 So let me start with modular 7. So I'm going to keep my notes so I don't stray. So what is a modular 7? Well, so you start with a subgroup of GL2Q. Sometimes it will be GL2Q. But usually it will be a congruent subgroup of SL2Z. M is a gamma mod. Let's say, let me say, they usually have your ring sitting around. And so R will be summary. And then delta zero, that's my notation for the separate divisors of degree zero on P1 of Q. So P1 of Q I imagine as being, start with a half way. P1 of Q is the set of rational numbers on a real line together with the point at high and fit. Okay. And so then let me just tell you now, so that's what I'm thinking about P1 of Q. And then, you can see that board so well, I'll move over here. And then, so what are the modular symbols? So the modular symbols over gamma, with values in M, sometimes I'll write it that way, that is the set of homomorphisms equivariate over gamma from delta zero to F. Okay, so if I have a modular sample, I'll be thinking about things like this. I'll tend to write things like this. Phi of R to S. When I write that, what I'm, you might, in this interpretation, that would be phi of the divisor of degree zero S minus R. So I'm thinking about R and S. It goes like that. And so when I write phi of R to S, I'm imagining it as an integral from R to S. And so that's the reason for this notation. And of course, there's a remark that, in fact, this captures an important part. So a remark that I've been asking myself a long time ago that there's a natural isomorphism from the contact you're supporting called logic gamma with values in M to the modular symbols with values in M. And so that maybe is a justification for this picture. Now the way of Jekker sort of visualizing these things, it's fun to visualize these things too. Let me try. I'll do it here. What you might think about is zero. You have I infinity up there somewhere. And if you think about the path of one, two, minus one, minus two, you can draw these geodesics connecting those guys. Take a half. You make what's called the fairy tessellation. My picture's not great. But you can think about what's called the fairy tessellation of the path. And the idea is you draw the fairy tessellation down on the real axis and then you connect up. So let me just say it very simply. If I take infinity to zero, which is this path, and then sort of apply the elements of SL2Z to it, there's a one-to-one correspondence for PSL2Z, I should say. There's a one-to-one correspondence between the elements of PSL2Z and the edges of this thing where if I have a matrix gamma, this is A over C, B over D where gamma is the matrix, A, B, C, D. And so those are two adjacent terms in some fairy sequences. The idea. So I'm going to leave this picture on the board. This, by the way, is something that I learned a long time ago when I was in high school about fairy tessellations and fairy sequences. And one of the first things that I actually fell in love with was this idea of continued fractions. And I think that is the initial reason why I got so interested in this project. Gary showed me how to use continued fractions to compute periods of modular forms, compute modular symbols. And so, of course, that caught my eye. Of course, now there's very little continued fractions in the theory anymore. But still, I like drawing that picture. That picture, I'm going to leave on the board so I can point to it now. Okay? All right. My principle with values in M is a way of labeling the fairy tessellation. When I say labeling, I mean just the entries. Labeling the edges, giving each edge associated with some element of the module M. But doing it in an equivariate way. Yeah, in an equivariate way. That's a picture of the module set. You'll see that if you do that, there are certain relations because we need to have, when we go down and over and then back up again, we better get zero if this thing represents divisors of degree zero. And so, in order to make such a labeling, such a modular symbol, there are certain relations that have to be satisfied. And they're the ones you see in the picture, and they're called commodity relations. Okay? And so you will use them without a lot of saying here, but they're important. Okay, so one of the things that Barry taught us is that when we're studying values of L functions, at least for modular forms, really what we're studying in some sense are these modular symbols, or if you like, the homology of the modular curve, and you're making certain homology classes in the modular curve and evaluating modular symbols on them, and you get numbers, maybe, or elements of a model that correspond to values of L, things like they're the universal. That's the path that you integrate a modular form over in order to get the melon transform of the modular form. And so that gives, I thought we'd compute L, maybe in my group. This you see is known as L to zero, or it's an element of the relative homology of the modular curve. Land of chi would be some sum, A equals zero to M minus one, A in Z mod M cross, then you'll get the value of the L function. It somehow changes the context for thinking about values of L functions, thinking of them as homology classes, which is very natural. Okay, so this is a basic example. If F is, say, a cost form for some group gamma, it's greater than or equal to two or even, we let omega F bring it equal to two pi i times minus two F of 50 tau. That's a differential form on the upper half, making values in some space that I will call wk minus two. Wk minus two is just our homogenous polynomial in two variables, x and y, to reach k minus two. Okay? And the key thing is that if F is a cost form, this defines a gamma invariant differential form. This is something that's invariant under the action of the group gamma. And so I can integrate it over paths and make a module assemble this way. And so I want to do that. Then we can define phi so that's the module assemble and notice that it takes value. Well, this is the degree k minus two. Okay, I want to mention. And then I'll pick up the picture. Okay, so module assembles the group gamma with values in w of the ring over F. This is the ring of HECA eigenvalues of the form F. The universal for the L value is something else, right? Pi sub F plus omega F. So the split of this guy over here are the algebraic parts of the module assemble associated to F. And when you take L to get something similar, you separated the L values of the cost form into the even and odd multiplied by, which is one that Barry taught us very well back, module C, L functions are something that we all know have something to do with arithmetic. So we think of some of the arithmetic logic like an elliptic curve or something like that. We make an L function and that then is a way of encoding local function complex something to tell us something about arithmetic. But you can actually do arithmetic with the L values. And because it's what I'm most interested in, in this context is Isenstein congruence, which is something that Barry taught me a long time ago. And then can you say something about the associated L values? So this is another very old story and this is a typical example, standard example or first example is one that we've seen already in this conference is to start with this Q times some product one minus Q. Square, square. Yeah, square. The Isenstein series, this is modular. There's an Isenstein series but we're not at Isenstein series. Of way two, the one point, the cuss point is one-dimensional and the Isenstein series are also one-dimensional and this is this. Plus some sum, n equals 1 to infinity, some modified version of the sigma function times Q to the n. And the thing that one notices here is that the f is not obvious but it's true that the f is congruent as a Q expansion to the Isenstein series, modular 5. So in particular you'll notice that the constant term of the Isenstein series is the visible lifeline. And so that's a congruence between two analytic objects and the question is does that tell something about these L values are they congruent and the answer is yes, they are. And so, literally, but I'll say more about it in a second. The size of f, size of e minus also reflects the congruence between the cuss form and the Isenstein series and there are two periods. There are two. But for Isenstein series there's only one and it's this. Maybe e minus is in fact a tie-up. There's no even period. There's no even part of the minus and seven. And so we can't even quite make sense to frame this. Okay now, so this is something that's interesting because in fact, people like Rob Pollack and Preston Wake have been thinking about congruences even in the even case and they've all write down congruences between micro symbols associated to f and micro symbols associated to e but when they talk about the micro symbols associated to e, they're actually out of boundary symbols. Our things that only depend they're actually extended to homomorphism not just from delta zero to m but also to gap from delta to m where you don't take dividers at degree zero. That's much easier to make such things and there is an there are in fact elements of that thing that have the same eigenvalues as the Isenstein series does and so that's what people when they talk about those Isenstein series that's what they mean by the Isenstein series and it's very significant so the question then is what happens what do you do, what happens when you're working with an Isenstein series and you have you're thinking about the even part of an Isenstein series okay and so that's the question, what does that mean how do we make sense of that so let me just think about an example of the actual example that's all described as best I can I want to start with a cusp form this, let me say right in front that this is the question, the original question I don't know what, where is Romyard actually starting with this whole stuff but Romyard, certainly the simplest case of his theorem or his conjecture which is now theorem mostly the conjecture he gave a conjectural formula for besides the best plus in the case where p is an Isenstein prime in the case where there's a congruence like this and so that's really what I want to talk about so for me he's a conjecture, let me just stick I'm going to start with a simple case he says much more than what I'm going to say right now but I'll try to say something that's his full conjecture in a moment so start with not 2z of way to care 2 and p a prime it will always be greater than 2 for me prime such that pk is an irregular pair position that it should be an irregular pair basically says by the way I want the p p would be bigger not just than 2i okay I don't want to worry about small primes and so this just means that there's a congruence between this i and this cusp form I want to say this is an i form a Hecker i form Isenstein series there's only one space of Isenstein series of way k on SL2z is one dimensional and so I'll take a generator of that thing and suppose so e then think of it as s bk over 2k plus again some sum n equals 1 to infinity some sigma k minus 1 star of n q to the address is q expansion and I want the cusp form to be congruent to e modulo some prime p or p of some prime say in the Hecker algebra q generated by the Hecker algebra by the Hecker eigenvalue of s and I want f then to be congruent to e as a q expansion and that's going to happen then the only way they're going to happen is it must be the case that the constant term is infinity so p that is some prime above okay and so the condition that pk is in your regular pair I'm thinking of what I really want is this congruence of the cusp form in an ison cycle and the ison cycle so then okay and so to rock our game with a nice conjecture for what the value of the algebraic parts of the L value should look like in this situation so I want to just describe the e-bracket through the unity and invertity okay the units in our and so that's just some fp finite dimension of fp that just takes a bit more collectively that is and so okay so now we take the u and we write it we can decompose it I'm going to use the same notation I used before I equal 0 to p minus 2 u 1 minus okay sorry for the for the weird way of arranging this but by this I mean the part of we can decompose you in even spaces for the action of the Gallo-Augre of q zeta p over q so that's just some type of a character so we're taking this as a decomposition of the components on which the Gallo-Augre acts by the 1 minus i power of the type of a character okay 1 minus zeta p this is the element of our cross q 1 minus i okay so that's just some some element in the 1 minus i p so okay um and so the conjecture then is something like this let me get it exactly right so what we think about we take u tensor u of course this maps down to by comer theory that's where we want a product to h2 f up to u p tensor 2 which is k2 okay and then in terms of page it says there's a niceomorphism from there to a a tensor u p group of the field f the field of fractions r of course is not 0 the 1 minus k component of this a reaction the same definitions is not 0 and so tensor a minus or whatever i want to say this then is an element of u tensor u i'm thinking of an element of u tensor u and then i take i call this not this map row i take row of that r that is in this a tensor mu p 2 minus k okay it lives in there now under special conditions and this this is a effective vector space if you like it will be one dimensional in examples as always that seem to be one dimensional could be higher dimensional what i'm saying is is too weak but the point is that that i can say take a linear functional of this thing and identify a piece of it as sort of a line so i'm going to think of the row of that r as actually being an element of fp of the choice of the basis okay of the choice of some linear functional if you like and so rawnor's conjecture says that let me actually get it exactly right rawnor's conjecture says that if you take size of f plus appropriately normalize evaluate on the universal special value we will get modulo p sum over even r of k minus 2 minus r okay so it's a bit of a strange statement so i have to have this ability to think of elements of the class group as elements of fp okay but other than that i could normalize things but the statement is that when i take that out of value that's what i'm going to get so maybe you can identify coefficients sort of the coefficients of this modulo sample with certain elements built out of k2 it's a nice statement so this was just to be honest i don't know exactly what motive is i find it an amazing statement and so i had a student at that time her name was Cecilia Bufuyo and i said well this is a fascinating conjecture why don't you do some computations let's see if we can see it and so she did some calculations she first had to figure out where it all went so she took a couple of months i didn't hear from her for a while a few months later i got an email from her i bought nine o'clock one evening and she said i've done the calculations for these two examples i don't remember which ones and it's checked out so that was great so i went to bed and i woke up the next morning and i had another email from her at 3am that said and i can prove it i was kind of surprised her statement about autism might have been a little bit strong but the point was she saw the idea of what had to be do to prove at least something about i should say upfront that i communicated with ronkyar Sharifi and he said he just discovered the same thing so it seems like a simultaneous close to simultaneous i don't know exactly discovery now to be fair the element is he defines a bunch of symbols in terms of the theory this other way have a matrix gamma in sl2z then if i take gamma times 2 to 0 and i apply gamma to it that will give me some edge every edge gets a horizon in that shape for exactly one gamma well if gamma looks like this if it's something x, y and it's congruent to that mod p then the volume can read off what it is from the bottom row it's just 1 minus zeta px 1 minus zeta py this is the Steinberg symbol happening in k2 of the ring r so this what she said is in effect put that label the edge of the very exhalation that way then it's equivari that's actually what she proved and it's not too hard to prove because the mod n is the second word the part that's interesting is that it satisfies the mod n relations and not just a matter of writing down some particular relation in k2 of that ring which was not very hard to do she just came in and showed it to me it worked beautifully, it was amazing and so this is the mod u7 so mod n relation I'll leave it like that for now Chalice's conjecture much more than this so let me try to state a little bit more what he did what he actually conjectured is that this should reflect the mod u7 that we're writing down should be corresponding to the Eisenstein series so in particular this mod u7 should be part of the statement is that it's a heck of an item and that the item values are the Eisenstein series and that's the part that turned out to be really hard okay the evidence was there and you can see as the connoisseurs worked so it was right in the examples but Cecilia sat down and she tried to prove to go down what it meant for t2 that we have the right item value and she found some relations in the Steindor examples that implied that and so she proved that this is actually an item sample the operators t2 and t3 impressive Romyard did the same I'm not sure but I remember somebody saying they had a proof like this for t5 but I'm not remembering but the point is that proving that this worked in general that it was an Eisenstein sample was really hard for some reason okay this can make more conjectures anyway and so let me just say let hn plus be equal to the homology of the modular curve t to the n with values in zp and I'll take the plus part of the action of context conjugation and it's some level pain level so these guys are relatively prime in this case where you have this irregular where pk is an irregular pair the idea then is to understand what happens as you go up the modular psychotomic count and so the omega n is defined to be we can define some math hn plus the same a n but now it's the class group the p power part of the class actually is the p to the it's the class group on p to the n of the field fn as you go up the top of the psychotomic count okay this is a class group okay it goes to a n minus and I'm just making a remark that even in the simplest case proving that in fact it was not object and wait then does he actually say that hold on a second I don't actually have that statement written I have to say that if I take a limit close to some projected limit with respect to the norm math of a n minus this is the math that he describes as a nice remark so a bit of a mess to read at all this is part a and that part b he said more he built some sort of a theatic L function as I was saying to all of this which is I'm sure important but I don't want to talk about it now part of this what happens here is that Sharifi I'm not Sharifi but Foucailla and Cato with about this time came up with a proof of this trajectory much as I've written it down under the hypothesis that the Fumoto-Leopold pietic zeta function associated to the level n to the tamed level that they needed they wrote down an inverse to the snap omega and said it's the identity of the e-liberation the statement that this guy was a nice insight class even in the simplest case and so we did actually do some stuff and I think I might have enough time to jot some notes about what we tried to do and this is the first one this is the first one but it's not so hard even and so as double coset operators and so we'll try to use some representation some ring I'll call it R1 the sign entity is an example really looks pretty much the only ones that we use but the other one that you know there there is one the point is that these are power series in Q where Q would be the two pi IZ you can specialize Q1 and if you do that you might have to be fixing it so we can probably compute K2 of this ring you can remember in a way that had an action of groups on it and so the group GL1 you can imagine what it is so we first made a distribution Q taking values in what I'll call K1 of R K1 of R is just R1 cross okay what a distribution is but it's kind of what you think it is you make test functions on Q and you make homomorphisms from the group of test functions to this thing and that's the way of expressing sort of distribution related and then there is a unique minus sample delta zero to distributions on Q2 taking values in K2 of some ring which I also won't write down some ring of trace vector functions into her okay now this is not exactly right we had to divide by some ideal I can tell you what it is let's try to get something on the board this is an ideal actually the normal K ring of the ring R2 that's a graded ring of head logic graded pieces I'm just looking at the graded two parts and there's a part of it and then taking the ideal generated by what I call Q1 and Q2 Q1 and Q2 are the two variables okay that's a pretty trivial submodular submodular sample where there's this unique where Q1 is gl2q of a gl2q gradient and the other thing is that the phi I just have to tell you what it is I should be saying something like p pi1 I think of Q2 have two projections to Q when I'm taking a distribution on this flowing back by one of the projections and then taking some convolution product of that we're going to pull that according to the other projection whatever I've written down is true so far but this isn't what we wanted to prove we wanted to use this to prove that the thing she's written down was an ideal ensemble for Hecker and we never got back to that we wrote this down so I'm not even able to tell you how the idea was that we would be able to specialize this at a Q1 equal to certain points and be able to say that instead of the naked class out of this it's a head-to-height ensemble way upstairs and then specialize it's a head-to-height ensemble downstairs we never completed this project because it's about this time that Foucaillet and Kato announced the general so it seemed like a good place to study Foucaillet and Kato were doing okay so the idea let me just since I'm out of time let me just say that what Hecker did was then made out of Ziegler units and so specifically this was actually in my world this was doing a contour off the Ziegler units G0U back with G G0V and when I write this let me be careful it's U over N P to the N this is V over N then P to the N take the Stein for example so this is from an element of K2 but I'm going to think of it not living at level N but it's fine to think of it let me write it at level N gives an element of K2 here's the then make a make a minor assemble from H1 X1 N to the N Y1N N to the N then specialize and then they specialize and finish in other words they make a map you can specialize these guys to make elements of K2 of these rings write it down I would call the same type of R U let me just write it down where it ends up A N minus O Y so specialization you take this guy and so it's an element of K2 of this thing but you specialize it at the infinity cost all right infinity here you specialize that element of K2 and when you do that the idea is that specialization infinity sort of is an Eisenstein map so it respects the Hecker Arbiter's, the eigenvalues of those of the Eisenstein situation in fact of the boundary assemble in fact which is the thought that just occurred to me recently but because you're only using one cost that's the thing that's interesting okay so but the body assemble is made as follows looking again at the various specialization I take the U and the V it's given by I take the U and the V and I associate to the same thing I take a gamma U V this is an element of SL this is the element of SL2Z one minus O N T to the N there's something like this you might as well assemble to be at this path this new line is a path to be I'm going to stop except that there's one more thing that I just want to at least say the thing that I'm personally am interested in right now which I'm not able to, sorry to talk about is a related module assemble it's built in the same spirit as many people have seen this thing that Kudal had an ice paper and we should wrote down the module assemble taking values in a certain he didn't describe it as distribution theoretically and so our formulation of this is to make a modular sample of this type it's a distribution instead of Q2 use matrices of 2 by 2 matrices with rational entries that's a four-dimensional Q vector space so you talk about distributions on that four-dimensional Q vector space taking values in K2 of the module tower all the way up the tower and this is due to Von Schroff by the way he didn't state it but it's equivalent to this and also Bruno gave a different proof of Von Schroff's theorem with some extra restrictions but the statement is in the form that I like it is that you're getting a distribution on M2 of Q taking values of Q of the module power it's built not so differently from this except that you don't require a zero for the first coordinate this is just so that you can specialize that infinitely you use A and B so the first column of the matrix over there is AB and the second column of the matrix that I'm thinking about over there is CD and you get some big micro-sample I believe that this micro-sample deserves in some sense to specialize to the micro-sample until you're on a road down but more, I expect it to be a lot more than that let me just say a little bit more about it when you specialize at the cusps one of these one of these bailiffs in samples say you have got to worry about whether or not the binary units are holomorphic and holomorphic after cusps that you're specializing and so when you take, even though this thing I call it the concovalence in model sample, following in your notes suggestion but it doesn't if you want to say, well it's honest but when you try to specialize at the cusps make something Isenstein from the boundary you have problems because you don't know that you can specialize at all points you can say something called a partial we might call a partial larger sample I'm sort of tending to calling them pseudo larger samples because there's something about them that feels like we should be able to smooth them in the same way that you smooth periodic measures to make them bounded, things like that so I'm interested now in this relationship between these two things and I should say that my main interest is right now depending on these those are spooked to conjecture and so the Darmolda spooked to conjecture the Darmolda spooked to make a periodic measure for a conjectural unit above some in some great last field of a real quadratic field and can be described in terms of this thing that is an easy description of it but it's just another description of the thing and it doesn't seem to be that the distribution somehow needs to be enriched that's the field but if they don't enrich it somehow in order to make these models you never make a unit you have to get some arithmetic data out of it and so that's basically my interest right now is just to examine this particle valence and by the sample to see if it has enough information in it in order to recover in some concrete form of the conjectural unit