Transcript

Euler Systems in the Twisted Friedberg-Jacquet Setting - Shilin Lai

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· 1 h 4 min · 5,769 words · 86 timestamped sections

This is the full, timestamped transcript of "Euler Systems in the Twisted Friedberg-Jacquet Setting - Shilin Lai", an episode of Institute for Advanced Study published December 19, 2025 (1 h 4 min, about 5,769 words). It mentions 14 people, works and ideas, including L functions, shimma variety, anti-cyclomic, CM field, Andrew GR.

What is discussed, and when

  1. – 10:00: L functions, shimma variety, Spencer Leslie, GGP
  2. – 20:00: anti-cyclomic, F prime, CM field, F0
  3. – 30:00: F prime, U1, Kivakan
  4. – 40:00: Andrew GR, F prime
  5. – 1:03:47: L functions

Notable quotes from "Euler Systems in the Twisted Friedberg-Jacquet Setting - Shilin Lai"

  • Fields we are going to start with the DM expression f0 is totally real. F is a CN.
    about CM field · at
  • So a real statement should at least say the central value being non zero is equivalent to this HP being non zero for specific inner form GH that's already in the GDP case. In this case, it's even more complicated.
    about L functions · at
  • But so I'm going to follow the notations of Spencer Leslie and Okay.
    about Spencer Leslie · at
  • So we have so we get a special cycle in the arithmetic middle dimension. So there's this code [clears throat] group.
    about shimma variety · at
  • Low high going from gamma group of f0 to go to let's just keep it and this.
    about gamma group · at
  • There it is. split anticyclonic oiler system for not the representation of real pi which in a twisted case is not the natural object that shows up but for the first of all restrict from F0 to B.
    about anti-cyclomic · at
  • And if you write it out, this is low pi restricted to F prime direct sum ro pi twisted by the quadratic character.
    about F prime · at
  • So in this case the kind of delta L was constructed explicitly by Andrew GR and car and I think the public version is 2023.
    about Andrew GR · at

Who and what is mentioned in this episode?

People

  • Andrew GR — Person · Referenced researcher (constructed local test vectors) · 1 mention · first at
    Credited with an explicit construction of the local element/test vector (in the split-split case) using a local theta integral approach.
  • Spencer Leslie — Person · Referenced expert (prior work/notation source) · 1 mention · first at
    Cited as a source for notation/framework used to set up the 'twisted' setting and biquadratic extension diagram.

Concepts & topics

  • L functions — Concept · Core topic · 3 mentions · first at
    Central mathematical concept discussed in the context of automorphic representations and their relationships to periods and special cycles.
  • shimma variety — Concept · Key argument · 2 mentions · first at
    Discussed in the context of arithmetic geometry and its relation to automorphic representations and special cycles.
  • anti-cyclomic — Concept · Strategy · 3 mentions · first at
    Key concept in the construction of Euler systems and their relation to automorphic representations.
  • CM field — Concept · Framework · 2 mentions · first at
    Used as a foundational concept in the discussion of unitary groups and their extensions.
  • F prime — Concept · Framework · 7 mentions · first at
    Central to the discussion of field extensions and their role in the construction of Euler systems.
  • F0 — Concept · Framework · 2 mentions · first at
    Base field in the discussion of automorphic representations and their extensions.
    I'm going to reserve for here for the fields we are going to start with the DM expression f0 is totally real. F is a CN.
  • GGP — Concept · Key argument · 1 mention · first at
    Referenced in the discussion of periods and their relation to L-values.
    But the reason this is only a approximate equivalence is first of all already in something like the GGP case. I can't just restrict to a single unitary group. I have to consider all.
  • gamma group — Concept · Key argument · 1 mention · first at
    Discussed in the context of automorphic representations and their relation to cohomology.
  • U1 — Concept · Framework · 1 mention · first at
    Used in the construction of Euler systems and their relation to automorphic representations.
    So h basically actually so to define this I need to define this [clears throat] I need to tell you the additional piece of data which is a co- character of h.
  • ZL — Concept · Framework · 3 mentions
    Used in the discussion of Euler systems and their construction.
    So we have classes over abil over anti-cyclomic ability extensions of F prime satisfy the norm relation.
  • H1 — Concept · Framework · 1 mention · first at
    Used in the context of cohomology groups and their relation to automorphic representations.
    Just to be clear, this is the H1 F of Right. I'm going to put all this representation row just from now on.
  • Kivakan — Concept · Referenced expert · 1 mention · first at
    Mentioned in the context of defining Euler systems and their properties.
    This is more like a Kivakan. This is more like an oiler system as defined by Kibbaken.

Full transcript with timestamps

Thanks to those of you who came to the last uh member 3 seminar of the semester. Uh we're having she live talk about um other systems in the twisted uh 33 jack case setting.

Okay. Thank you for the invitation. It's good to be back. Uh okay. Today I want to talk about the twisted free jack case setting all your systems. Everything I have to speak about will be joint work with SI and F. So to start with I'm going to recall the unitary free project period

in this case uh I'm going to reserve for here for the fields we are going to start with the DM expression f0 is totally real. F is a CN. uh that we consider unitary group G a united group in four variables define with respect to the cm extension at H is going to be a direct a product of two unitary groups and we assume the spaces are this space is the direct of those two spaces. So there is a natural embedding uh and then pi

is there any signature or are they any signature I'm going to restrict to the usual one symbol. So pi is going to be a automorphic custooric representation of gl2na of the base field and we are going to assume it's simple uh which concretely means it's adjuant air function. has a po equals to one assuming that is expected. Okay, if pi is simple then you can uh you can consider the base change of pi to from x0 to f and then we can transfer pi to the unit group g. IG uh automorphic web of G with the same base change as P.

It's the same base change. Then we would expect that the H period of IG is not zero. This should be rel this is not this is not true but I'm going to explain that this should be related to the central value not zero [clears throat] approximately equivalent to that. Okay. This is kind of the standard philosophy at least that the denomination of a period should be related to an L value. But the reason this is only a approximate equivalence is uh first of all already in something like the GGP case. I can't just restrict to a single unitary group. I have to consider all pure inner forms of the unitary group.

So a real statement should at least say the central value being non zero is equivalent to this HP being non zero for specific inner form GH that's already in the GDP case. In this case, it's even more complicated

uh due to the need uh to consider inner forms beyond the pure uniforms. So to make this precise, [clears throat] we need to already introduce the twisted setting. But uh so I'm going to follow the notations of Spencer Leslie and

Okay. So what those what those scope did is introduce a by quadratic extension. So we have originally a CM extension here. We are going to introduce a totally real extension by a totally real quadratic extension. Uh then take their composite. This is by quadratic. There's a third quadratic field and this middle extension is CF. Okay. the field diagram. The more precise conjecture is H period of pi G is non zero for some inner form of H and G if and only if two two different L functions happen. One of my path change

uh I'm not going to specific ify what kind of inner forms show up. So those are not pure interforms. They are unitary groups of certain quarton algebbras. Uh the exact definitions can be found in their equivalent. Um another thing of interest is it's not just a single value. It's two L functions but it's not that both of them are not zero is one of them is not zero. Um, and this twisted setting naturally shows up if you try to use the relative trace formula approach to even study design twisted setting because this shows up in the stabilization process.

Okay. So the goal of today is to try to study some kind of arithmetic picture of this. So even if both are twisted as twisted settings assum you impose the right signature condition. So in the untwisted conditions the setting we use the signature where g is u 2 n -1 and h is u n - one one.

So we are imposing it to be have this signature at one arch media place of fur and compact at all other archimedia places and in the twisted case you impose something similar to it. In this case we have sha varieties h invalidated sha variety of g. This has dimension 2 n minus one. estimation n minus one. So we have so we get a special cycle

in the arithmetic middle dimension. So there's this code [clears throat] group of shimmer and this statement is just true in both the twisted and untwisted case. I just don't want to tell you exactly which quarterly algebra which unitary group we are talking about.

Okay. So to the automorphic representation high we kind of touches pic ga representation. uh low high going from gamma group of f0 to go to let's just keep it and this appears or this is expected to appear Here is the coology in the coology

the shimmer vity of G. So not necessarily this representation but at least it's restriction to the smaller gawa group gamma over F. So restrictions low restrictive gawa f will appear in the homology of the shimma variety of h and our question is so if you follow through with the usual I philosophy of translating from period statement to arithmetic version you would say maybe the height of this special cycle should be related to a certain L derivative. We are not going to make such a conjecture since it's already quite intricate on the pure side. So we are just going to ask the question.

Uh the question Yes. Extend this base. This class this special cycle shimmer H maybe the P certificate component to a boiler system. or row pi and I'm going to define what I mean by an oiler system more precisely

is there a group floating about there is a sar group floating about which will come here by define what our system for me is Okay. So just to fix ideas I'm going to state the main zero zero. This is our result. There it is. uh uh split anticyclonic oiler system

uh for not the representation of real pi which in a twisted case is not the natural object that shows up but for the first of all restrict from F0 to B. I then induce it from E to F prime. F prime is the middle CM field. And if you write it out, this is low pi restricted to F prime direct sum ro pi twisted by the quadratic character. restricted right. So we have an oiler system for the direct sum of two representations. Each of the summit is exactly all those L factors that shows up. So this is some kind of arithmetic version of that.

I will not be I will not say what I mean by that but this visually resembles that. And what do I mean by the system? Uh who is who base class? Yes. What I said earlier is Shimra. Uh just to be clear, this is the H1 F of Right. I'm going to put all this representation row just from now on.

So more precisely for each place L each place L of F zero which splits in F prime. So split andic means we are looking at the cm extension frime over f0 and only considering the split places frime. So for each split place uh there exist a class uh let's call it set L in H1

where here this is a plus Dr. B. So we have classes over abil over anti-cyclomic ability extensions of F prime satisfy the norm relation.

What do you mean by anti-loic? F prime is a CM field. N prime is a CM field. It has R2 + one ZP extensions, ZL extensions.

Yes. Okay. What's the anti-csychoic one? There are lots of them. I'm working over

Yeah. A non-cyclomic one. Yes. Okay. I'm Yeah. All I'm doing is I'm working over all of them.

All of them. I'm working over all of them. Yes. But you are you are No. So I'm working over all the rein class fields. So away from this. Yes.

Away from the psychoic form. Yes. Yeah. Okay. So all the dimensions that are conjugate access minus one.

So if you hold this true, you have one or two of them in this. Yes. Okay. But really we are just going to be working over one place at a time. So

No, I understand. Yes, I understand. But the point is we have classes over class fields uh satisfies and this should really be for a fixed integral lus uh the point is I don't want them to don't have arbitrary denominators. I fix the integral notice and require all my classes to be in the exact choice doesn't matter but it has to be fixed up front uh such that the co restriction of Z L is this base class that's just called Z But with a certain forbaneous acting oil

here is the characteristic polomial. uh you know and by forbadius I mean I secretly mean we fix one of the places of frime above L take that forbadius so there is a choice of a place above L this that I'm hiding in the notation and that's This is arithmetic. So this is geometric verbius. The actual definition of a system you are allowed to take a couple of different but this is kind of already is enough.

Is it more like a polyaging system? This is Rubin. This is this is more like a Kivakan. This is more like an oiler system as defined by Kibbaken.

So not the Kolivian systemed by Maser and but not a Kivakan system defined by Mason and Ruben.

I understand. This is this is the split anticomic oiler system was defined by GHF uh risker at negro.

Do do you get the other layers of the oiler system or do you just I I get all the layers. I just can't be bothered to introduce notation.

But this is enough. This is not you this this gets the idea across. You get all the other layers too. Okay. So the source of all Z and all the higher layers is highly translates. of the embedding shimmer H into shimmer G. So first I'm going to just reduce this problem of constructing those higher classes. Okay, I'm going to reduce the problem of constructing those higher classes to one that's purely functional analysis harmonic analysis. Then we are going to solve that harmonic analysis problem. So yes know that there is a map going from all smooth compactly supportive functions.

There exist a map to uh CH group. I'm not going to the dimension is yes sheet with CP this is not strictly speaking correct well I don't know standard contexture so this is not correct we use some u talk realizations of it well let's not go into that detail and this is a f group. So here's a shimmer variety G really means the immers limit of also shimmer varieties. So this is the direct limit of a bunch of sh group a bunch of continuous tokenology groups actually. So it's known that there exist such a map going from test vectors of identic test vectors on basically gmod age to special cycles of the human variety.

Okay. The original cycle is constructed at finite level. Right? So all the cycles all of them are constructed at finite level. So this is just an way of packaging. So

so if you just quotient on the left by some finite you could take in variance you would so if you have a k0 it will just be k this is just convenient. So the point is this is GF f equivalent and to get integer coefficients here all the all that you need is integer coefficients there. So integer value function CP value function gets mapped to CP value cycle. You do not introduce any extra denominators in this construction. And they are very standard Kobi constructions giving you uh elements here. Well, in this case it's actually over f F prime. Okay, this part is standard. I don't really want to talk about that more. Um, so first thing to do is I'm going to explain how to get all the opinion extensions.

So to get a extensions Take this augmented group. I didn't tell you what G is. You can keep operation is U2 and I'm going to cross a U1 but this is a F prime over F0 U1. So h basically actually so to define this I need to to define this [clears throat] I need to tell you the additional piece of data which is a co- character of h.

So once I have a co- character of h I can define this uh augmented map. um and the law of this co- character is really to tell you how to twist my characters. In this case, because I'm using U1, I'm looking at antiatomic characters. So you can continue using the same formulas you get h mod cross u1 cp value functions mapping to this is just a collection of points numerical variety of g times the shu variety of u1 was h again

h is basically u cross but some inner forms of it for the extension f over f0. So both of them are inner forms of unitary group for yes.

So so what does this character new look like? I'm going to tell you what it looks like in the local setting.

So in the in the split setting h is u cross u and the character new is the quotient of the determinant.

Okay. Uh, but I'm going to explain the local settings soon. So now the shimmerity of G is defined over F. This is defined this shimmerity of U1 is basically the spec of the anticyclomic extension of F prime. So once you fix a level here you get shimmer of G base changed to an anticyclomic extension. So fixing you want your response to taking extension.

So basically the problem of constructing classes over a billion extensions comes down to instead of studying G over H studing G cross U u1 over H which on the functional analysis part doesn't change all that much. Uh

but could you say this without the U1 just by keeping track of the field of definition of each component?

You probably can and that's annoying to write. Okay. So, yeah, uh this is probably the nicer way to look at it. Okay. So, now that we have all the setups, we wanted to construct something that's defined over the ring class with conductor A. That means the UA level should be unmified everywhere and just slightly ramified at home and I'm going to I never had to think about the G level really. So really they want

yeah I think there's a more detail I mean the answer that you gave is not just that it's more comp more problematic to keep track of the deals of definitions is actually from some of the harmonic analysis point of view it's this translates the GAWA action into

yes okay yes okay you I I'm just going to ignore that point for now. I don't know if that's absolutely correct. Okay. So, we want uh to construct an element let's call it delta L in this function space.

So, I'm going to just start with a fixed level structure on G and the level structure on U1 will be the aromi everywhere but as a place L I'm going to force it to be in this instead of oil cross uh such that the trace of the L I'm taking the trace from this 1 plus L O L to O cross. So this is going to be in the same function space. So this is a trace of degree L minus one. We want this trace to be uh basically this polomial times a basic element which is just the indicator function of the maximum compact.

Well indicate the indicator function of the maximum compact subgroup. So under this map the basic element will be just set no hack translate standard shimmer ID special cycle. Uh and this element under this map will be set to the higher level class Z and this relation plus the comic is exactly going to translate to this co-restriction relation.

Okay. So now we have translated everything to the construction of this element and this is a purely local statement. Everything up there zero is now I'm just going to secretly pretend I have taken FL points. So uh let's look at the place L. So L was a place of f0 which splits in F prime. So now there are two cases whether it splits in F or did not. So there's what's known as the split split case where L splits everywhere. Well, not everywhere like all three fields. In this case, whether you're twisted or untwisted, the local setting is G is G2N.

H is GL cross Gn. The character new is set since the matrix matrices A and B to zero the quotient of the determinant. It's very important that this is the quotient. the product does not work. There is from uh basically local harmonic analysis. There is a good reason why the quotient doesn't work. Uh why the product doesn't work. There's a good reason why you had to choose the quotient. So in this case uh the kind of delta L was constructed explicitly by Andrew GR uh and car and I think the public version is 2023. So in this case this is the standard frequenc model. There's a local theta integral. They constructed the correct test vectors verified it works using theta integral.

So and because of multiplicity one that relation was hold. And in our case because we reduce it to a local picture everything is still the same. So we don't really do anything. Well, we don't get anything new here. Our method is completely new even in this case. The next one which the LX set group would call inert inert case is L is inert F. In this case the local set is a quantity split unitary group over f over f0. The subgroup is restriction of scalar of G. Okay. And the character new is just the norm of the determinant.

In this case we we don't know any local za integrals for those pair of groups. In fact the local multiplicity for the iron restriction problem is 2 to the^ of n. So we do we have no Z integral we have no multiplicity boy uh so uh we use a new method to approve the existence of this element directly we use aromite thresh formula

and method uniformly applies to both settings provide design pressure formulation. Okay. So in the last part of my talk I'm going to explain this approach. Are there any questions so far about this reduction step?

So remember there are two things. So I'm going to introduce some notation. X is going to be G. Let's say a logic points. I'm just going to run XS G H. There is a difference between F points of G mod H and F points of G mod F points of H. This is the whole reason we need to stabilize the trace formula in in any case but I'm going to ignore that. That's doesn't really matter for us. So x is just going to be the variety. Uh so there is this vector bundle g cross u1 which because we are working at a split place just becomes a gm.

So this is a GM bundle and remember our local translation was looking for an element delta L in as a smooth compactly supported function X2 with Z P coefficient and here because L is a rep we are just going to keep it at the uh hyperspaceial level at G and sub level on GM is still the same. uh we need to find this such that it's trace from 1 + L or L invariance to cross invariant is equal to a happy polomial times the basic element. So this is abstractly the problem statement with all the geometry is gone. It's just purely bal function in this test vector space. So now we have two steps. One, characterize the image of the trace map.

So the image of the trace map, the trace map would be subjective. If this were QP coefficient, you should think L is so this trace map is of degree L minus one, you should think that's very divisible by P. So because it's CP coefficient, this is not going to be subjective in general. uh so we are going to give a characterization of this answer and through just the right hand side is satisfaces. If I were to make a naive guess it would be that that image was generated by the t times everything and of the norm or the index.

How close is that? I don't know. So I we did a characterize it that way. Our characterization is very clear. It has nothing to do with PR or anything like that.

Does it for PL of T*? It does. That's the whole point. But that's actually the goal of step two. So it contains what I was guessing.

It contains what you are guessing. Not all of it. Right. I cannot say anything further than that because our characterization is completely separate from

the KO stuff. So step one is we we want to calculate the image of trace. Well the image is lines in this function space. Uh so the image lands in this function space. We are looking for a sub set of this function space which is just x mod this open action and not any space I see here. So this is some space with a group action and we are mod action of the maximum compact subgroup you get a discrete space. There's such thing there is something called well I guess here the generalized corresponding position.

So classical opposition is basically the maximal compact acting on G cross. So G cross G acting on G will give you classical caronic composition. This is generalized cartoon decomposition. It tells you this space can be reason canonically indexed by the Taurus by biolus which in this case looks just like bio by bioone

also our teacher has actually because X2 always carries this GM bottle there's always going to be some kind of completely unrelated set here but generaliz composition tells you that this space is just indexed by it one for the proposition is the trace I please contains the element. We suspect this is if but all we need is it contains all f such that obtains all functions f such that f of this pup of integers is divisible by L minus one if a is zero and just no condition otherwise. So we prove that if you have a function which is divisible by L minus one on a specific wall on this specific wall then is in the image of the trace and there are and all other n minus one balls doesn't matter only one wall matters.

Uh the way we prove this proposition is by exploiting the proof of the generalized cartoonic composition by gay greenler. We consider the asmtotic degenerations of the spherical varieties. Well they they exploited that to prove the cottonic opposition. We considered x cubed over x. So degeneration of this EM bundle at infinity and shows that in quite a large generality this kind of characterization always holds and the difference for the experts are those walls are type T. This wall is type A. goes the other way since the only ball of type T and all the other are type n.

So we can prove that the trace at least contains all the functions that are divisible by L minus one on the wall type T. So given that characterization the second part reduces to show this PL IP basic element evaluated. on one of those tuples where a n is equal to zero is divisible by a minus one.

So this pt is not given concretely as a double coat operator. This plt is given concretely by its static transform. is given by subject. So it's our annoying problem to even compute this plt as a double coat operator. So we bypass that completely zero. This is due to Omar in the split split case I guess. Yeah. Yeah. General but not general enough to cover the inert inert case. And uh as But our method is basically just okay they all of them use basically the same method uh but and our proof is just copying offense proof at various points we need to change a side it's not anything particularly it's like a master's thesis level project by described that serum is exist this isomorphism from this instead of CP we are going to use C coefficient

you can back. Yes, this would be L plus minus F I and then there is the and then there's a TUS piece the S there. There is this one which is equariant with K and that looks different depending whether we are in the SL or inner case is but it's just s the convolution algebra G mod K mod kx on this and on this side tells you it's The real functions of a Taurus but the Taurus is different depending on whether we are in the split or inert case uh from this you can already if I if I write down exactly what I mean by equivalent s you can tell that it's going to have multiplicity one in the split case and multiplicity two to the in the inert case just because in the inert case the Satic isomorphism evolve terms of the form L + minus 2 Z uh and while variant and the while group is a type C while group that's part one part two which is more important well equally important is there exist an immersion formula

formula. So if you have a polomial L hat in the image and I want to compute its inverse formula of the form part times interpret polomial multiply it by a very concrete rational function. Restricted lambda x plus okay this requires a lot of explanation. So L hat we view it is a function of the polomial. Uh so we just view it at the zero includes the definition of this rational function which is basically the partial measure uh half the square root of it. Yes, this is basically the square of this will give you the measure on this space. [clears throat] So uh this is also basically the value of spherical functions at u specific elements. But in any case we construct a very explicit concrete rational functions uh multiply that by l hat. Now we have a rational function which we expand in a long run series in L to the plus minus one along a certain cone that yes along a certain cone that is opposite of this

cone. So now the x is some cone like this uh we have a lauron series that's with coefficients always here. So if we restrict to lambda x plus uh our virtual velocity plus so once we restrict to lambda x plus I get a compactly supported function on lambda x plus which is that function. So there is an inversion formula of this form uh and all you really need is to specify this rational function. Then it's just the coefficient of a certain series expansion.

Now you just need to compute a long series expression. Uh except I don't even do that. That was too much effort. So followally is easy to show coefficients are via Z that P well they're actually integer coefficients I'm allowed to invert P but inverting P is fine because P is cor to L. I do not. No, that's I'm allowed to invert L. Uh coefficient is P. I'm allowed to invert L. That's completely fine. Uh yes we we can show formally that's a coefficient that's there and we need to prove divisibility by a minus one. So what we do is so we specialize formula at L= 31.

I'm going to write out the shape of this rational function in a maybe mysterious way. [clears throat] This integral kernel, this rational function looks like a product of two things. In the numerator is some kind of co loops. In the denominator, it's some kind of stop. So first series is going to be L to the power of minus one/2 by some kind of so gamma gamma uh check and set check are both some kind of very concrete cor roots that I don't want to write down and this is plus or minus depending on whether I mean split or inert case so it has this shape and I'm not going to explain I'm not going to give you the exact formula for this.

The point is first of all this part has sub where I need to prove that the L to the 1/2 cancel. So we get integer power of L. That's easy. Now the point is if I specialize at L equals to one this becomes a product of 1 minus E to the gamma check over product of 1 minus well plus minus E to theta check. The L of the one half disappears. Most terms cancel.

So after most terms cancel there are a bunch of terms left remaining terms multiplied by the specific L factor. The specific L factor is a polomial that we have been carrying around this whole time is a polomial. So first of all most of those terms already just cancel within the kernel and now once I multiply by this L hat all the denominators disappear and this polomial is anti-ymmetric under one group action.

So this is easy. The denominator is completely written down is you just observe all the terms already appears in L hat. This you just look at the shape of this resulting polomial is visibly anti-ymmetric with respect to a well group. meaning if I apply a well reflection to the resulting formula it changes sign under a well action cor so under let's be more precise simple reflections

but it only changes signs if it's of type T symbol reflection across walls of T if you if you do a reflection across the walls of type n you do not change sign it's symmetric across those walls and antiymmetric across walls of type T. So this formally implies that it's coefficient on a wall of type T is zero. and the coefficient is zero means it's divisible by L minus one. So that's how we prove part two which is this P this high polomial applied to the basic element satisfies the criterion established by part one. So this the kind of the distinction between what's of type T and uh that's not type T that's type T not not that it really means much but so there's a dichotomy between type T and type G t that's seen in both kind of this is geometry part and this harmonic analysis part uh we don't know how to handle both of the type n but nobody does So, okay, that's it.

Any questions? So, what does the oil system do for nothing? Well, I don't know.

Yeah, I was asking you earlier, is there a Z group? Yes, whatever you want to. I also see it's it's a it's a role plus a role visit by ATA

and all your system requires your rep to be irreducible. Yeah. What if you project to one of

I we talked about this. I don't the whole point of this talk is not that we got okay you got an order system. The point is we got an order system. Stop asking questions you do the rest. for the moment. Yes. Uh it's more like we have this new technique of doing things. Here's a case where we got something new

related question. What what do you know about the base class? Nothing. Okay. Are there other possible cases where the new cases for this?

Yeah. So I'm working on incorporating other side series. So this recovers most of the previously normal inner systems like the one for elements. uh all the cycles stuff. So I'm looking for new cases invented by Chris. Uh and we are also kind of looking re looking at corn's case again. Cornu's case is orthogonal group embedded into unitary groups

other way around. Other way around. Yes. Yes. Do you know if there's expected to be a formula for the free burn cake?

I asked my jaw. He said not yet. It's not it's not really an answer. Like maybe it's expected.

Well, if he doesn't know if he doesn't know or refuse to tell me, I'm not going to guess.

It seems to be more complicated because this is two different L functions through up. You are not obviously product way.

Thank you. Any more questions? Oh, let's thank on the organ.

Questions about this episode

What is discussed in "Euler Systems in the Twisted Friedberg-Jacquet Setting - Shilin Lai"?

In order: 0:00 L functions, shimma variety, Spencer Leslie, GGP; 10:00 anti-cyclomic, F prime, CM field, F0; 20:00 F prime, U1, Kivakan; 30:00 Andrew GR, F prime; 1:00:00 L functions.

What is mentioned in this Institute for Advanced Study episode?

14 people, works and ideas are mentioned, including L functions, shimma variety, anti-cyclomic, CM field, Andrew GR, F prime. Each one on this page links to its own page with every episode that mentions it.

How long is "Euler Systems in the Twisted Friedberg-Jacquet Setting - Shilin Lai"?

The episode runs 1 h 4 min; the transcript is about 5,769 words in 86 timestamped sections.

More from Institute for Advanced Study

About this transcript: the episode's spoken words as captioned on YouTube, split into timestamped sections. Reclip.Me locates every person, work and idea in the text, keeps the time it is first named, and quotes the words around it verbatim. Last updated January 24, 2026.