Thank you very much for the introduction, and I'd like to say thank you for the invitation as well.

The results that I'm going to talk about today are part of a collaboration of 10 mathematicians, none of whom as it happens are French, but nevertheless it's impossible to imagine proving the kinds of results that we prove without building upon the achievements of the French School of Automorphic Forms, so I'm very pleased to be able to talk about this work here today.

I'm going to begin by establishing some notation, which I apologise is going to be rather similar to what Arno told you at the beginning of his lecture, but it's impossible to talk about these things without being precise. So throughout my lecture today, k will be a number field, and pi will denote an automorphic

representation of gln of the dels of k, and we'll use freely the usual set of mostly

unrammified data associated to such a thing, so there's always going to be factorisation, pi is a restricted tensor product of local representations pi v, each pi v is an irreducible

admissible representation of the corresponding local group, where you interpret this in the correct way at the Archimedean places, and then for all but finitely many, so

the number field, well the first thing you know is that v is non-Archimedean, almost always, and pi v is unrammified, and that means we can define its Satake parameter,

and here I'll diverge slightly from the notation used by Arno, and I'll write t of

pi v. Okay, so this is going to be a conjugacy class in gln, complex numbers of a diagonal matrix defined up to conjugacy, and of course in the case of gln this is particularly simple,

it's just telling you how to recover pi v, the unrammified representation as a subquotient of parabolic induction. Okay, so the conjecture that I want to talk about today is the Ramanujan conjecture, and we already have enough to state it, so let's do that.

So this I suppose is what really should be called the generalised Ramanujan conjecture, because Ramanujan himself certainly didn't use the notation that I've just written down,

and it says, what does it say, suppose that our automorphic representation pi is furthermore cuspidal, and let's say if I take the absolute value of the central character of pi, then I get norm of k to the r for some real number, so this

conjecture is often stated just for representations of unitary central character, but of course there's no reason necessarily to make such a restriction, and then the assertion is for almost all v, so T of pi v is

defined, every eigenvalue alpha v of the Sate parameter satisfies absolute

value of alpha v equals qv to the r over n, if I've got my normalisations correct. Okay, and qv here is just the size of the residue field at a

finite place v. Okay, so this is a conjecture, it's quite a famous one, and

it's very often rather hard to say anything about it, except in certain special cases where you can prove instead another conjecture that I want to state, so I think I'll do this on this blackboard, and this is what you might call the reciprocity conjecture. I'll clarify the relation between these

two after I write it down, and this says, well suppose again that pi is cuspidal,

and further that it's cohomological, and just to be precise, in order to

have to avoid introducing too much notation, I'm going to restrict cohomological to mean that pi has cohomology in the trivial coefficient system, so that would be the algebra cohomology of the infinite component

without tensoring with any finite dimensional representation is non-zero. Then the assertion is, this is the reciprocity part, for any prime L and

isomorphism iota between QL bar and complex numbers, there exists a Gala representation, let's say a continuous irreducible representation, let's say

rho pi iota, so that's the same notation as in the earlier talk, from Gal K bar over K to GLN over QL bar, which is a ramified at almost all

places, and again at almost all places satisfies rho pi iota of Frobenius at

that place, equals, and this is more or less the same condition that Arno had, it's just a twist due to normalizations, so I take the

satake parameter and then I multiply by a fixed constant. So I guess little k is the same as capital K up there? Uh, this is a capital K. Oh it is, okay, looks a bit different from that capital K.

Yes, I don't think any small k's will appear in this talk, so hopefully the risk of confusion is minimized. Okay, so this is the first part of the reciprocity conjecture. The second part, which I want to add on, is that I

want this representation to be motivic, which I will give a precise sense to. So this means there exists a smooth projective variety which realises rho pi i in its cohomology. Let's say X over K, such that rho pi iota is a

QL bar, Gal K bar over K sub quotient of the group which I can finally fit

the entire cohomology of X space change to K bar with QL bar coefficient.

So these are two conjectures that we believe are true, and we know some cases, as I'll get to later on. And the first remark that I want to make is that if pi is cohomological and cuspidal, then

reciprocity for pi implies Ramanujan for pi. And this is essentially,

thanks to Deline's proof of the vague conjectures. And this is the obvious generalisation of a remark that was first made by Deline in the case of GL2 over Q.

Second remark, just in case you haven't seen the reciprocity conjecture before, this is not the most general or the most precise thing that one could write down. And the first formulation that I'm aware of in literature is in Clauzel's article in the Ann Arbor Proceedings, where you can find a more precise version of

this reciprocity conjecture for any so-called algebraic automorphic representation pi.

So being algebraic is a condition on the Langlands parameter of the component at infinity. I'm not going to say exactly what it is, but suffice it to say today that if you are cohomological, then you're also algebraic. So this conjecture is a special case of this one that's been around for quite a long time.

I don't think I understand the mark one. If you rotate twist, you need to know an extra information. I'm being a bit optimistic here. No, but you write z and you write x, a implies b. So, well actually no, I think it's certainly true that this one applies this one without

assuming anything. You may object that this is stronger than motivic. No, I don't think so. Well, let me explain my thinking.

So I've stipulated that rho has to be irreducible. That means that it appears in a single cohomological degree if it appears at all. So that means rho, if it is motivic in this sense, certainly has to be pure of some weight. So maybe it's the wrong way.

But then you look at the determinant of the representation. That will have weight n times the weight of rho. And I think probably what I've written down means things have to line up at the level of determinants. It's also cuspital and therefore generic, and therefore there's the purity of those

that basically fixes the possibility. Yes, okay. Well, we can continue to discuss afterwards. Okay, so basically, almost every case of the Ramanujan conjecture that we know,

not absolutely every case, but most cases are proved by establishing first reciprocity and then deducing Ramanujan as a consequence. Anyway, let's carry on. So let me write down what I just said.

Most cases of Ramanujan that we know are proved by first proving reciprocity.

And this is certainly the case for the original conjecture made by Ramanujan, for example, about the Ramanujan delta function. That was proved by Deline, who constructed the

compatible family of Galois representations that one expects to exist. But I want to talk about a more general example than just the case of GL2 over Q. In particular, I want to allow any n, because then the theory of photomorphic representations is a bit more interesting.

So I'm going to describe a particular set of hypotheses, which I will denote with a star for the rest of the lecture. So let's suppose k is an imaginary CM field.

So that means it's a totally imaginary quadratic extension of a totally real field k plus. And let's suppose as well that pi is cuspidal,

cohomological, and finally, conjugate self-dual.

So conjugate self-dual means that if I take pi and act on it by the Galois automorphism, which is the non-trivial automorphism of this quadratic extension,

then I get a representation that is isomorphic to the conjugate gradient of pi itself.

So in this case, we can say something and indeed have said something using a strategy that generalizes in some sense the original one of Deline. So in this case, one can hope to descend pi to an automorphic representation,

let's say capital pi, of let's say g of a k plus,

where g is a well-chosen, you don't want to say any more than that, unitary group in n variables over k plus. And for example, if in particular you choose g so that when you look at the real points,

so that's the points valued in k plus tend to r,

are isomorphic to one copy of the unitary group of signature n minus 1, 1, and then some number of compact unitary groups, then the general theory of how we think the cohomology of Chemore varieties

should look leads you to expect that you should be able to construct the sought-after Galer representation inside the Ataka homology of

the Chemore varieties which are associated to g.

But of course, actually carrying this out is really rather non-trivial, and how many of these do I have?

The first case where the strategy was carried out for general n

was done by Clauzel again in, I guess, about 1990.

And this included the condition that pi v is square integrable for some finite place v. The reason that that condition is important

is because it allows you to choose the unitary group g so that the associated Chemore varieties are special cases of the so-called simple Chemore varieties first considered by Kotwitz.

So this is oversimplifying a bit, but the word simple means that there's no endoscopy, and that's why it was possible to carry out this program without knowing the fundamental lemma for a unitary group in a general number of variables. And of course, this was almost 30 years ago now, and a lot more has been done since.

So we now know the Ramanujan conjecture for any representation pi satisfying the hypothesis star.

And we also know that the Galer representation exists as well.

And this is more or less the content of the first volume of the book project that Michael mentioned earlier. So let me mention some names. So certainly, this is contained in, well, mostly contained in the article by Clauzel,

Harris, and Lebesse in that volume. And there was independent work of Shin that established something rather similar. And then this constructs the Galer representation in a motivic way for most pi. And to get all of the representations, you do need to use

a interpolation argument using an Eigen variety in a similar way to what was done during Arno's lecture. So this gives you the representations. It doesn't quite give you the Ramanujan conjecture because, as I said, not all the representations are known to be motivic, so you can't apply the vague conjectures to those ones.

So the proof of Ramanujan for all pi of this type was completed,

well, again, pi, but wrong, sorry to keep mentioning you, in, I guess, 2009.

And this was by showing, even if you can't always show that the Galer representation is motivic,

you can always show that either it or its exterior square is motivic. And the reason that there's an obstruction, well, it's the same kind of sign condition

that Arno alluded to earlier that causes problems in the setting of the symplectic rib 2. And the point is that if you can use the Schmoe variety of type u1n minus 1, then you're happy and you have the Galer representation. If you can't, then you have to use something like u2n minus 2,

and that's why you find yourself with the alternating square. It's also possible to use u1n minus 1 squared, and then you'd get essentially this representation tensed with itself, or a Galer conjugate.

Okay, so that is the Ramanujan conjecture for all orthomorphic representations satisfying this condition that I've notated down here. But I want to talk about something new today. I want to talk about what happens when we drop the hypothesis of conjugate self-duality.

And I've described this context because I think it's interesting to see what you need to do at each stage in order to actually prove the theorem. I mean, we already have the Ramanujan conjecture for some representations, which we don't know to be motivic, although they are close to being so. But we'll see for the theorem that I'm going to describe,

you're very far from being motivic indeed, in the sense that I make precise. Okay, so we now consider a more general situation.

So let me call these hypotheses star prime. And this is, suppose again, that k is imaginary Cm.

And suppose again, that pi is cuspidal and conjugate self-dual. Sorry, cuspidal and cohomological. But don't suppose that it's conjugate self-dual. So now, and I guess this is relatively recent,

we again know that the Galois representation that you expect to exist really does exist. So that's half of the reciprocity conjecture as I stated it.

And this was proved, well, first by Michael Harris, Kywin Len, Richard Taylor, and myself. And then afterwards, another proof, I think arguably better, was given by Scholzer. And that's the proof that I'm going to resume briefly now.

And I think the strategy to prove this is quite interesting, because it seems to fit nicely with some other trends

in the language program that have appeared recently. So what is the strategy? Well, you're again going to have to use a detailed understanding

of automorphic representations of unitary groups. But in this case, you need a much bigger unitary group. So you want to take the quasi-split unitary group in 2n variables. And just for definiteness, let's say it's defined by the matrix

0, 0, and then two n by n identity matrices. And let's take P to be the block of a triangular subgroup of G star.

It's a parabolic subgroup, or what we might normally call the Siegel parabolic.

And let's take M to be the block diagonal matrices inside P.

So that's a levy of P. And of course, if you just look at the definition, you see that this is isomorphic to restriction of scalars of GLn from K to K plus.

And I want to introduce some symmetric spaces as well. So let's take XG star. So I want that to be the real manifold, which is just the complex points of the Chemin variety of some tame level.

And let's take XP to be the symmetric space for P. So if you want, you can present this as some kind of idyllic double quotient. I won't say what UP is.

And let's take XM to be the locally symmetric space for M, which we can again think of as double quotient of some kind. And if you know about such things, you can think of XM as being, in the case where K is imaginary quadratic and N is two, as just a Bianchi manifold.

So that would be a quotient of the usual hyperbolic three space by a congruent subgroup of GL2OK or a finite union of such things.

P is over K plus, everything's over K plus. Oh yes, AK plus, sorry. Thank you.

So the idea here is that if you know something about G star, then you can know something about M. And in fact, the strategy that is used in Scholtz's paper is astonishingly naive. It's just amazing that it works because he's able to prove so much about

the cohomology of XG star. So what's the idea?

So we have a diagram of spaces. So topological spaces or real manifolds, if you like. So I can think of XG star as being contained inside

its Borel-circumpactification. So that would be a manifold with corners. That contains its boundary, which is just the complement of XG star. And this you compute in terms of parabolic subgroups. So this contains XP as a locally closed subspace.

And this has a map down to XM, assuming that the level subgroups are chosen appropriately. And using the theory, for example, of Cuswell cohomology,

you can first of all show that pi, representation of M of AK plus, or at least its finite part, embeds into the cohomology of XM, let's say with complex coefficients.

And then using the boundary exact sequence associated this diagram, so this is the long exact sequence of XG star embeds into XG star Borel-circ contains boundary.

You can even show that if you take the parabolic induction from P to G of pi infinity, then this appears as a subquotient of the cohomology of XG star.

And this is where something quite interesting happens. And I think the idea to apply a strategy like this in order to get the Gala representations that I'm trying to construct is quite an old one. But the reason why it doesn't immediately work

is because what you can look at, let's say the Atal cohomology of small variety of G star base change to K bar QL bar coefficients, and then take the part where you're finding these Hecker eigenvalues.

So because the Hecker action computes with the Gala action, this will give you a Gala representation, which should have some relation to the Hecker eigenvalues by the Eichlich-Moor relation. But this will never contain, or let's say almost never contain

the Gala representation that you're looking for. This, the original idea of using this code, is also due to you, 30 years ago. Sorry?

The original idea of using this inclusion of the cohomology is also due to you. Was it, what? It was due to you. No, it was you, I think. Why do you say almost, it never does, it's one-dimensional? Well, you may know that, but I don't know how to prove that.

It doesn't work anyway. Yeah. If you think it's wrong, then it doesn't work. O pi i is n-dimensional for me, is that correct? Yes. Okay, so you think that this is always one-dimensional?

Yes, you make the computation by using the yoga form, which you have right here. Okay, yeah, it's basically proved by pink. Yeah, yeah, anyway. Okay, well let me say that I've- I've asked people to show me how to prove that it's one-dimensional, and multiple people have asserted there is a proof, but no one's shown me a proof yet. So if somebody can show me a proof afterwards, I would be delighted.

Okay, anyway. But it was Laurent's idea, not that it was one-dimensional. Regardless, we all agree that there are many different ways to think about this,

and there are good reasons why this can't contain the representation that you're looking for in general. So that might look like you're kind of stuck, because then you're going to ask, well, where else am I going to find the Gala representation? But of course, after Vincent Lefort, we know that there are more ways

than just looking at the cohomology to construct Gala representations, and if his theory of excursion operators could be generalised to number fields, then you'd expect to find an algebra of excursion operators acting on the cohomology of the Shmuel variety,

and that would be- what would it be? Well, it would give you a pseudo-representation valued in the Hecker algebra, and in fact, that's exactly what Scholter constructs in his paper. So, I'd like to think of Scholter's construction as evidence that excursion operators do exist over number fields.

Okay, so what does Scholter do? He constructs a pseudo-representation, let's say a 2n-dimensional pseudo-representation,

let's say, TG star, so as we saw in Joel's talk this morning, this is a map from the group to,

so let me write it like this, so our TG star, this is the Hecker algebra, which acts on the cohomology of the Schmuel variety, and here I ran out of space.

Okay, so again, what is TG star? It's the algebra generated by unramified Hecker operators, that's capital TG star, little tg star is a pseudo-representation which is compatible with the Hecker operators.

So here, little tg star satisfies for almost all v,

tg star is unramified at v,

tg star of Frobenius is the correct unramified Hecker operator Tv, which I won't define.

Yes, if you like. Well, this is the Hecker algebra of the Schmuel variety. G star is the unitary group,

so tg star is the Hecker algebra which acts on the cohomology of the symmetric space for the unitary group which is the Schmuel variety. So the fact that double n is then a log of extra Schmuel? No, no. So this so far is just in the context of the Schmuel variety, we haven't kind of passed to the Levy yet,

but I guess that's the next thing to do. So then the point is that

the appearance, the parabolic induction in the cohomology of the Schmuel variety,

let's say xg star c, gives a homomorphism from the Hecker algebra of g star to ql bar, which is associated to the Hecker-Ancom values,

let's say of induction from p to g of iota inverse of pi infinity.

So then if you just compose to get a pseudo representation of a ql bar, let's say t pi infinity,

so this goes from the gallard group to tg star to ql bar, is the one associated to,

well I guess it'd be something like rho pi iota direct sum rho pi iota conjugate dual twisted by epsilon to the one minus n, if I've remembered my normalisations correctly.

So with a bit more work, once you have a pseudo representation, you can reconstruct just the single direct sum and rho pi i. All right, so the key thing is to construct this pseudo representation, which looks like the kind of thing you'd get

if you had access to the theory of excursion operators. And let me just say two ingredients that go into construction, into the construction of the pseudo representation.

Well, one major one is Shulcher's theory of perfectoid shimor varieties and the Hodge tape map.

But another equally important one, from the point of view of proving unconditional theorems, is the classification of automorphic representations of g star.

So in particular, this means base change.

So I don't want to attempt to provide a list of attributions for this statement, because I don't think I know enough to get it completely correct. But suffice to say, this could never have been attempted without knowing at least the fundamental lemma and the stabilisation of the trace formula for the unitary group.

Okay, so that's how you make the Gala representation. But that's not all of the reciprocity conjecture. You want to be able to prove it's motivic. And unfortunately, I have no idea how to do that. Neither does anyone else I know. But we can prove something that you might not expect to be able to prove,

which is the Ramanujan conjecture, at least in certain cases. So this is where the list of 10 authors is finally going to appear. So that's Patrick Allen, Frank Caligari, Anna Cariani, Toby Ji,

David Helm, Bao Leung, James Newton,

Peter Schulze, Richard Taylor, and myself. So the question now is, can I fit the theorem as well as the list of names? So the theorem is, suppose pi is an homomorphic representation

satisfying hypotheses star prime, and n is equal to two. All right, so star prime has disappeared from the blackboard now.

So if it's not in your notes, this means pi is a cuspidal and cohomological homomorphic representation of GL2 of the Adels of k. And let me remind you that for me, cohomological means cohomology with coefficients in the trivial coefficient system.

Then the assertion is that pi satisfies the Ramanujan conjecture.

So k can be any imaginary CM field. Well, I guess it could also be totally real as well, but in that case, this has been known for quite some time.

And let me note, in contrast to the case of cusp forms which descend to unitary groups, when we always know that either rho pi or the exterior square of rho pi is motivic,

in this case, we don't know that any sense of power of the associated Gala representation is motivic.

So this appears to be a case of the Ramanujan conjecture that doesn't rely, at least directly, on the proof of the vague injectors. Okay, so how do we actually prove this?

Well, the thing we actually prove is that for any, let's say, m at least one, the nth symmetric power of the associated two-dimensional Gala representation

is what we call potentially automorphic. What do I mean by this? Well, just to say something correct, let's assume, without lots of generality, that pi is not of dihedral type.

And the statement that we prove is for all m at least one, there exists a Cm number field, let's say, Km over K,

and an automorphic representation, capital Pi, of Glm plus one of the ideals of Km, satisfying the hypotheses star prime for Glm plus one, and that if I look at rho Pi m iota,

this is isomorphic to the restriction of the symmetric power to the Gala group of this Cm extension.

Now, you might guess that the fact that you're replacing K with Km here might cause problems with the usual application of symmetric powers to control the size of the eigenvalues.

But in fact, that's not the case. So you can use the results of Jacques-Ishalaika or Tadic on classification of, let's say, unitary generic representations

to show that if, let's say, w is an unramified place of Pi m,

and beta w is an eigenvalue, the Satake parameter, then the absolute value of beta w

is bounded above by the square root of the size of the residue field and bounded below by one over the square root.

But then you observe that if v is an unramified place, let's just say v is a place of K

below w, which is unramified in Pi, and alpha v is an eigenvalue of T of Pi v, then you can take beta w

to be alpha v raised to the mth power times log Qv Qw, that's the degree of the residue field extension. And then if you just plug this in, then you get Qv to the minus one over two m

bounds below absolute value of alpha v bounds above absolute value of, sorry, bounds below Qv to the one over two m. And then if you just let m go to infinity, you get the result, namely that alpha v has absolute value of one.

So that's why potential automorphies is enough. So then the question is, how do you actually prove that these representations are potentially automorphic? And well, to do that, you need to prove automorphies lifting theorems and try and generalise the machinery that's been developed for

conjugate self dual Gala representations to this new context. And I just want to write down two of the main ingredients that go into doing that.

Okay, so what goes into this? Well, an awful lot, as you might imagine.

So the first thing I want to mention is work by Frank Caligari and David Garrity. So they have this paper, which I think is called Beyond the Taylor-Wild Method,

something similar. And they showed how you can attempt to generalise existing proofs of automorphic lifting theorems beyond the setting of Shmuel varieties.

Now, they stated theorems which were conditional upon many hypotheses. So they didn't prove anything unconditional in this paper, which I think is now published in Inventiones, but they did show somehow the path that one should attempt to follow. So then the question was, how can you actually attempt to check the conditions that need to be true?

The next major contribution is work by Cariani and Scholza So they proved vanishing theorems for the cohomology of Shmuel varieties.

So the first paper that they proved has been published in the Annals is about vanishing theorems for the cohomology of compact Shmuel varieties. And they show, in fact, that if you localise at a maximum idea of the Hecker algebra, which satisfies a relatively mild condition, all of the cohomology goes away except for the cohomology in the middle degree,

which is an extremely strong result for something like well, any Shmuel variety that's not one-dimensional, basically. And in work in progress, they're going to generalise this to Shmuel varieties which are not compact, although the statement is slightly different then.

And this, one of the main things you do in the 10-author paper that I've alluded to here, this allows you to prove some cases of local global compatibility for the Gala representations constructed by Sholto,

even with integral coefficients and not just rational coefficients. So a major part of this is proving, for example, that if the level is maximal at the prime dividing L, then the associated Gala representations are crystalline. And this is, as I say, made possible by the vanishing theorems

of Karajan and Sholto. So those are just two ingredients. There's much more that we need to do in order to prove this theorem unconditionally, but I think that's everything that I want to tell you today. So I'll stop there.

Well, I'm going to take over from the president of this afternoon's session because he seems to have had a train to catch. So are there any questions? Yeah. You might mention that there is another very important consequence of this construction.

What do you have in mind? Or maybe you didn't think about it. Saturday? Yes. So you can also prove a potential modularity for elliptic curves of a CM field and start at eight as well.

Are there questions, observations, suggestions? There were previous results about the local global compatibility in short, so they are not used here? This student of Taylor's?

Yes. So you're referring to Ilavarma. You use these also? We use related techniques. So I said we use related techniques. So the first paper that she proved, she published, proved local global compatibility at the prime to L places for the Gala representations

attached to automorphic representations. So that's good for all possible local representations? Yes, up to controlling the monodromy operator in the Vadeline representation. But we need to know it for the Gala representations attached to torsion classes here as well as rational classes.

The technique that one uses to do that is very similar to what Ela does in her thesis. There's also the question of proving, for example, that the Gala representations are crystalline when you expect them to be crystalline. No, I don't wish to create that impression. So I believe that Ela is working in a work in progress, although I haven't seen details myself.

We are really interested in torsion phenomena. So we only prove anything in the case where we know how to formulate it, which is for Gala representations, which are in the Fontaine-Lafay range. So crystalline representations with Hodge tape weights are all in a bounded range less than the residue characteristic.

And do you know the value of such a thing? I don't believe we do at the moment. And I think I've obviously picked something else on the rack. In the Caligari-Kiraki argument, you also need the representations associated with torsion classes, is that correct?

Yes, and those were constructed in Shelter's paper. So the existence of those has been known since the paper of Shelter in the Annals 2014.