so the strategy is to try to adapt each of these steps

above x of v. So the first step is to associate to our situation

a crystal with some Frobenius structure. So the choice of the crystal is obvious. So we will denote by M the following crystal.

And this is equipped with the Hodge filtration. So more precisely, we have an Hodge filtration only on the evaluations of these crystals where we can on which

above which we can deform the universal abelian scheme. But this will not be a trouble for us. And we also need some Frobenius structure.

And this is provided by the canonical subgroup. So if v is less than one half or one third when p equals 3, then we have the canonical subgroup H1 that defines a map from x of v to x of p times v.

Since the Hodge 8 of the quotient of the universal abelian scheme by the canonical subgroup is p times the Hodge 8. And composing with the natural map to x, I get a map phi.

I will also denote by Iota the natural map from x of v to x. And we call this pair an over-convergent lift

of Frobenius. So it has the property that modulo this lift,

the Frobenius modulo p to the 1 minus v roughly speaking. In fact, this lifts, it lifts modulo p to the 1 minus v. This lifts the composite of the Frobenius on xv composed with Iota.

And then by functoriality, we get a Frobenius map from phi upper star of n to Iota upper star of n. Right. Also, we have the similar properties as above,

namely that the filtration when it exists is mapped to something divisible by almost p. And such that the quotient is close to the eta.

So more precisely, we have that phi of phi upper star of, so here is some evaluation of the crystal. Tensor is w bar. I call w bar the normalization of w

in some algebraic closure of k. So this lies inside p to the 1 minus v, Iota upper star of mz tensor w bar. And moreover, so we have an induced Frobenius map on the quotient mz mod fill mz tensor w bar mod p to the 1 minus v.

Right. And the property is that p to the v belongs to the determinant of phi bar.

So as very small, this is close to be eta. Right. So now I come to the second step.

So the second step is to extract some close to unit root parts in this crystal. Unfortunately, this is not possible to do this at the level of crystals, because Dwork's arguments

uses fixed point process. And here, the Frobenius shrinks the radius of convergence. So we have to do this after computing cohomology. So second step.

So this will be getting an almost unit root phi module. So here, we have to do some local computations. So fix an affine formal sub-scheme of xv.

We assume that s is integral and normal.

So as usual, we can associate to this ring s bar. So this is the union of all the finite extension of s in some fixed algebraic closure of the fraction field of s.

So those extensions that become eta generically. So we have a local Galois group. So here, the index k means that I inverse p. And also, we have the periodic completion of the string s bar.

This is perfectoid. So we can take the tilt. So I recall this is a productive limit of s bar and p s bar under the Frobenius map. I will call it rs for historical reasons.

So this is a ring of characteristic p that has an action of gs. And it also contains elements that will be useful in what follows. There's an element p tilde, which is a compatible system of p to the n-th root of p.

And zeta, this is a compatible system of primitive p to the n-th root of unity. And as usual, we have a map theta, which is a subjective ring

homomorphism from the width vectors with coefficients in rs to s bar hat. So as usual, the kernel of this map is a principle ideal generated

by xi Pech-Muller of p tilde minus p. And then we define a Chris-Nabler of s as the periodic

completion of the divided power envelope of this width vector ring with respect to the kernel of theta. And so this has various structures.

So it has a Frobenius and an action of gs. And we will use it to compute cohomology more precisely. So we will denote by ms the productive limit of the

crystalline cohomology like this.

So this is naturally an a Chris-Nabler module. And this has an extra structure. So it has Frobenius, an action of gs.

And also, it is equipped with some filtration coming from the Hodge filtration. Right. And starting from this, we have to extract some almost unit

root parts. So there is an analogue of Dwork's theorem. So there exists a sub-fi module, call it us in ms.

So this is equipped with a Frobenius, an action of gs, which is close to being etal, which means that there is a small, small power of p tilde belonging to the determinant of the reduction

mod p of the Frobenius. So acting on us bar. Right. So also, this us is almost a direct factor

of the filtration. This is not completely exact, but yes, it is an a Chris-Nabler sub-fi module. Right. And the third step is to associate

to this a Chris module periodic representation of gs. So yes, I should mention, of course, that there are some gluing data, all those constructions glue. So it's enough to construct periodic representation

locally. Excuse me. So this map of phi is induced by? It's by functionality.

You have the Frobenius on this mod p to the 1 minus v c. So this provides a map between the pullback of the, sorry, yes, the pullback of m by this Frobenius to m.

And then you compose by taking the pullback of iota. And as I mentioned before, the Frobenius here, modulo p to the 1 minus v composed with iota

is precisely this. But this is not a quotient by the economical subgroup. No, no, no. The map phi here comes from the canonical subgroup. The map phi, yes. This one. Yes. So I was thinking that if it is a quotient modulo economical subgroup, shouldn't it respect to the concentration?

That is, by phi, shouldn't it be in the field of the? Sorry? So if this is induced by taking the quotient as a canonical subgroup, so it should be like a misogyny.

So shouldn't it preserve the hot filtration? This is the case for the excellent Frobenius, for example, of Korkovich, which is a? I'm not sure.

The excellent Frobenius is characterized by this property to be an assault and essentially a level of phi, because it's a quotient. Well, I don't know. But anyway, under this weaker assumption, we can produce the US.

So the third step is to build a periodic representation. And this will be meant using a symptomic map. Yes, it is unique. And in fact, it is not really, it

is kind of independent on the filtration. So it's enough that one has filtration here that has some similar condition here at the level of MS to get US. It is unique.

Right. So here we use some periods to build the representation. So the trouble, of course, is that aqueous Nabla is not big enough, which is logical, because the overconvergent

Iguza tower is not associated to a PD-visible group. So the first thing we do is to make it smaller. And so I start to get rid of trouble with the divided powers. I denote by lambda 0 the quotient of aqueous

by the ideal i p minus 1, which is the ideal of those elements in aqueous Nabla, whose iterates under Frobenius all belong to the p minus 1's divided power of the kernel of theta. But this ring is nothing mysterious, because this is simply the quotient of the width

vectors by a very simple ideal. Right. We get in trouble at each step. So I didn't count them, but all the time.

So let me also mention that the reduction of lambda 0 mod p is very explicit. This is the quotient of rs by p tilde to the p,

which in turn is isomorphic to s bar mod p s bar. So in the computations, at some point, we need to consider to divide by some powers of p tilde. So that's why we introduce the following ring.

So let alpha be the rational between 0 and 1. Lambda alpha, so roughly speaking, this is just lambda 0 to which we adjoin p divided by the touch miller of p tilde

to the alpha. And so with more details, this is simply lambda 0 with one variable t alpha. So I have the relation, of course,

at p tilde to the alpha times t alpha minus p equals 0. But this ring has p torsion. So we have also to kill p torsion. But here, this is not a serious trouble, since p torsion is very explicit. And then we take p-adic completion.

So here, we are happy, because we have an action of gs. And we have these elements. But the drawback, of course, is that we have troubles with Frobenius. Because if you apply Frobenius to p divided by touch miller

of p tilde to the alpha, the denominator will explode. So the situation is that we have something similar to what we had at the beginning.

We have two maps. So here, I assume that alpha is less than 1 over p. So we have two maps from lambda alpha to lambda p alpha. I call them v and phi. So v is lambda 0 linear and maps the variable t alpha

to the touch miller of p tilde to the p minus 1 alpha times t p alpha. And phi is phi linear. And phi of t alpha, of course, is t p alpha.

Right. And having this, I should also mention that those two maps are gs-equivariant. We can define the following, p alpha of us. This is the kernel of the following map.

So it goes from xi lambda alpha tensor us over a Chris Nabla, and goes to lambda p alpha tensor us. And the map is the Frobenius divided by p on this factor,

because the Frobenius of xi is divisible by p, tensor the Frobenius of us minus v tensor the identity. This is the so-called syntomic map.

And the hope is that when alpha is appropriate, this provides the periodic representation we are looking for. And indeed, this is the case, this theorem.

So alpha is rational. So it shouldn't be too small and shouldn't be too big. So there's some bound that is explicit in terms of v. And less than 1, 1 over p.

Then this v alpha of us is zp to the g. And since the maps v and phi are equivariant, this is handled with an action of gs.

And this is precisely the periodic representation that, I mean, locally, over the maximal spectrum of s is inverted, corresponds to the overconvergent Igua tower. So, well, this is a bit complicated.

So we want to proceed by the visage. The trouble is that, well, defined like this, there's absolutely no the visage. The naive sequence modulo p to the n minus 1, modulo p to the n modulo p is not exact. So the first step is to slightly deform this map.

And then you have a beginning of exact sequence. And then you have a second trouble, is that modulo p, the kernel of this map, is not finite dimensional over fp. So you have to use a trick by considering those elements

in this kernel that come from the representation in characteristic 0 to select the right elements modulo p. And also another trouble that comes from the fact that the map is not subjective. So you cannot use the simple snake lemma arguments to get

the exactness of the sequence. But anyway, you can show that. So before I say some words about the H-type map,

I would like to mention a few facts about this representation, at least mod p. These are the following. The first one is that one can consider what I denote v1. This is v0 of us mod p us.

This is a stupid, a symptomic map using lambda 0, but mod p. In fact, this does the job. This is isomorphic to fp, fp to the g. And indeed, the reduction mod p of v alpha is precisely this v1.

So the arguments to show this are the usual one. One has to solve a matrix equation like this one

inside s bar mod p to the 1 minus 1 over p s bar to the g, assuming that the determinant of A divides some small power of p.

So these are the usual computations when one constructs a canonical subgroup. Also, there's some funny facts. For all eta in 0 p minus 2 divided by p minus 1,

one can reduce v1 modulo some power of p tilde. So recall, this is the kernel of the following maps. This is us bar.

So here is a reduction modulo p tilde to the p minus 1 to us bar, which is just a Frobenius minus p tilde. And here, one can look at its image inside p tilde us bar modulo p tilde.

So I always forget the exponent. Right. p tilde to the p divided by p minus 1 plus p eta. Then this is a bijection.

And the second observation, which is related to what I said at the beginning about canonical subgroups, is that you can show explicitly that the first floor of the overconvergent Igo-Sattauer

coincides with the torso of basis of the canonical subgroup. So you can see that explicitly, for instance, using the description of this by Andrea Tagaz-Bali.

OK. So now I will come to the Hochtet map. So here, so far, we constructed the overconvergent Igo-Sattauer.

So the construction of the associated Hochtet map is as follows. So by definition, v alpha of us lies inside xi lambda alpha tensor us.

So if beta is rational, larger than alpha, and less than 1, so we can extend the scalars to lambda beta.

So we get a map that I call r beta from lambda beta tensor v alpha of us to xi lambda beta tensor us. Also, using the representation v1 over there,

we have a map mod p that I call a bar from lambda 0 mod p tensor v1 to p tilde lambda 0 mod p tensor us bar.

Right. So we have the following properties. The first one is that the kernel of a bar is killed by p tilde to the p over p minus 1.

So to prove this, you can, by normality, reduce to the case where s is a DVR, which is explicit. Another thing is that there exists a constant c between 0

and p minus 1 such that the co-kernel of this map is killed by p tilde to the c. So to show that, it's a bit tricky.

You have to lift your application into a map between three r modules of rank g and then control the determinant. You play with the Hansel arguments.

And using those two ingredients by successive approximation, you can deduce that p tilde to the 1 over p minus 1 kills the co-kernel of a bar. And then now, coming back to a characteristic 0,

you get that the tesch-Müller of p tilde to the 1 over p minus 1 kills the co-kernel of r beta

when beta is bigger than 1 over p. How do you raise to the power of 1 over p minus 1? Well, in fact, it should be put inside. If written like this, it has no meaning.

It's to be kind to the typist. And OK, so here, you had this condition here so that the element p divided by p tilde to the 1 over p

minus 1 is topologically nilpotent in lambda beta. OK, so as a consequence, you obtain an isomorphism.

I will call this a proposition. An isomorphism, r beta p inverted from lambda beta p inverted times V alpha of us to psi lambda beta p

inverted times us. So here, this is over z p and over a crease lambda. Right, and so under the same assumption, of course.

And now, we have to relate this to the co-normal shift. And this is done as follows. So recall that it's still on the blackboard. We define Ms as the crystalline cohomology of M.

And this can be explicitly described in terms of horizontal sections of a big module. So let me explain this. So we choose what I call a presentation of S. So this is a subjective homomorphism of w algebras

with a t formally smooth over w. So this defines du over S. So this is the PAD completion of the divided power envelope of t

with respect to the kernel of u. So this provides a divided power thickening of the formal spectrum of this. So I call it z. And we can also define a ring a crease as follows.

So we start with the tensor product of t with w of rs over w. And we can extend the map theta by a linearity. A t linearity gives the map theta u.

And so the ring a crease of u gives the PAD completion of the divided power envelope of that ring with respect to the kernel of theta u. And this is d of u algebra. So it has an action of gs.

It has a Frobenius. So well, the Frobenius structure is a bit complicated since Frobenius changes the street neighborhood. There is some kind of Frobenius structure.

And there is also a connection. And the point is that we have an isomorphism which is given by Taylor expansions between a crease of u.

Sorry, first, yes. So ms is isomorphic to the horizontal sections of the tensor product of a crease of u

with the evaluation of m at d of u. And we have a comparison like Taylor series expansion like this, a crease of u tensored with ms with a crease of u tensored with m d of u.

And so using the decomposition theorem, this is step two. This relates a crease of u tensored with us.

Well, let's say with p inverted. And a crease of u tensored m d of u modulo its filtration.

Yes, with p inverted. OK, so we can tensor with lambda beta and inject here. So denoting by curly a beta of u, the tensor product of a

crease of u with lambda beta p inverted. We have an isomorphism, a beta of u tensored with v

of us with xi a beta of u tensored m modulo fil m.

And then taking, so the map theta extends to that ring, theta u to s bar hat by sending one tensor t beta to p to the 1 minus beta. And taking the first grade of this, we get an isomorphism

between s bar hat tensor v alpha of us with xi s bar hat k tensor m, well, let's say s, the conormal shift.

Right, so in order to, so this is not quite the Hodge state map. This is a dual Hodge state map. And the Hodge state map is just the inverse of the transpose of that map.

And so this goes from, well, this provides a map from the dual periodic representation and omega tensored with s bar hat. So in order to shiftify this, as we did on the ordinary locus,

we need to have some descent here. Typically, if we call H the kernel of the periodic representation attached to this,

then we would need a statement like this. Well, in fact, we need more, but we already would be happy to know that. Or something similar. We need some descent property like this,

or maybe with reducing a little, there is a reduce of convergence. So, well, we have some strategy to prove this using the Tetsen formalism. But to do this, at some point, we need to construct some perfectorization of s.

So I don't know how to call this perf. And well, making things perfectoid tends to shrink the reduce of convergence. And so this is a big trouble when we want to build normalized traces.

Well, so this is still in progress. And in the few minutes left, I would like to come back to the observation I made at the beginning.

Namely, the relationship between the overconvergent Igusa tower and the canonical subgroup when both are defined. So I will, for simplicity, denote by ln the shift

for the finite et al topology on Xn that attach to the nth level of the overconvergent Igusa tower. And we call that we have the canonical subgroup.

So this on X of v when v is less than the minimum of v0 and 1 over 2 p to the n minus 1.

So the point is that they coincide on the ordinary locus. And so we want to show that under this they coincide on X of v. And so, as I said before, this is true for n equals 1. And then we use induction.

The argument is the following. So this is like obvious exact sequences.

So we already know that there is an isomorphism like Yota 1 here. So by the induction hypothesis, we have an isomorphism here, Yota n minus 1. So those isomorphism induce an isomorphism between the X groups here.

And I...

Those two exact sequences provide classes here, classes that agree on the ordinary locus. So then by analytic continuation, we deduce that, in fact, they agree everywhere.

So it shows that they are equal in the ordinary locus. Also, we want to compare the associated prototype maps. So we have the map from ln dol to omega tensor at z mod p to zn.

We have also the architect map attached to hn. So here, we had this map, this isomorphism uta-n.

So we get an isomorphism uta-n dol. And here, we have the natural map here. And this square commutes on the ordinary locus. And here again, by analytic continuation,

this commutes on xn. So this shows that the H-state map that I constructed before, this overconvergent H-state map, modulo this descent trouble, agrees with the usual H-state map for Hn.

Any questions? Carmen? So it's just at the end in this analytic continuation

arguments. Well, you see that you have to shrink the lead. Because the lead is getting smaller and smaller on the n. Sure. I mean, both objects have to be defined. So this ln has to be defined, so it must be less than v0. And this is for the canonical subgroup. It's just the definition.

Yes. I want to compare the two objects. OK, no other questions, let's thank our speaker again. Thank you.