 I would like to thank the organizers for the invitation. It is a great honor for me to speak in this conference for Arthur Gabbard. Over the years, I have benefited tremendously from him. Just yesterday, I received some corrections from him for a paper by other people that will be using this talk. But fortunately, there was nothing serious. As another illustration of his tremendous generosity, this talk will be based on applications of a couple of marvelous theorems of Gabbard that were written up by other people. Here the keywords are after Gabbard. So the subject of this talk is about some old questions on L-independence. So Sarah certainly made many conjectures on L-independence. And the ones that I'm going to talk about are those he made in his definition, in his recipe of the Hasselweig data function of smooth productive varieties over global fields. So he made these conjectures C1 through C8. So eight of them. Actually, a couple of them already appeared earlier in his paper with Tate that were stated as problems. So here's the definition of the Riemann's data function. This can be generalized without any problem to a rhythmic data function of a scheme of finite type of respect z. I just have to replace a set of integers by the set of effective zero cycles on x and to replace n by the norm of the cycle. And with the Eulerian product, you can write it as a product of local zeta factors. As we all know, Groening gave a chronological interpretation of these local zeta factors. So now let x be a variety over a field. Actually, I'll take a variety over a finite field. I'm going to define this analytic homology with compact support equipped with a continuous action of the Galois group. Then the data function can be written as alternating product of characteristic polynomials of Frobenius acting on the compacted supported analytic homology as a consequence of the Groening trace formula. So a natural question here is whether these polynomials, which are a priori with coefficients in ql, actually zl are actually integer coefficient and independent of l. So in the case where x is proper smooth, very conjectures tell us that the action of Frobenius on the i's homology is actually of weight i. So you can extract these pi l from the zeta function. So as a consequence of this, these polynomials are actually has coefficients in z and independent of l. And these were Sers conjecture c1 and c2. So these are now known. Another consequence of the weight conjectures is that by spreading out, you get that for proper smooth variety over an arbitrary field, then the better number of the analytic homology is actually independent of l. So this spreading out argument will be a major theme of the rest of the talk. So now let's get back to the has a weight zeta function. So here's Sers's recipe. The local factors are defined as characteristic polynomials of Frobenius acting on the inertia and variant part of the analytic homology. So in order for this formula to make sense, he has to assume that this polynomial is actually independent of l. So that is his conjecture c5. Actually, he stated it in a slightly more generality. So now we work over a local field k, by which I mean a complete description field of finite residue of fq. And so now assume only that x is defined over k. Then Sers conjecture says that the Frobenius acting on the inertia and variant part of the analytic homology has characteristic polynomial with coefficients in z and independent of l. Related conjecture is also a conjecture concerning the full structure of the local Galois action on HIL. So it's a more refined conjecture is the full. So you are this one, this C8? OK, C8. Yeah, so yes, there is. He also made this conjecture C8 for the 4HI. The action of the wave group actually has traces independent of l. So has it implied the previous one? Yeah, I'm coming to that in a moment. So to explain the relationship between these two conjectures, let me recall the monogamy wave conjecture, which says that the eigenvalues of Frobenius and liftings on the graded pieces of these homology for the monogamy filtration are pure of wave i plus n. So combined with the conjecture C8, this implies that all these graded pieces are actually independent of l. Because you can extract them, they have different weights. And from here, actually, you get that all the primitive parts are independent of l. So it follows that this conjecture C5 on the inertia invariant part holds. So actually, Ser didn't make this monogamy wave conjecture, but he made conjectures C6 and C7 between C5 and C8 that were special cases of the monogamy wave conjecture. So these are Ser's conjectures. Now, for over a local field, a finite residue field. Now, if we now allow the residue field to be an arbitrary field, then while there's no wave group, but we can still consider the action of the inertia group on the chronology. And Ser conjectured, his conjecture C4, says that for any element in the inertia group, the characteristic polynomial on the ice elliptic chronology has coefficients in z and independent of l. So to put this into context, let me also recall the local monogamy theorem of a granitic, which says that for acts of variety, by which I mean a separated scheme of finite type over k, not necessarily proper smooth, the action of the inertia is actually quasi-unipotent on the comparatively supported chronology. And this has several variants due to DeLine, Gabber, and UDZ, which says that, first of all, you can replace the compactly supported chronology by a euro-alleted chronology. And this subgroup, which acts unipotentally, can be taken to be independent of l. And also, the index of unipotency can be taken to be i, which is the chronological degree. So this is some other problem of independent of l. Now, let me concentrate in the case of equal characteristic for a moment. So let me recall that in equal characteristic, the monogamy weight conjecture was known, it was due to DeLine teresoma. So DeLine in the case where it is defined over a curve and teresoma, maybe in general, and vetoed for a generalization to rescue fields of finite type. So here's the statement of our results. So still in the equal characteristic case, let's recall k is a completely discrete-valued field of residue field k of characteristic p greater than 0. So both of characteristic p greater than 0. Let x be a proper smooth variety over k. Then conjectures c4 and c8 actually hold. So let me recall that for c4, it says that for every f in the inertia group, the characteristic polynomial of f acting on HIL is in z bracket t and is independent of l. And for c8, it says that if you assume the residue field to be finite, then for the lifting of Frobenius, the above holds. So this was known, this c8 was known in special cases to DeLine and teresoma. Now as mentioned before, so this combined, so this c8 combined with monogamy rate conjecture implies conjecture c5 that the inertia invariant part of the cosmology is independent of l. Now let me make a number of remarks. So previous results on in general characteristic without equal characteristic assumption, which holds for not for individual commons, but holds for the alternating sum of the traces. So first of all, there were this theorem by Gabber over a finite field already that for f in the way group, the alternating sum of the traces is a rational number independent of l. And then for local fields, first of all, for general completed square regression field, the action of the inertia is actually in z and independent of l when you take the alternating sum. This was a theorem of V-dall. So these are variants of the conjecture c1 and c4. And finally, there's this variant of conjecture c8. So if k is now a local field, a finite residue field, fq, then for the action of the elements of the way group are independent of l, which was a theorem of RGI in the proper case that can be extended to the general case. So now let me explain the strategy of the proof of such serums. So basically, the proof is a spreading out argument. So when never axis defined over a field f of characteristic p, we can spread it out to something of finite type over fp. So there is this chronology on the left-hand side. It's now spread out to least sheaf on the right-hand side. And when we vary l, we get a system of least sheafs on the base b. So we are led to study, instead of this system of galvanized representations to the system of added sheaves on the base scheme. So now our basic setting is a slight generalization of the previous one. So we work not just over a finite field, but over the ring of integers, OK, of a local field. So here I do not assume that k is equal characteristic. So k is just a Hensselding. So OK is an excellent Hensselding discrete-valuation ring of residue field. Fq is no restriction on the characteristic. And we take x to be a scheme of finite type over spec OK. And I denote by k the growth in a group. And we should fix a number of primes, a system of primes, li. Now given a system of elements of the growth in a group, a system of virtual sheaves, li. So each li is a qli bar added sheaf. This system is called compatible if for every point x in x. So here by a point, I consider only locally close points. For every locally close point x and every element in the wave group, the local wave group, the trace of f on the stock of li is a rational number independent of i. So basically the local traces are rational numbers independent of i. So let me recall that this being a locally close point here just means that it is either a close point in the generic fiber or a close point in the special fiber. Of course, there is a more general notion of compatible systems with traces in a field extension of q when you fix it in beddings. So now the fundamental theorem of Gabber says that the compatible systems defined in this way, and it's over a finite field, are preserved by growth invariant duality and growth in six operations. And this can be extended over a spec OK over the ring of integers of a local field. Now for our purpose, it will be important to consider not just points on the curve, but also not just points on the variety, but also points off the variety. So points on the boundary of the variety. So let's take the simplest case. If C bar is a smooth curve or fq, and I take a zarsky open dense subset C, we are interested in just having compatible systems on C. But we look at points on the boundary, on the complement of C and C bar. And then of course, we have these local fields. So we look at these local Galois groups, which has this short exact sequence. And this can be defined algebraically by taking the handsalization of this curve at this point, and then removing this point itself. Now this generalizes to this general setting. So now let x bar be an enormous scheme of finite type over S. And we take a zarsky open x in x bar. And for any point of x bar, we take the handsalization of x bar at x. And then we remove the boundary. So this gives us an open sub-scheme, which gives a surjection. So this pi1 of this gadget is an analog of the local Galois group. And the kernel is the inertia, which actually appeared in Tekashi's talk this morning. So here's the key definition. So now given a system on x, we say that it is compatible on x bar, or it is compatible along the boundary of x bar minus x. It could be least. Yeah, I assume that these LIs are least sheets so that I get maybe representation of the pi1. So just take the inverse image of the wave group here. So no, but a normal scheme of finite, what is S? S is S. So I only consider locally close points. So it's just the euro wave group, I think. Yeah, I only consider. So this bar means locally close points. So the residue fields are either finite fields or local fields. So there's a well-defined notion of wave group. Well, not quite locally, because you're only in the Azzelian. Yeah, yeah. Hanselin. The paper is the same. So everything goes to S. And we can take the map from pi1 of this thing to pi1 of S. And that gives a way of defining this wave group. So the definition is that for every point, every locally close point on x bar, or actually, for example, on the boundary, the action of the local wave group has traces in q and the independent of i. Now, a natural question here is, given a system that is compatible on this open subset x, is it automatically compatible on x bar? So is it compatible along the boundary? So I don't know the answer to the question. But it does hold up to stratification or modification. So let me explain what I mean by this. So what we actually proved is the following. So let x be a scheme of finite type over S. And we take a system that is compatible. So I also assume that the index set i is finite. Then there exists a stratification of x into locally close up schemes such that each stratum x alpha and it's a normal compactification x alpha bar over S, such that the restriction of the system to each of the stratum is compatible along the boundary. So this is what I mean compatible along the boundary up to stratification. A more or less equivalent formulation is that it is compatible up to a modification. So now the setting is slightly different, but the conclusion is that there exists a modification. So I take x bar and I take open subs, there is a dense open x. Then there exists a modification of x bar such that the pullback of the system is compatible along the boundary of y, of y bar. So you... What do you check? Is it an isomorphism? Is it a good thing for the x? Something new. You don't check the good looks. I think I don't know. So in the proof we may have to change the good looks. So of course these are serums of deline in the case where x is a curve over fq. In the case of a curve you have no need for stratification or modification. It is simply compatible implies compatible along the boundary. Now what is perhaps, at least I find it funny, that these statements imply a evaluative criterion for a compatible system. So I have this system that I assume... Well, for the moment I don't assume it's compatible. And we consider this all-valuative criterion commutative squares like this where all L is a hands-on evaluation ring and L is a field of fractions. Does that seem to be discrete evaluation? Not necessarily. Not necessarily discrete, yeah. Then a system Li on x is compatible if and only if the restriction of the system to spec L, sorry, with all quasi-finite over S is compatible. Because for such L you can also define the wave group. Of course one side of the implication from right to left is just a definition. So the only content here is from left to right. So it's a consequence of being compatible is that it satisfies this evaluative criterion. Now also in the strictly local case, so if you just restrict to the inertia, then the action is also independent of L. These are the consequence of theorem up to stratification or modification. Let's come back to Sarah's conjectures. So actually a consequence of this evaluative criterion combined with spreading out argument is the following slight generalization of Sarah's conjectures in your characteristics. Now all L is just a hands-on evaluation ring not necessarily discrete. And for extra proper smooth scheme over L, the second point of this evaluative criterion implies that the action of the inertia is independent of L. So here I have left out the part on the indigrality which are given by some other arguments but at least the evaluative criterion implies that the characteristic polynomial is in cube bracket T and is independent of L. And the first part of the evaluative criterion gives the generalization of Sarah's conjectures. See it in your characteristics. So for any element of the wave group, the trace is in Q and is independent of L. So this evaluative criterion was very much inspired by Gabber's evaluative criterion for the ramified part of the fundamental group. So let me explain this relation with wild ramification. The ramified part of Pi-1 was defined in the first paper of V-DAL perhaps with the help of Gabber. So the setting is over any... Okay, so this is V-DAL's definition. So now we take any excellent Hansel and Describes evaluation ring of residue characteristic greater than zero. Now V-DAL defines these subsets of the fundamental group for a normal integral scheme separate the final type of rest. There's this subset of ramified part, this ramified part of the fundamental group and there's this wildly ramified part of the fundamental group. These are closed subsets stable under conjugation. So the definition is a bit technical. So for every normal compactification there was this subset Pi-1R subscript X-bar which is the closure of the union of the conjugates of this image for X-bar running through geometric points of X-bar. So basically these are the local inertia groups that appeared in Takeshi's talk this morning. So we take the image of all these, take the union, take the conjugate and take the closure to get a well-defined subset and then we take the intersection with respect to all X-bar with respect to all normal compactifications. Now to define the wildly ramified part is just the intersection of the ramified part with the union of the propriocelos. Now Gabor gave this evaluative criterion for this Pi-1R which says that Pi-1R is the closure of the union of the conjugates of images of Galois group of L for L appearing in these evaluative diagrams where OL is a strictly hands-on evaluation ring and L is the fraction field of OL. No, it's just a subset. Yeah. So there's also a evaluative criterion for Pi-1WR for the wildly ramified part where you just restrict to the pro-order of elements. Now one can define using this wildly ramified subset a notion of compatible wild ramification. So given a scheme X over S, a system, now I take a system of FL bar sheaves, virtual FL bar sheaves because now we restrict to propriocelos so it suffices to consider torsion sheaves. So such a system is said to have compatible wild ramification if for every, first of all we have to restrict to pieces where all the sheaves are least and then we can talk about Broward traces for G in the wildly ramified part of the fundamental group and we ask that these Broward traces are rational numbers independent of L and as Takeshi mentioned this morning so there's also this notion of same wild ramification which is actually more general, it's a less restrictive notion and that was studied by Takeshi and his students Yuri Yadagawa and Yadagawa herself and there are some old work by Deline and not so old work by Vida and some very recent work by Saito Yadagawa, Yadagawa and Kuo there's also a related work by Hiroki Kato which is not exactly studied like this which says that this notion of compatible wild ramification is preserved by all six operations while this notion of same wild ramification is preserved by four operations and the corporations upper star, lower star, upper tree not preserved by duality yeah, not preserved by tens preserved by duality preserved by duality, sorry, yeah it is up yeah, but not preserved by tens or harm so actually there's just a quick explanation of this non-stabilized by tenser but I didn't put it in the slides basically this is what Takeshi explained this morning so a way of defining the same wild ramification is to ask for the trace to be in... so if you take the trace so this broader trace the upper area lands in a bigger field than Q and then you take the trace to Q and you ask that to be independent of L but by doing so you are actually getting some sub-spaces which are not sub-algebra so that means this won't be compatible this won't be preserved by tenser product okay now we have this extension of scalars morphisms this maps as in modular representation series so there's an isomorphism between Kx with coefficient in ZL bar with Kx with coefficient in QL bar and there's also a reduction from Kx with coefficient in ZL bar to Kx with coefficient in FL bar composing these two maps we get the composition map from Kx QL bar to Kx FL bar then by combining our evaluative criterion with that of Gavir we get the following consequence that for any compatible system Li so this decomposition map carries compatible systems of static sheaves to systems of mod L sheaves with compatible wild ramification so this is well this gives some examples of systems with compatible wild ramification where the recent work can be applied to that will be all of my talk thank you Q&A please can you give a hint on the multiplication of the equation here well so well at least one ingredient is to use an equivalent form of of Gavir theorem so we have to work with Dion's alterations in the equivalent form I have the paper stashed in my backpack I believe that if you have some alteration something that makes the ramification tame for one for one KL KL and probably you can prove that the other ones are also tame and then you can is it the case of if one is tame and suppose you have a normal grossing situation and you have the compatibals so we are speaking about this different notion of compatibals in the boundary compatibals means that all locally close point you have compatibility let's say for this sheaves so you have a compactification with some normal grossing device or maybe some of us so suppose you know that one of them is tame then it follows that the other are tame is it the case the others are tame but in any case we only we only have a finite number of else so we can make everything tame so then you can study the action of the group and look at I believe that probably the restriction to finite number is not needed if you can indeed get the I mean if you prove yeah the one tame this implies the other yes and then you can probably put all of then you can find a comb on the alteration if you want alteration it was for everything else okay so another technical question about this kind of problem we said and so on so the bright bright is the same proper smooth or the field of fraction but we can instead work with digital analytics space as a formal scheme so of course you can put the same conjecture for proper smooth digital analytics space so of course some of the results on the way monodromes thing is used as a bright nature because you are so you don't have this but the other part of the kind of independence so is it how much does it extend to this case did you so there were some results by media I think but we're simple but otherwise I don't I don't know about these particular questions are there places where the trace the greater trace is rational but not integer well integrality is another problem because you can naturally have for example when you take deal it will not preserve the integers when you take deals or just take twists so they naturally occur you just take twist so you can have q to some negative power as eigenvalues estimates on how the integrality is preserved both under the other operation for both the for both the integrality and for inverse integrality for single file fit and also for the local which the thing where I would deal with you so if we localize that so if we invert p then both integrality and inverse integrality is preserved by all the operations sorry except that which is the two then for the p-divisibility you have some bounds on all the operations any more questions so if not it's time to thank the organizers of this beautiful conference so first Ahmed who I think is the main actor who is organizing