We are happy to welcome you to one more session of the Math Associates, and not from the ITP. Our speaker today is Ayel Pachukti from the Universidad Nacional de Córdoba, and he's speaking about modularity of some geometric objects, transmission, reasons, and the state of the art. Please go ahead. Okay, so first of all I want to thank Alejandra for inviting me to give this talk. I was told that this was supposed to be a talk mostly aimed for PhD students and postdocs, so I'm going to try to give some ideas and examples, and not so many proofs. Actually, I don't think I have any proof in my talk, so I apologize for that. So let me start with some simple example. So let's consider the projective line. So I'm going to denote it by P1. And I'm going to take P and E prime and N a positive number. And we know that if we want to count the number of points in P1 of F, P to the N, then such number is P to the N plus 1. I think everyone is more or less happy with this fact. I mean, we know that points are always of the form. We can choose the points of the form A1 and the point 0, 1, sorry, 1, 0. And then you have P to the N possibilities for the first case. And you have an extra one for the second one. And then there has to be a set of function. What you have, what you want to do is when you have a geometric object, you want to count the number of points over the different extensions, all extensions starting from the finite field where your variety is defined. And then you want to put all this information in some analytic function. Well, this is just, at first, it's just a formal power series. And then you want to see if you can extract information from there. And in this particular case, I mean, using this formula over here, we know that we can split the sum. We're going to have a term with the shape P times T to the N over N plus one term, which is just T to the N over N. And then if we use the formula for the logarithm, I'm just talking about formal power series. Then we can see that when you compute this theta function and it has only two times, one term, one minus T, which comes from this contribution, one term, one minus P times T coming from this contribution over here. And this is something that is related to base conjecture. I'm going to say a few words later. But this works for a particular prime. So this is like the first part of the story. And now what we would like to do is to consider all different primes at the same time. So this is the second baby example. What we do is we gather the information from all the local contributions together. And now we don't want this just to be anything, something formal, but we want this to be a natural function. So we have to evaluate and try to put some constraints so that this keeps a complex function. And so what we do is we consider the local pieces we defined before, and we evaluate the variable T in P to the minus S. And this is for conventions. And then we take the product of all primes. And I'm cheating here a little bit when I'm going to talk about the general picture. We have to be a little more careful. I mean, what happens here is that the projective line has good reduction at all primes. So this is why we can just do these without any extra hypotheses. And then remember that we had one minus T and one minus P times T. So when we replace T by P to the minus S, we get one minus P to the minus S times one over one minus P to the one minus S. And this is something that is well known. The first product corresponds to the zeta function at S and the second product corresponds to the zeta function at S minus one. And now if you study the properties of the zeta function, you know that this is a function which converges in some half complex plane. So with a half complex plane, I mean all points whose real part is bigger than something. And in which half plane it converges, it depends on which of the two parts you are looking at. So for example, the first one converges for real part of S greater than one. But what is really interesting of this zeta function, which is something that Riemann proved, is that it can be extended to the whole complex plane. And it satisfies some functional equation. And actually it has a pole at S equals one. That's why you can, the series doesn't converge to the left of S equals one. And it has preceded one. And the other part has a pole in shifting. And we're going to talk a little bit about these shifts and how they appear in different situations. But so what is important here is that you can extend it and you have a functional equation. This is the two important properties of this example. What happens if we want to go to a more general situation? So as I said, I started with the projective line, not the affine line. So we're going to look at smooth projective varieties. So these are given by equations in some projective space of dimension D. And I'm going to work over any finite field. And we can do something similar to what we did before. We start counting the number of points over FQ. Now Q can be a power of P. It depends where you're working. And then you look at all its extensions. So it's given by powers of Q. Then you count the number of points and you put this into this power series. And you look at the exponential. And then you ask what properties does this zeta function satisfy? Remember that in the previous case, it had a really simple expression. It was just a portion of polynomials. It was given by 1 over 1 minus T and 1 minus P times T. So you might wonder what happens in general. And so this is part of waste-based conjecture. So it has more or less three different parts. The first part of the conjecture is that it's given by a rational function. So by a rational function, it means that it's a quotient of polynomials. This part was proved by Borg around 1960. And note that it has some indices, the polynomials, P0, P2D, P1, P2D minus 1. So the even ones goes in the denominator. And the odd ones goes in the numerator. And each one of these polynomials have some particular properties. They're integral polynomials, so they have integer coefficients. And the zeta function satisfies a functional equation, something similar to what it happened before. And what's most important is that in each one of these polynomials, the roots have absolute value q to the i over 2. And here, let me explain just a little bit what I mean with this part over here. So this is something that is relating to different objects. So suppose you start with some variety that it's defined over some number field, for example, over the rational or something like this. Then you can look at its reduction and then you're working over a finite field over fq. But also you can look at it, for example, as a complex variety, and you can just compute some other comologies, for example, like the veticomology. And what it's telling you here is that if you compute the veticomology of your global object, then its dimension coincides with the degree of your polynomial. So recall that we started with P1, right? So it's the projected plane. So P1 has dimension 1, and we can compute the veticomology. H0 is 1, H1 is 0, and H2 is 1, right? So what it's going to say, this conjecture is that when you want to compute the theta function, it has to be a quotient, well, d is equal to 1, so you have P0, P1, and P2. Since H1 is 0, it means that P1 has degree 0, so there is nothing in the numerator, which is exactly what happened in a previous situation. Since the H0 is 1, it means that you need to have a degree 1 polynomial, and if we look at the third condition here, it says that the root has to have absolute value q to the 0. So it has to be the polynomial 1 minus t. And the last fact over here means that the P2 is also degree 1, and its roots needs to have absolute value q to the 1 to the first power, so then it's given by 1 minus p times t, or q times t, right? So the root is 1 over q, which has absolute value q. So what this base conjecture says is that you can read all the information, and it has this prescribed properties. Of course, knowing just the absolute value of the roots of a polynomial doesn't give you the polynomial in general. So the projective line is a very particular, easy example. But also, for example, you can construct this data function just by counting points, and in particular, you can recover the beta numbers from this formula. So it combines a combinatorial part with a geometric part, so this is really interesting. Okay, and then we can do something. Okay, so let me show you an example, and not so easy example. And as I said, most of the talk will be about examples. So let's start with an elliptic curve. An elliptic curve is just an equation of this form. It's y squared equals a cubic polynomial, and I'm going to assume the characteristic of our base field is not two or three, so I can write it in an easy way. It really doesn't matter. Now, if we go back to base conjecture, then we can compute the beta numbers of our elliptic curve. The h0 is equals to the h2, which is one, but the h1 is two. So what this means is that if I want to compute the theta function, then I'm going to have, well, two parts in the denominator, one corresponding to the h0 and one corresponding to the h2, which are already given by this information and the information on the absolute value of the roots. And then in the numerator, I'm going to have a degree two polynomial. Okay, and I know that the roots have absolute value q to the one-half, so the product has half absolute value q. And in particular, for the theta function to be given by this particular portion of polynomials, it implies that if I want to compute the number of points, then it has to be given by this particular formula, where alpha and beta are the roots of the polynomial appearing in the numerator. So this somehow, you get this for free from base conjectures, but you don't know which are the numbers alpha and beta. And so what you can do, know that, okay, know that if you call a q of e, then the sum of the two roots. Now, if you compute the number of points, you look at this formula here and it tells you that the number of points is going to be one plus q minus the aq of e, the sum of the two roots. And in particular, if you just count the number of roots, so the number of points of your elliptic curve mod q, then you know the number of roots over any extension of fq. So you can recover, it's not that you can only recover your theta function, but you can also give a formula for the number of points over any finite extension of fq, okay? And of course, the way to prove it is going all the way around, right? You start counting points and you prove that the formula holds, and you can see from how I'm doing like reverse engineering. So once I know what the expected results should be, I can show you how you get this information. And I want to stress something important here is that for computing the whole theta function, you only need to count the number of points of your elliptic curve over fq, okay? In general, if your polynomial has degree, has higher degree, then this will not be enough, right? Because in this case, we want to compute two numbers, alpha and beta. And the point is that we know that the product has to be q, and we're computing in sum. So since I only have two numbers, if I know the sum and the product, I can know the two of them precisely. But if I have, I don't know, ten different roots, then I need to start counting over many extensions, and then these computations becomes extremely challenging from a computational point of view. Okay. I don't know if there are questions if I'm supposed to ask for questions, or I should go ahead. Alejandra? Go ahead. Okay. Okay. If there is any questions, just post it, and Alejandra will let me know. Yes, anything. Okay. So what do we do now? So we just do the same thing. We define the global has a bail, a theta function. And what we do now is we just take the product of all the local contributions and we do exactly the same thing as we did for the projected plane. We substitute variables at variable t in p to the minus s. And here we have to be a little careful with the primes where our smooth projective variety has bad reduction. Okay. So this is something that I'm not going to talk about it, but you have to think that if you, if you start with a variety, then there are many different instances that may occur. For example, it might be that you start with an equation which gives you something which is a singular when you reduce mod p, but it is something, but the issue is that the question is not good enough. Enough. You might take a change of variables and get a better equation where now your curve or your variety has good reduction. So this is something that might occur. And so you have to be really careful with this. But at least for all the primes of good reduction where, where you take your equations and you reduce mod p and you get the non singular variety, then you do precisely the same thing. Okay. And now what happens is that now the theta function, it converges for some half plane. Okay. So the situation is pretty much the same as it happened with the primes theta function. And it comes from the fact that we know base conjecture. And so, so we know on these parts here that they appear different polynomials and we know the absolute value of its roots. And this will give you, give us enough information to be sure that this converges in some half plane. And so the conjecture is that what happens with this theta function and we expect all of them to extend to a metamorphic function on the complex numbers. And we also expect it to satisfy some functional equation. Okay. So remember that here we have to be a little careful. I'm being a little vague here. For example, when we computed the theta function of p1, remember that it was given as the second function of s times the theta function of s minus one. Okay. So when we saying a functional equation, so, I mean, I cannot put the function equation for the two pieces together, right? I mean, each one satisfies a different functional equation, but still they do satisfy functional equations and the functional equation should be something that has to do with the different in the comology degrees that appear. So I will come back to this later. Okay. So let's go back to elliptic curves. If I start with my elliptic curve, then I can compute the theta function and remember that when I computed the theta function of the elliptic curve in terms of t, then it had a part which was like one minus t and one minus p times t. And in the numerator I had a degree two polynomial. Okay. Do you want to add something Alejandra? No. Okay. So in the numerator I had a degree two polynomial. So these two parts here will give me again the theta function at s and the theta function at s minus one. And the part coming from the degree two polynomial is what I'm going to call the L series of the my elliptic curve. So by definition, it is given by the product over all primes, just the good ones of one divided by one minus a p of e times p to the minus s. Remember that I had a t here and then p to the one minus two s because I have p times t square. Okay. And this converges for the real part of s greater than three halves. And these three halves here comes from the estimate that the absolute value of a p of e is less than a two square root of p. So this is has a spell. And now the natural question is that so I know that the two terms in the numerator, which comes from the theta function, I know that they do extend to the whole complex numbers and they satisfy the functional equations and this is due to Riemann. And what happens with the denominator? Can I say or prove that this is a series extends to the whole complex numbers? And furthermore, I mean, is it automorphic or it has poles because for theta function, we expect poles to appear. But what happens with this series? So the story goes somehow this way. If I start with a curve which has complex multiplication. So complex multiplication means that the automorphism rings. So all the maps from the elliptic curve to itself is bigger than just multiplying by integers. Then what happens is that the series in this case is related somehow to something similar to the theta function, which is you have to add a character, a character. And it factors as the product of the series of two characters, one character needs conjugate. And what happens in this case is that you get the extension from the extension of the series of the character. So I think this is due to Heke mostly, but I don't know the precise reference. So using a similar proof than that given by Riemann, you can prove that if your curve has complex multiplication, then the series satisfies expected properties, but it is automorphic. This is something that happens with any character. So if you start also with a Dirichlet character, then the series extends automorphically. It doesn't have a pole. The post comes from the theta function, not from the series given by Dirichlet characters. Okay. And this will happen for high dimensional things, but if the curve doesn't have complex multiplications, so these ideas of Riemann are not enough. We really don't know how to extend this series or how to prove anything out of them. So I want to change subjects a little bit completely, and I want to mention what we know how to do. Okay. So I'm sure most of you already read something on this, but let me just recall what is a modular form. I'm being baked all around the talk, so forgive me for that. So I'm going to take K to be a positive integer and a weight K modular form. It's just a form from the upper half plane. Okay. So the complex numbers with positive imaginary part to the complex numbers would satisfy some functional equation, which is given by if you take a matrix A, B, C, D and S2 of C, we call that S2 of C at two by two matrices with integral coefficients and determinant one. Then this group adds by a omotetes, I think it's called. So sending C to AC plus B over CC plus D. And what you want to do is when you evaluate the function F at an image of an omotete, you want it to be related to the function to the value of F at C. Okay. And it changes by CC plus D to the K. So the story of these functions come from then to understand all of the differentials on quotient of the upper half plane. And for, I just wrote it for S2 of C, but we'll also be interested in subgroups of this, like gamma naught of N or gamma one of N, where gamma naught of N, you ask, you impose the extra conditions that the place two one is divisible by N. See, it's a little smaller subgroup. So it doesn't matter. And also you have to impose some olomorphic condition at CASPs. Okay. So I will say just one word about this. So if you look with this definition, if you look at the matrix 1101, it acts sending C to C plus one. And this part here is trivial. So it means that your function is invariant and the translation. And in particular, it has a fully expansion, which is given by some sum of powers of e to the two by I N. If you take another subgroup like gamma one of N, you have to be a little careful and then something may appear in the denominator, but it's not important. And the condition of binolomorphic implies that I want the sum to start from zero. This is the definition. Okay. And for most of the things I'm going to say, I will only consider forms which are called CAS forms, in which case you also impose the condition that the a zero is equals to zero. Okay. This has to do with the olomorphic differential being olomorphic at the CASP, but I don't want to get too technical. And then to a modular form, you can attach what is called the melin transform. It's given by a transform, but it's not important how it's defined. It's an integral that you have to take a convolution with some exponential function. But the result is what's called the N series of the N function, which is given by the same sum. Note that the coefficients are the same ones as I had before. I start from one because I only consider CASP at once, but then you have N to the S in the denominator instead of an exponential. And again, you can prove that if you, your form is a modular form of weight K, then this sum converges for some half plane. Okay. Real part of S bigger than one plus K over two. Okay. I don't want to, to give all the definitions, but there's something important that you, you have an involution in this space. It's roughly speaking given by the matrix zero minus one and zero. If you look at the level gamma zero of N and it's, since it's an involution, so it squares the identity, then you can split the space between the forms which has eigenvalue one and eigenvalue minus one, or you have to add something if you look gamma one. And then looking at, if you start with a form which is an eigenform for this involution, then you can prove that it's a series extends to allomorphic function on the whole complex plane. Okay. And of course, since if you want only extension, then you don't need to be an involution because any form can be written as sum of eigenforms. So if you can extend the eigenforce, you can extend all of them. But the important part here is that it also satisfies a functional equation, which is what we are expecting coming from the Riemann theta function. Okay. So extension holds for all of them. But if you want some functional equation, you need to stick to some particular ones. Okay. So this is something which is somehow related to Riemann's theta function because you have something which is like a Dirichlet sum. And then you know that you have functional equation and you have that extension. It can be extended. Okay. And so what is the goal? The goal is to try to relate these analytic functions with geometry. So in particular, if I can prove that if I start with, I don't know, something like an elliptic curve and the series matches the series of one of these forms over here, then I get extension and functional equation for free. So somehow the analytic side propagates its nice properties to the geometric side. And this is somehow the goal. So let me go back to elliptic curves. So I didn't write it here. K is going to be equals to two. So suppose we start with an eigenform for the headquarters. It doesn't matter too much what it is. I would say a few words. And let's suppose that it has rational free expansions. Okay. So remember that when we started with a modular form, you can write it as a free expansion. A n times e to the two pi i n. So I'm asking for these numbers here to be rational. So you can always normalize your form so that all these coefficients lie in a finite extension in a number field, the finite extensions of q. And I'm asking them for precisely be rational. And in this situation, then there is a construction due to Eichler and Shimura. So it comes from an old construction of Abel and Chakobi that to a modular form, you can attach an elliptic curve. Okay. And this is pretty explicit. It comes from how the complex points of an elliptic curve are at Taurus. So they have genus one. And so what you do is you just compute this by integrating some periods against the differential form, the olomorphic differential coming from your modular form. And when you compute this integral, it gives you a period lattice. And then the elliptic curve is just the quotient of the complex numbers by this period lattice. And an important property that Eichler and Shimura proved is that the L series matched. Okay. So they prove that you have to be very careful with the price of bad reduction. But that's still okay. And what these Hiccup readers have to do. So how what happens is that you have this nice modular curves, which are the quotients XC of N, which are just the quotients of the upper half plane by gamma naught of N. And this is a curve. So you can look at this Chakobi and when you look at the Chakobi and it's a variety of really high dimension. But what happens is that this Chakobi has a lot of endomorphisms appearing on them. And so here you have something acting, which is what is called the Hiccup readers. And so since you have all these endomorphisms in your variety, then you can try to split it into different pieces according to these endomorphisms. And what happens is that this part here is a search in us to products of different billion varieties, which are most less useful. And your elliptic curve appears somehow in this way. So this is where the Hiccup operators appear. And as I said, I don't want to get too technical. So the point that here is that you have a rule that if you start with an eigenform with rational Fourier coefficients, you can attach to it an elliptic curve. And then the Schumann-Tanijama conjecture or questions, but it was a conjecture, is whether all curves or rational curves appear in this way. So is it true that you can construct any elliptic curve from modular phones? And so this has a lot of advantages. Of course, one of them is that we can translate since DL series matched and we know that the left-hand side can be extended, it has an elliptic continuations. We get the same for the right-hand side for free, but also it has other implications that maybe they're not so trivial. What happens, for example, if you want to make tables of elliptic curves? So if you want to make tables, you want to order them in some reasonable way. There's something called a conductor.  And somehow the point is that computing modular phones is easy.   And so in particular, if this is true, you can always also prove that you have complete tables of elliptic curves, which is what happens, for example, in the LMFDB where we're looking at all Cormona's tables. And the answer to this question is yes. And this is pretty much the theorem of Weiss and Taylor Weiss. And what they proved is that this is true for semi-stable elliptic curves, but this will later generalize to all elliptic guys by a real Cormon and Taylor. And in particular, the Hasselbeckon texture is true. The series of an elliptic curve does extend and it satisfies a functional equation. So what happens if we want to generalize this strategy? So we started with objects, with geometric objects. We only consider genus 0, like P1, genus 1, like elliptic curves. And we didn't do anything else. So if we want to generalize these results, we can go in two different directions. So somehow the point here is we have an analytic part and we have a geometric part. And the question is when we want to generalize some, in one side some computations are easier than on the other one. For example, we can instead of working over Q, we can say what happens if I take an extension of Q. So on the geometric side, then it's clear what we have to do because we start with a schema of a variety of a K, which has some points. So it's clear what we're doing. But what happens here? What should we put instead of modular forms? So if we want to change the base field, the analytic side is a little harder to describe. So it's convenient to talk about the modular forms in the automorphic language. And what you have to do is you have to look at representations of some groups, of some redacted groups. And in the case of elliptic curves, they correspond to automorphic forms on the other groups of Q. And then once you understand this, which I think it was a huge step in history, once you understand what you're trying to do, then if you want to increase your base field, then what you have to do is you look at automorphic forms or automorphic representations and you just change the Q by K. But it's not completely clear from the picture of modular forms. But I mean, it comes from, you can look at the upper half plane as a quotient of GL2 of R, and then this is more or less where this GL2 group appears. And so what do we know in this particular situation? Well, there are some interesting results during the last couple of years if we have a field K, which is totally real. So totally real means that in turn, you can look at any field, you can write it as adding some element to Q number fields. And then what you're asking, this is going to be root of a polynomial and you ask all the roots of this polynomial to be real. This is what a totally real field means. And if it is quadratic or of degree three, then we do know modularity of ellipticals. So for real quadratic fields, this is a theorem of Freitas, Leung and Sechsek. And for cubic fields, it is a result of Derrick, Nachman and Sechsek. And this is from 2018, I think, or something like this. So has a conjecture R true, but if we take a totally real field or a CM extension, a CM extension is a totally imaginary extension of a totally real field, then what we do know is that any elliptic curve over K is what's called potentially modular. So the notion of potentially modular, I think that came, I think this was an idea of Taylor. And it means, I don't want to give the precise definition, but it means that it is attached to a modular form, not over K, but over some field extension and discussed to do with how you prove modularity. You don't get risk of modularity otherwise. But there is a really nice result and all results of Braga that implies that if you prove that something is potentially modular, then you can prove extension of the Hasse-Bael theta function. But the part that you do not get, you do not get allomorphicity. You only get that you can extend it in a meromorphic way. You do get functional equation, but you might have poles. So the idea is that you write your Hasse-Bael function as a portion of different Hasse-Bael functions. Each one of them are allomorphic, but the denominator might have zeros. So we do expect things to be allomorphic, but in this case, we only get meromorphic continuation. Okay? And this is... So the CM case, it's really hard one. It's also from the 2018, and it has many authors, so I'm going to mention all of them. Okay. And if you want to look at imaginary quadratic fields, then you can somehow try to prove modularity for each given case. So this is something that Cremona has been doing, computing tables of Bianchi modular forms. And what we did with Luis Dulefe and Lucio Geberov was to prove that most of them, many of them are modular, giving an algorithm to check modularity, and then you can replace meromorphic by allomorphic. Okay. So let me talk a little bit about what happens when we try to compute the data function. So we're going to increase the genus. So if we start with a genus 2 curve, and we try to compute the data function, then the formula is going to be similar. We're going to get two data functions in the numerator, but the denominator is harder. Now it's going to be a degree 4 polynomial. So it's even challenging to try to compute it. Okay? It's not so easy, and it takes more time. The way to think about it is that when you start with a curve and you look at its Jacobian, it's a surface of dimension, the genus. So it's going to be an ability of surface. And the modularity of the Jacobian was conjectured by Yoshida, and Brumer and Kramer somehow gave a refined conjecture, where the main contribution was to give a specific description of the level of the modular form you are searching for. In this case, you're searching for signal modular forms. So you can think of this as automotive representations of GSB4, or you can think of it as functions in some generalized upper half plane. I don't want to get too much into details, because I want to say a few more things. And so they gave a recipe for the precise level where this should appear, and they also made many computations on surfaces giving a list of the ones that appear for the first values in the semi-stable case. And so more or less the idea is, the idea is the same as I mentioned before, you have an analytic object where you know that the series can be extended and it's a very functional equation, and you have a geometric object, and somehow you want to match these two ones. And the way everything starts is that if you know where to search, you can just start playing with examples. So you compute some of the surfaces, and then you go and try to search for the automotive forms and see whether they match or not. So there is a big issue here, which is that all these forms are going to be of weight two, and automotive forms for GSB4 of weight two are non-comological. So this is like trying to compute weight one forms for the ones who know this. And the problem is that they are extremely hard to compute. I mean, you cannot just desolate some variety or something and try to compute in this way. You need extra tricks, and it's really time consuming. So computing the first 100 free coefficients might take a couple of weeks. And what we do now nowadays is that all ability surfaces are potentially modular. This is a result of a boxer, a Gallegari, Guillain Piloni from, I think, 2018 also at the end of 18. And what we did in this joint article with Bruma, Myself, Puer, Tornaria, Boyd and Yuan is to give a recipe to adapt an algorithm, which is faulting search method, to prove modularity of ability surfaces. So if you give me an ability surface with some, not so restricted hypotheses, you need the image of the residual representation at two to be not too small. And you give me a candidate for the automorphi form to prove that the two series actually do match. Okay. And for cups of cheese greater than three, then not much is not. Okay, so this is really a huge jump and there are some reasons. There is a post by Frank Gallegari explaining that nowadays techniques are not enough to do this. Unless the Jacobi has many automorphisms. Remember that if we started with an elliptical with complex multiplication, then it comes from some Hecke character. So in this case you get modularity, but because it's not something genuine, so somehow it's something weird. Okay. And the extra five minutes I have or so, I want to talk about some other examples that appear and I think it's interesting and maybe interesting for the audience particularly. So as I said, I don't want to be too technical in anything, so I'm not going to give the precise definition, but I'm going to say what a Gallegari variety satisfies. So a Gallegari variety of dimension D, what it satisfies is that when you want to compute the dimension of the homology groups, then you have a Hodge decomposition. And what it satisfies is that, so when you do this Hodge diagram, then all the elements in the orders of your diamond, the dimensions are zero. Okay. And the middle one is going to be one. So I'm going to talk about examples mostly. So for example, if you start with a one-dimensional Gallegari variety, then it's going to be something like this. I mean, this condition here imposes no condition at all. D is equal to one, so there is no integer between zero and one, so it doesn't impose any condition. And this condition here imposes that the H10 has to be one. So the Hodge diagram is precisely that of an elliptic curve. So these are precisely the elliptic curves. Well, if you have a two-dimensional Gallegari variety and you look at the Hodge diagram, then you have to put here zeroes. This is the first condition. And the second condition is that you have to put here ones. And, well, the 20 comes from some criteria. It doesn't matter. And note that. So I would like to compute the theta function of this Gallegari variety and see whether it extends or it doesn't extend. And so I'm playing here with something related to the proof of the base conjecture, which is there is a relation between counting points and computing comology. So we already mentioned the beta comology. Now this has to do with the tau comology, where representations appear. And so your theta function is going to, these two parts here are going to give you the terms set of s and set of one minus s in our previous example. And then this part here is somehow like the new interesting and hard part to compute and where we don't know whether they have a better function extends or not. And what happens that the non-severic group, so this comes from cycles inside your variety modulo equivalence.  So this part here is the H2. This is H0, H1, H2, H3 and H4. And you have a duality. So that's why it's symmetric. And these contributions, you know that all these parts there contribute something which is like a theta function. It might not be divine over Q, but it is like some extra theta function factors. This is called the algebraic part. And then you have something new, which is called the transcendental part. Okay. So if you never saw this non-severic group, it really doesn't matter. But if this non-severic group has maximum rank, the maximum number it can get is 20, then know that the remainder part, which is something that we do not understand, it is two-dimensional, right?  I have 20, then I have two remaining. And then when I look at this color representation, then I'm going to have a 20-dimensional part coming from the Neuronsseveric group, which I know it extends and satisfies some functional equation. And then I have this extra two-dimensional part, which is like the weird one. And if I can prove that this extends to the whole complex plane and satisfies the functional equations, I get the whole Hasebel conjecture for free. And in this case, so this is something proved by Leibniz, this two-dimensional part is actually a modular. It comes from a modular form of weight three, and some Neven typos. But nowadays we have something even better. We have something which is called the SES conjecture, which says somehow that any two-dimensional representation over the rationals, with some minor hypotheses, comes from a modular form. You need it to be... even the determinant... the determinant at complex conjugation has to be minus one. You need it to be odd, sorry. And you see... so for this particular... these ones, which are called extremal Calabiyao two-dimensional, when the rank is as big as it can be, you also get modularity. Okay? So again, you have something which comes from an algebraic part, which is some sort of a theta function, and then you have a remaining part which comes from a modular form. So you see this is really interesting because the whole picture here, this is what I was saying before, when you want to count the number of points, you have some parts coming from theta functions and then the middle ones, I mean, it depends on the dimension, then it might be harder to understand, but sometimes this... it splits as different parts and some parts are easier to understand. Okay? And also... there are... there are some results of Lybne, Matthias Schultz and Yui, which prove that for some other smaller ranks, the modularity is also known, and what happens is that... I mean, if you have... as I said before, if you have a four-dimensional representation, it's not clear whether it should be modular or not, but if you have something which has many endomorphisms, then it should come somehow from a Hecke character. So what they prove is that in many instance, these L-series come from Hecke characters, so we know the whole picture. And let me talk... I'm almost done. Let me talk about three-folds. So before I was talking about Calabria's surfaces, so dimension two, and I want to talk about dimension three. So a Calabria three-fold, it's called rigid, if the H2-1, so this number here is zero. Okay? Note that if this number is here by symmetry, this other number here is zero as well. So it tells you that the middle part, which is in general the hardest one, it's going to be two-dimensional as well. This H1-1 somehow is algebraic, it comes from cycle, so you do know that it has some analytic continuation or functional equation, but so what happens is that when you have a rigid Calabria three-fold, then the H3 is going to be two again. And so, again, we get that it is modular, and it comes from a wait for modular four. And this result was mainly proven by Julefe and Majoharmayoum, but again it follows from seskin checkers nowadays, so this result is older than seskin checkers. Okay? And let me go back just one second. Note that when you had the surface, the weight was three and now that we have something of dimension three, the weight is four. So this is one plus D in general. Okay? And for some non-reached Calabria three-fold, what happens is that sometimes so non-reached means that these numbers are non-zero and then this part here might be the dimension of this might be quite large, but there are some instances or many examples in the literature where the H3 representation is reducible. So somehow it decomposes at smaller dimensional parts and then so you might expect to prove modularity as well. So there are some really interesting examples of Bernet and Bangui examples on this sort of phenomenon. Let me finish with another example, which is the Kontani-Scholten-Kintiq. So you look at this polynomial, which is HHF polynomial into variables. And then you look the Kintiq given by p of x1, x2 equals p of x3, x4. The question is not so important. You have to desingularize this, like I think it has 121 singular points or 120, I don't remember exactly. So you have to desingularize this and then you compute the Hodge diagram and know that this is huge. Okay. So this has to do with what I said before, this number 141 here is telling you that some polynomial, which we have degree 141. So if you want to compute the theta function, it's extremely expensive. Because even if you know that the dimension of each different piece, you need to count the number of fp fp square over many different finite fields to be able to deduce precisely what are the polynomials. So this is also a computational challenge to try to compute the theta function of these guys over here. And the interesting part, the H3, so somehow this guy here is algebraic. So this guy here, it is H3. This is four-dimensional. So you have a four-dimensional representation. And what Konsenial Schulten realized is that when you look at this four-dimensional representation and then you restrict this to q square root of 5, the other group of q bar by q square root of 5, it splits as a direct sum of two two-dimensional ones. So now you're in a nice situation because it is something of dimension two, but over a quadratic extension. And what we proved that it actually comes from a modular form, but it is a Hebrew modular form and the way it is 2 and 4 and the level is 30. So again, since it's related to some object, so this is again a case where you have a four-dimensional representation but it's not as big as it can be because when you go to a quadratic extension, it splits as two two-dimensional ones. So we proved that it comes from a Hebrew modular form and so again, you get that this function satisfies the conjecture. Okay, that's everything. Thank you. Thank you so much. So are there any questions? You can either unmute yourself or write it on the chat. And in the meanwhile, I'm going to also send a poll so that you could answer before what you think about your questions. So I ask a question, but so for example, in the case of the K3 surfaces can you say something about the ophthalmorphisms when you have the modularity or when you expect the modularity, can you say something about the ophthalmorphisms of the K3 because it depends on the lattice, so I don't know if that's the thing that comes to my mind. So that would be my question. You mean the K3 surface? Yes. So when you say this path here, so I think so I didn't look at the proof of these ue and chute these results where they get the CM forms, but what happens in many instances that your surface, you look at the endomorphisms, I mean you see for example in this equation here you see it's pretty symmetric, right? So you know that you have a lot of endomorphisms already appearing there and so sometimes what you try to do is you try to see whether you have enough endomorphisms to ensure that your variety should come from something which has complex multiplication. So I mean the way to prove this is that so they give some conditions for the ranks of these endomorphisms, there's a very group to be exceptional and so on and in one of these cases they somehow present families and they prove that any such variety belongs to one of the families and then they study the families on themselves. So it has to do, I think this is what you were asking, it has to do with equations. You have to know the equations to be able to say something. So I don't know if there is any theoretical information that from extraordinary endomorphisms, but I think what they do is they just compute the family and they do it on the family itself. Okay. Thank you so much. Are there any other questions? So if not, we thank Arielle again.


Welcome to toffee TV. It is the Everton Newsday the Everton have been linked with a move for Barcelona centre back Ronald Aruku the 22 year old Yoriguayan as Apparently just rejected a new contract with Barcelona a pretty big contract as well It's out of contract next summer. There are a lot of Premier League clubs looking this player and Everton are apparently one of those clubs. He's played seven times for Yoriguay Made this big breakthrough this season into the Barça team, but has rejected a big contract so Everton obviously in the summer will probably be in the lookout for For a centre back a long-time successor to Yeri Mina who's Really going into the final year of his contract You know his future looks like it's going to be elsewhere ever and probably won't be offering a new contract and Everton will be looking at players. So this is the latest one. We've been linked to that was by the athletics So it's actually a reputable link reputable source Yeri Mina himself is He will not be having surgery on the injury that he sustained against Newcastle He's still likely to be out till probably the back end of April But ever will not be having you know put him through that surgery Which obviously means probably he would be out for the whole of the season if they did have surgery. So Well, after waiting to see if anything comes of these links a place being linked with a move away from everton is Jean-Philippe Gabam and he's been linked with a move to CSKA Moscow Now they transfer window doesn't shut till the 22nd of February so next week So he played last Friday for the under-23s and to get some fitness So we'll have to wait and see what ever to prepare to do and whether that you know Whether they look to maybe get him out on loan for the rest of the scene obviously evidence We feel they still pretty like with the likes of Fabian Delph Still not back Tom Davis who's out for the rest of the season Decorate is only Maybe a week away or two weeks away from playing so we'll have to wait and see on that one But we'll have to wait and see what everton decided to do whether obviously this this is a genuine link For the player who obviously played a handful of games since signing for everton obviously had a couple of really Bad injuries as well, so we'll have to wait and see what happens with that Evan have announced a couple of fixture changes the game against wolves at home is now sunday the 13th of March with a two o'clock kickoff and The game against Newcastle which was which has obviously been rearranged from over Christmas There's now Thursday the 17th of March with a 745 kickoff at goodness and park now Everton are due to play on the Saturday against Wofford, but if everton were to be bored and wood Then that game would be the quarterfinal of the FA cup So that game would likely be a sunday on Monday now And it's it is slightly disappointing everton are having to play on a Thursday only three days before a potentially huge quarterfinal game If everton do be bored and would of course Which you obviously have to wait and see whether that happens or not, but it is it is disappointing, but It is the way it is and we have to get on with it It is set in stone now So the 13th of March two o'clock kickoff for walls and the 17th of March 745 kickoff on a Thursday for Newcastle and talking of a boring wood apparently both of the Kits the home and away kits clash with everton's home kids and Apparently everton are prepared are prepared to pay for a commemorative kit to be made for The for the team for the fixture of goodness and park So that's that's nice nice to hear nice to hear not sure what that Color that kit will be but there you go. They are prepared to pay for born and was to have a kid made Yeah With the charity once again as our ticket prices go up. That's okay. That's okay. Say none There you go. Make sure to check out our Match preview for tomorrow's game at Southampton and our starting 11 show which is out now as well There's lots more exclusive content over available on page and link is in the description and it's on the screen Right now check that out with you daily live shows over there as well. Thanks for watching. See you later


We're in James chapter four. We're gonna be looking at verses seven through 12 as we continue our verse by verse traveling through and going through the book of James, our exposition of James. James is dealing with situations now in the church. Chapter four, we'll be looking at that, but we've already looked at it, the first several verses by introduction. And there are what are called church wars. There are problems that are taking place that in this portion of scripture, James has to begin to address. He's addressing, according to verse one, wars and fights that are occurring in the church there. And the church is really that he's addressing in the book of James. And so that's what we're gonna be looking at because I'm gonna be laying the foundation by speaking about the context of division, division in the church and all, and we'll be looking at that in just a moment. We'll be looking today at verses seven through 12. And again, chapter four is dealing with conflicts and wars that are taking place, church wars that are taking place amongst the believers. So beginning in verse seven and reading to verse 10. James writes, therefore submit to God. Resist the devil and he will flee from you. Draw near to God and he will draw near to you.  You double-minded. Lament and mourn, weep. Let your laughter be turned to mourning, your joy to gloom. Humble yourselves in the sight of the Lord. He will lift you up. And so James again is dealing with strife, division. He speaks of wars and I mentioned in verse four when he said, where do wars and fights come from? I mentioned to you that the word wars can also be spoken of as quarrels. And when the word fight is used, it can also be translated strife. So where do the quarrels and the strife come from? And so this has begun to be commonplace among the churches that he's writing to, quarrels and strife. Now that kind of church life reveals carnality. It reveals spiritual immaturity. But it's something that occurred in the early church it had to be dealt with. When you begin to read your Bibles and you start looking at the book of Acts and you start going from chapter one and just flow through the book, you're gonna notice that the church was birthed. It came to be in the second chapter of Acts. It records how that on the day of Pentecost that the Holy Spirit fell upon 120 who were in an upper room there in the city of Jerusalem. And that they were baptized in the Holy Spirit and they flowed out of that upper room. And they went out there in the area surrounding it and began to speak in languages they hadn't learned. They were speaking in tongues. And as they came out and they were speaking in these languages, there were people who were beginning to be drawn and they assembled and began to mock them and they began to say things about them including, oh, they're filled with new wine. And you remember how that the apostle Peter stood up amongst them. And he said, men and brethren, he said, these men are not drunk as you suppose. It's too early in the morning, the bars haven't opened. He says, but this is that which was spoken of by the prophet Joel who said in the last days, I will pour out my spirit upon all flesh. He began to preach out of the Old Testament scriptures given an explanation for something spiritual that was happening. And as he did that and as he proclaimed this message of the gospel, people began to listen and many were saved. Acts chapter two verse 41 tells us that around 3,000 souls were saved that day. And so you have 3,000 or so brand new believers in Christ. And so in verse 42 through 47 of Acts chapter two, we're told that the new believers continued steadfastly in the apostles doctrine and fellowship, the breaking of bread and in prayers and goes on to say the result of it was that many people came to know the Lord Jesus Christ. According to Acts 247, the Lord added to the church daily those who were being saved. So they began with 3,000 but on a daily basis people were being added to the church. And so you see that in the original church in the first days of the church, the church began to explode. When you get to chapter three, the third chapter of Acts records how that Peter and John performed a healing on a man who was crippled. He was there at a gate that is called beautiful, the beautiful gate. And Peter and John were about to walk in and the man looked at them as Peter looked down at the man and scripture says that Peter said to the man, look at us. And the man looked up expecting to receive something because he was there at this particular site because the pilgrims who had come in would be generous to those in need. And they would give what is called an om.  And so he looking up expecting to receive something was I would assume at first a bit surprised that the words of the apostle when the apostle Peter said silver and gold have I none but such as I have give I thee in the name of Jesus Christ they say to you rise to your feet and walk. And the Bible says that Peter reached down and took the man by the hand and began to lift him and the man received strength in his ankles and throughout his legs was able to stand. And he began to walk and he began to leap. And the Bible says and he began to praise God. And when this miracle happened because this man had been there for so long and people knew him. A crowd begins to gather around and once again that gives the apostle Peter opportunity to speak to them concerning Jesus Christ. Men of Israel why do you look upon us as if this miracle was done because we're good people is how he goes about speaking to them. He said this was not done because we're good people. Let it be known to you that it's at the name of Jesus Christ whom you crucified. It's by him that this man has made completely whole. And as he began to preach once again sharing with them about what Jesus has done. The result was that this crowd began to listen and according to the book of Acts in chapter four verse four many who heard the word that was being preached believed and the number of the men came to be about 5,000. And so now you have an initial 3,000. You had people being saved. Another 5,000 apparently are added and the church is exploding. Miracles are being performed. The church is growing but in the midst of all of that opposition begins to exist. And now the apostles are being persecuted. They were arrested, they were beaten, they were forbidden to speak in the name of Jesus Christ but instead of closing their mouths they began to rejoice that they actually were suffering for Jesus Christ. They didn't stop them from preaching it energized them to continue to do so. You see the devil did not succeed by inspiring persecution by an attack from the outside. So if you can't beat them, you join them. And so in the initial stages of the growth of the church from chapter two, three, four, five now you enter into chapter six and there's a division. A division begins to occur between cultural Jews. Those who were from Israel and they were culturally Jewish and what they called the Hellenized Jews are the Greek speaking and culturally Greek Jewish people. And they were in the church and there was a division that began to occur. It was related to how widows were being treated differently. The Jewish widows were being treated differently than the Hellenized Greek Jews. And the Hellenized Jewish widows were being neglected in what it was called the daily distribution. And because of that that's where you begin to see a little division start and the church at that time actually was led to form the deacons who began to serve in the church leadership. The apostle said it is not right that we should leave the word of God in prayer and wait on tables. It wasn't because they were too good to wait on tables. It was just that this is not our call. What we need to do is we need to be in the word and we need to be in prayer. We need to spiritually lead but we need help. And that's how the deacons and they chose out from amongst themselves seven men filled with wisdom and the spirit and all who were able to do the work. But it all began because division occurred and that was very early in the church. All the good things that were taking place people being saved, miracles that were occurring even though persecution came. The fact of the matter is when the persecution came it only strengthened their resolve to remain faithful to the Lord whether it is right for us to stop preaching in his name you decide but as for us we ought to obey God rather than men was the response of the apostles to that and so what do you end up with? Well if I can't beat you by coming on coming against you on the outside I'll just infiltrate. I'll just inspire division. I'll inspire argumentation. I'll inspire these kinds of things because a house that is divided cannot stand. So the enemy has tried for so long to divide from the outside but he began to work from the inside and that's what James is beginning to speak about here in the fourth chapter of the book of James. He says that there are wars and there are fights among you and he's rebuking the churches for that. You see, love for Jesus Christ is intended to produce a unity not competitive divisions amongst believers. In Ephesians in chapter four verse three we are to be endeavoring to keep the unity of the spirit in the bond of peace Paul told the Ephesians and when Paul was writing to the Romans in chapter 12 verse 18 he said, if it is possible as much as depends on you live peaceably with all men. Now as much as depends on you do your best seek peace pursue peace live at peace with all men. Now we know as believers that the fact is and it's a simple fact but the fact is sometimes when you come to faith in Christ it does create a division. It creates a division in your family not every family member is happy.  Not everybody's that way.  Not to say your family's bad but sometimes it's tough isn't it? Sometimes it can be tough to be with your family. Grandma's not always nice. You know, that's a fact, you know? She's not always nice. Sometimes she's been putting a little brandy in this and a little more in her and then at the end of the day, she's not too happy. You can have conflict at the table. Sometimes it's nice, sometimes it's not. Well, you kind of expect that in some ways. Marie and I before in earlier days in our relationship, you know, there were times that that she and I would pray before we went somewhere. You should always keep what you're doing in prayer anyway but we would say, Lord, you know, help us but we're here to remember who we are. That was kind of our prayer to remember who we are. We're believers, help us remember that because sometimes it may get a little conflict, a little, it could be a little tough sometimes so you pray and I can remember we'd go and I'd say Lord, Jesus, help us remember who we are. Then I'd drop her off and drive away and let her deal with it. No, I wouldn't do that. Well, you kind of expect, you're prepared and Jesus said in Matthew 10, 34 through 36, Jesus taught us. He said, do not think that I came to bring peace on earth. I did not come to bring peace but a sword for I've come to set a man against his father, a daughter against her mother. And here you go, ladies, and a daughter-in-law against her mother-in-law. Underline that one. Man's enemies. I have a great mother-in-law, I better say that. Marie's here. A man, no, she is, I love her. A man's enemies will be those of his own household.  They're not always happy.  Now, that division, if you will, is between a believer and non-believers. That's gonna happen. Jesus didn't come to bring peace but a sword. There's gonna be a division and he prepared us for it. You see, people can resent having friends leave them and they can get mad because that friend came to faith in Christ and so they condemn their faith in Jesus Christ. And that kind of division is a result of believers being rejected by non-believers. And that kind of division is common in the world but it's not to be common in the church. You see, division isn't birthed by the Holy Spirit. Division very most often is a result of our sinfulness, our flesh. Paul identified this kind of contention and dissension as a work of the flesh in Galatians 5, 19 through 21. And he described it, he said works of the flesh include hatred, tensions, jealousies, outbursts of wrath, selfish ambitions, dissensions, heresies, envy, and also told you in time past that those who practice such things will not inherit the kingdom of God. So that is works of the flesh is what the scripture says. So because of their carnality, James is calling the church, adulterers and adulteresses. In verse four, when he says adulterers and adulteresses, do you not know that friendship with the world is enmity with God? You see, they have adopted the world system as their standard for behavior. And so doing it reveals an unfaithful heart towards the Lord. You know, we live in the world, but we're not of the world. It's not that we should run around and change the way that we dress and everything and try to dress quote unquote Christian because that's kind of a hard thing for me to be able to define. What does it mean to dress Christian? You know, I'm sure that you can dress with modesty and things of that nature. Obviously that's true. But I'm not saying that if some style's out that I can't wear that jacket or that pair of shoes or whatever. I'm not saying that and I'm not saying that to you. And there's sometimes in the church, the church has adopted such a distancing from the world and in terms of its appearance that you'll see that even to this day, you'll see the Amish will wear a certain way or dress a certain way and wear certain mirrors and don't use electronic devices and things of that nature because they're saying there's trying to stay separate from the world and all of that. There are some churches that will tell the women that they need to wear a certain kind of dress and they have to wear their hair in a certain way and that they can't wear makeup. Makeup's a good thing. But they'll say you can't do these things, you can't wear jewelry and all of that and they'll take scriptures and out of context apply them to you and put you in some rigid format that is all on the outside. It's all outside. The short hair on the guys and the suits and the ties and the nice shoes and all of that. Sunday dress we used to call it, we're in our Sunday best and you dress like that and it's all on the outside. It's like dressing a chimpanzee in a tuxedo. I mean it's a nice tux but it's still a chimp. It's the nature that has to be changed and that's what God does. God changes the heart. He works on the inside and there's so many of us that are caught up trying to clean the fish before it's caught. But the Lord says no, once you get caught by the Lord then you begin to read your scriptures and you begin to understand what it means to stumble somebody else and begin to understand what the word modesty means. You begin to learn how to use your finances properly. These are the things you learn. And it's not because if somebody's got a tattoo, I remember that years ago, my dad hated tattoos and my dad was from the Navy. My dad served in the Navy during the World War II and he got out of the service and my dad hated tattoos. He just thought it was a mark of a criminal. It was a mark of something he didn't appreciate. And now that was my dad's opinion. You know and I honored and respected my father for his opinion but that isn't something that I necessarily had to follow my life saying all tattoos are bad because I don't think they are. And I don't have a problem with them at all. Some people have them all over the face. That's a little extreme but sometimes they needed it. You know the piercings and things like that. So we get caught up on the outsides when I'm saying, we get caught up in the way they're looking and all of that but I discovered a long time ago that sometimes a person who gets saved from a lifestyle and he has the taggings of that lifestyle will say on his body or her body that they are people that actually have a whole mission field that they speak to who are not put off by that and can respond to that. So who am I to say, oh go out and get it all lasered off and this and that? I don't do that. I don't do that because God wants to change the heart. It's the heart that he works on and it's a heart that he wants to fix because it's the heart that's evil. And so in their carnality because they're adopting the world system James says this is revealing an unfaithful heart towards God. So he points out in verse six that instead of judgment, God actually has provided his grace because without his grace the church would never accomplish what it's intended to do. It's through his grace that we master our carnality and it requires humility to receive the grace that is offered by God. He says God resists the proud but gives grace to the humble. So it requires humility to receive the grace that is offered by God. So God's grace extended to us gives us opportunity and time to repent and turn to him. So how are the churches as well as modern believers to overcome carnality? What are we to do to experience and obtain his favor? Well he begins to tell us in verse seven and he says in verse seven first therefore submit to God. Resist the devil and he will flee from you. Since God opposes the proud and gives grace to the humble, submit to God. Instead of resisting God, submit to God. James suggests that the unfaithfulness of the believers is due to the enemy's influence. I want you to note that. Submit to God and resist the devil. He's saying that the enemy is provoking them. He's provoking this and that's what he does. He brings division, he provokes differences. And so he says the answer to this is first and foremost submit yourself to the Lord. Now the word submit speaks of a voluntary attitude of yielding, cooperating or assuming responsibility. So submission is the yielding of one's will to God which leads to obedience. In the attitude of humility, yield to what he has said is necessary for our good. Because it's when we submit to God that we're able to resist the devil. Now that word resist, he says resist that word resist is to withstand or oppose. So he's saying withstand or oppose the devil instead of resisting God's will for us. And we do this in faith. We wear the armor of God. We're aware we are in spiritual warfare. The Bible tells us that we're in a spiritual war and that the weapons of our warfare are not carnal but are mighty in God for the pulling down of strongholds. He says casting down imaginations, every high thing that exalts itself against the knowledge of God and bringing into submission every thought and to the obedience of Christ. So though we're in the flesh, we do not walk according to the flesh and God has given to us spiritual weaponry in Ephesians 6, 10 and 11, my brethren be strong in the Lord in the power of his might. Put on the whole armor of God that you may be able to stand against the wiles of the devil. So it's a spiritual war that he's speaking about. We submit to God and that way we resist the devil. We don't resist the devil in our own strength. There's no way we could defeat him. We resist him with the armor of God and faith in God knowing our sure victory is in God. Having done all we stand is what Paul said. And when he said we haven't done all to stand, that word stand in the Greek language means they have your feet planted in a pose of victory. We already are victorious because of what Jesus Christ has done. You need to understand that. You're already victorious. We're more than conquerors in Christ. A lot of times the enemy will whisper propaganda into your mind saying you cannot win and that's not true. We've already won. We stand in the position of the victor. Now that does not mean that there aren't still battles that continue but we always are on top in Jesus Christ. We always end up on top. We are more than conquerors. We are victories in him and nothing can take you down. We resist the devil. He will flee from you. Now it's not because you're so bad because you're not. It's because he's bad. I was six years old once. I went to school for a little while in Manabello at a school called Fremont Elementary School.  I didn't like his tone so I turned around and I kicked it in the opposite direction. He didn't like that and he jumped on me. I still remember it. He's right on top of me.  All of a sudden his expression changed to one of pain and he's coming off of me. See, I had a big brother who went to the same school and my brother Frankie saw this kid jump on me and Frankie came and pinched his neck. I still remember he had him by the neck and picked him up and put him off of me and the kid ran away. Well, you know what? I got a big brother who does that to the devil all the time. All the time, so do you. The enemy may come after you and he does and when you're trying to serve Jesus Christ, you know this. When you're serving Jesus, he comes after you even harder. If you're backslidden, you're on his side. You're not doing any harm for his kingdom.  So I've got him already. That's no problem. But when you stand up for Jesus, when you say here I stand and I will go no further, this is what I'm gonna do for him, then he comes after you and the weapons of our warfare are not carnal though. They're mighty in God. I have the victor on my side, Jesus Christ who said it is finished.  He stands beside me. I'm walking through that valley but I'm never alone. He carries me in my troubled times and he gives me victory. I am always victorious in Jesus Christ. I am more than a conqueror because of him and that's how you need to understand it. So you resist the devil. You resist him. Not with puffing your chest out and posing like whatever you saw Bruce Lee do in one of those movies that you like so much. But in Christ, in faith, you stand up and say here's the shield of faith and here is the sword of the word and my feet are shod with preparation of the gospel of peace and I've got my waistband of truth. My helmet is on, the one of salvation and my breastplate is of righteousness and in Jesus Christ I can do all things and here I stand and I'm not gonna fall for this. That's what he's telling us to do. James is making that clear. He says submit to God, resist the devil and he will flee from you. It may take a while. It's not always as fast as you want is it? It's not always as fast as you want. Sometimes I wish that he just wouldn't bother me at all. I always wish he wouldn't bother me at all. But guess what? I'm always a conqueror in him and every wound that I've ever received has only deepened my love for Jesus Christ. That's what happens. That's what happens. And so he is your strength. In First John 5.18, the Bible says, we know that those who have become part of God's family do not make a practice of sinning. For God's Son holds them securely and the evil one cannot get his hands on them. In Romans 8.35 through 37, who shall separate us from the love of Christ? Shall trouble or hardship or persecution or famine or nakedness or danger or sword? As it is written, for your sake, we face death all day long. We are considered a sheep to be slaughtered. No, in all these things, we are more than conquerors through him who loved us. And so we submit to God, we resist the devil and he will flee. He says in verse 8, draw near to God and he will draw near to you. Cleanse your hands, you sinners. Purify your hearts, you double-minded. Lament and mourn and weep. Let your laughter be turned to mourning and your joy to gloom. Humble yourselves in the sight of the Lord. He will lift you up. And so draw near to God, verse 8. He will draw near to you. In setting their hearts on pleasure, they have ceased drawing near to God. But when a soul sets out to seek God, God sets out to meet that soul so that while we are drawing near to him, he is drawing near to us. Now what do we do in order to draw near to God? Well, he tells us, verse 8. He says cleanse your hands, you sinners. Purify your hearts, you double-minded. When he says cleanse your hands, that's simply another way of saying clean up your life. When he says purify your heart, he's speaking of your thoughts, your passions, your desires, your purposes, your affections. Clean up your lives and purify your hearts. What you dwell on is what you desire. And what you worship, you become like. So what do you do? Well, you cleanse your hands. You say, Lord, I know that this is not something that you are pleased with. And you purify your heart. God, I wanna settle my attention and affection of those things that matter. And you start today. You start today. You start keeping short accounts with the Lord. You start learning to confess and to forsake as a habit. You begin to say, Lord, I don't want that thought. And Lord, do you know where that desire will take me? I'm talking about being a Christian now. I think a lot of people who are Christians have failed to actually put into practice what James is saying right now, to be honest with you. I think that a lot of believers have ceased drawing near to the Lord. That they have ceased cleansing their hands. They have ceased purifying their hearts. They've filled their hearts with so many things that not to be there. And they haven't learned verse nine to lament and mourn and weep. They haven't let their laughter be turned to mourning and their joy to gloom. They think, if I do this, that sounds like a bummer. What a bummer life that is.   Not know what he's describing here is repentance. And repentance is a word that a lot of us think is a bad word, but it's been said that the word repent is the first word of the gospel. It's the first message Jesus gave in the book of Matthew, repent. The kingdom of God is at hand. It's the first word that John the Baptist gave, repent, the kingdom of heaven is at hand. It's the first word of the gospel. It's a turning away. And a lot of times we fail to realize that that is what God has called us to do.  It's your whole life, really, a life of repentance. And he describes what repentance looks like. The word lament, the word lament there means to grieve. It speaks of being miserable or even wretched, afflicted. The word mourn, mourning is a passionate grief that people can see. You see mourning at funerals. You're going when your mom, your dad, grandma, grandpa, one of your favorite friends or relatives has died. That's mourning. You see, it's a passionate grief that people will look at you and know that you're in mourning. Weeping speaks of strong tears. It's the times that some of us in this room have actually wept. There's times you've cried. You watch one of those YouTube videos of kittens and it's so cute that you cry. No, weeping's an entirely different thing. Weeping, some of you have, not all of you. All of you will if you haven't. Weeping is when you roll up into a ball on a carpet and you just are so intensely in pain that there's nothing you can do other than just pour out your tears to God. How many of you know what I mean? I'm not asking to raise your hand, some of you will. But you know what I mean. It's that moment in your life where you think that the floor has dropped out and you don't, who's gonna catch me? I don't understand. What is going on? Oh God, I've done that. Have you done that? I have done that more than once. More than once. I think sometimes my life has been filled with it over the loss of this and the loss of that. The sorrow of this or pain for that, you know? But again, you resist the devil and he will flee from you because sometimes when you're mourning you think there's no hope and the enemy's whispering and saying there isn't any. You got a friend that's unsaved, you got a mom or a dad that's unsaved, you got a kid that's not doing well to you parents or grandparents, you have a grandchild that's being raised in an environment that you know is destructive for that baby and there's nothing you can do. There's nothing you can do. I've been there where I've been on a carpet somewhere on my face and I've been saying God, unless you do something, I feel like I'm gonna die. God, unless you move and then the enemy, oh he's not listening. Why do you even pray? So waste your time. That one's mine. You can't have them. I fought for my kids when they were in high school. I still do but not to the same degree because they have come to follow Christ but in high school I have to tell you it was hard for us to have kids. I weren't doing well and I'd be in the room and I can still remember being in the room speaking to the Lord. Some of you parents know what I mean. Some of you do. Others, you've raised great kids. God bless you. I had mine. They were a little tough. They could be tough. Just because I'm a pastor doesn't mean they didn't have a sin nature. Just because they gave them devotions, brought them to church, took on missions and did all that I did for them doesn't mean that they didn't have a choice that they had to make for their own life and it doesn't mean that they weren't number one on the enemy's target list because they tell you pastor, kids are. The enemy is after those kids in a way that he's not after other kids and that's just a fact because if you take down the kid he can take down the dad. That's how that works. And so I can tell you there were times in my life that I would be on the ground. Marie wasn't even around. My wife wasn't even around where I would shake my fist into the heavens, not at God but in the direction of the enemy in my heart and I still remember doing that and I still remember saying they belong to God. You cannot have them. They belong to him. You cannot have them. Resist the devil. He will flee from you.  You hear my cry. You know my desire. I love these children but not as much as you do. In Jesus' name, save them. In Jesus' name, touch my mom. In Jesus' name, touch my father. In Jesus' name, Lord work in my neighbor.   I lament before you. I weep before you. And he says in verse nine, let your laughter be turned to mourning, your joy to gloom. In other words, become sober-minded. Take sin seriously. Guard your walk with Jesus Christ. In 1 Thessalonians 5-6, let us not sleep as others do. Let us watch and be sober. Now you have this attitude because you see sin for what it is and you begin to see what sin has done to this world if you watch the news at all. Besides the fact that you have your friends who share with you the pain that sometimes they go through. This is a tough world. It's a lot of pain. If you look around this room here, there's a lot of pain in this room. And some of you know that. Some of you are experiencing it. Some of you don't know that. And the person you don't know who may be seated near you, you don't know them. They may be going through some pain that is unbelievable. When I go out and I talk to people and I do it after first service and I stand out there before service and I walk amongst anybody who's here. Then I talk to people afterwards and I hear their stories every week. I hear their stories. Can you please pray for my husband just this morning? I don't think I'm violating a confidence. One of our ladies, can you pray for my husband? And she starts describing what he's going through. Can't be here. This guy's been in our church a long time. Can't be here. He's ill right now. Think of somebody that you may know that's going through something right now, something painful. Maybe an illness. Maybe a diagnosis. Maybe something that is personal pain. All of us know somebody like that. And the enemy whispers and the enemy whispers. And he may be saying something to you. You might as well give up. Nothing's gonna happen.  Is he God's not listening to you? There's no such thing as a God. He doesn't answer prayer. A friend of mine was speaking to me about an atheist and a Christian evangelist. It's a true story. This guy's one of these preachers who likes to share with people on college campuses and all. And he was on a college campus and he was sharing. And some guy walks by and calls himself an atheist. This evangelist on campus says, do you mind if I pray for you? And the guy says, I don't believe in God. If you wanna pray for me, there's no God. If you wanna pray for me, go ahead and pray for me. But I don't believe in God. I don't believe God's gonna answer your prayers. And he says, so you don't mind if I pray for you? And he says, no, pray. So this guy looks at this guy. This is a true story. It happened on one of the local college campuses. So the evangelist looks at the guy and says, God, in Jesus' name, I ask that you just destroy this man. I ask that he may have many accidents that he might break his legs than it being so. And the guy's, wait a minute. What are you praying like that for? And the evangelist says, then you do believe in prayer, right? So you do. Gotcha. But there are people who do that. And the enemy can whisper your heart can conspire. And so what do you do? You say, God, in Jesus' name, I ask that you would work in me. I humble myself because Lord, I don't want to be causing problems because I'll way back into the context in chapter four, verse one, Wars and Fights. And he's saying, do you want peace? Submit to God. Resist the devil. He's provoking these things. Draw near to God. Cleanse your hands. Purify your heart. Lament, mourn, weep. Let your laughter be turned to mourning. Enjoy the gloom. And then he says, humble yourself in verse 10. We've sinned against God. Humble yourself before him. In verse 11, he goes, do not speak evil of one another, brethren. He who speaks evil of a brother and judges his brother speaks evil of the law and judges the law. But if you judge the law, you're not a doer of the law, but a judge. Instead of engaging in wars and fights, he's saying, humble yourselves. Live in peace. Do not gossip. Don't backbite. You're family in Jesus Christ. Peace and unity are evidences that you belong to him. It's all built on the foundation of loving God and loving someone else. He says in verse 11, he who speaks evil of a brother and judges his brother speaks evil of the law. You see both the law of Moses and the teachings of Christ forbid believers from judging harshly. And by disregarding God's word, you're saying that it's not true and that you're above it. When you look at the law in the Old Testament, the law of Moses is called, it's really summed up in loving God and loving one another. In Galatians 5, 14 through 16, it says all the laws fulfilled in one word, even in this, you shall love your neighbor as yourself. But if you bite and devour one another, beware lest you be consumed by one another. I say then, walk in the spirit, you shall not fulfill the lust of the flesh. The whole law is summed up in love. And anyone who claims to love God but doesn't love other people, that person doesn't understand what a Christian is. According to 1 John 4, 7 and 8, beloved, let us love one another. Love is of God. Everyone who loves is born of God and knows God. He who does not love does not know God, for God is love. And that'll help you when you begin to hear of these people who blow themselves up for the love of God. And they kill people as they do that. Well, John would say, if you don't love people, you don't love God. You don't really have a love for God. You see, the essence of love is the willingness to sacrifice your interest on behalf of somebody else. And that's best demonstrated by God's willingness to send Jesus to die for us on the cross. And it's that kind of love that God gives to us and it empowers us to give that kind of love to somebody else. In John 5, 12 and 13 it says, this is my commandment that you love one another as I've loved you. Greater love has no one than this that it'll lay down one's life for his friends. And so love is the birthmark. It's a birthmark of a believer. It identifies us as God's children. It's the kind of love that God gives to us and uses that kind of love to reach others. In John chapter 13, 34 and 35, a new commandment I give to you, that you love one another as I have loved you, that you also love one another. By this I will know that you're my disciples if you have loved one to another. That's how they know that's your birthmark. So a person who says I love God but hates people doesn't know God. You know, my mama used to say that, David, I love God. It's people I have problems with. I say, yeah, mama, but that's not the way it works. I love ministry, but I don't like people. No ministry, people are ministries, mama. And mama didn't understand that. If you ask a non-Christian to describe a Christian, sometimes the answers can be interesting. Many of the people who respond to such a question will say something kind of negative. Some will describe a believer by what we're mad about today or what we are against. But if you were to ask an ancient pagan to describe a Christian, something else might be said. Chuck Swindall says, one of the most profound comments made regarding the early church came from the lips of a man named Aristides sent by the emperor Hadrian to spy out those strange creatures known as Christians. Having seen them in action, Aristides returned with a mixed report, but his immortal words to the emperor have echoed down through history, behold how they love one another. That was the mark of the early church. Listen guys, that's the kind of thing that you heard me before the service began where I was saying to those who came on Thanksgiving and served here, made meals, served, put tables out, took the tables down, cleaned everything up. That's the kind of thing I'm talking about. That kind of thing that on Tuesdays when we have our men's studies, you know the study begins at 6.30 a.m., that's early for it, pretty much for anybody, 6.30. But we have guys here at 5.30. And these guys who are here at 5.30 are out there making burritos and things so that they can serve their brothers who are gonna come later on. To me, that's the tremendous thing. I love that, I love that. And like I mentioned, we have guys who took off right after Thanksgiving, drove all the way up north to the Oregon border and they're there in a prison right now, ministering to people who could not get out on Thanksgiving but they came and brought the love of Christ. You see, that's what the Bible says we're to do, is to love one another, to get along with one another, to put up with one another sometimes, you have to put up with one another, to receive one another, to pray for one another, to comfort one another, to care for one another, to show compassion on one another, to be patient with one another and not to cause divisions, James is saying. Not to have wars and conflicts, not to be that way. Not to speak evil of one another in verse 11. Not to speak slander. You can destroy people with your tongue. And when you do that, God's love isn't being revealed. It's been said that slander destroys three people, the speaker, the one spoken to and the one spoken of. In Proverbs 16, 28, a perverse man stirs up dissension. A gossip separates close friends. So instead of tearing one another down, we build one another up. He says not speaking evil, that refers to any form of speech against somebody else. It includes speaking what may be true but with a heart to destroy or hurt somebody. Christians are different. We're to encourage and to love one another. Our goal should be grace and love, truth. It should be unity. Our speech should reveal that. And our goal as believers must be for the body of Christ to be edified and unified. And that's evidenced in the way that we speak to and of one another. And to love is the decision that we make through the power of God's spirit. Listen, we all know people who are difficult to love. We may be married to them. We may be raising them. But we all know people that are hard to love. If you don't know anybody, you must live in a box because we all know people like that. And yes, they can be irritating, of course. And of course, notice how I say they. I'm probably that person to somebody else right now. To a lot of you. Yeah, I know what you mean, Pastor David. How do you handle that? How do you handle that? When this church was very new, I was standing in front of the building that we used to meet in back in the 80s, about 83. And somebody came up and they were one of these difficult people. And I stood there and I listened, talked to him and all of that. And I had my worship leader standing right next to me for some reason he was there. Well, this person was speaking to me. And they were one of these irregular people. You know, irregular people, you know that term, right? You go to a store and there's something on sale and they're called irregulars because maybe one of the legs is an inch longer than the other and they're irregulars or maybe the sleeve or maybe it's just sewn. But they call them irregulars. They used to be called irregulars. You go to the store and they had irregulars and they're regular sales. And there are people who are regular people. They're kind of like, hey, how you doing? They're just different. And they're hard. It's not hard to love the lovely, is it? It's not hard. They think you're the greatest. And you see, tell me more. They're the greatest. But what do you do with the one who comes and complains and whines and cries and every time they're on Facebook you have to pray for them because something bad just happened? You wonder what happened before Facebook? Who'd they complain to? Pray for me, my feet hurt. Okay, pray for me. I have the wrong color of nail polish and I'm dying. I'm in such a trough. Okay, I pray they, I'll pray for you.  Human beings are trippy. They really are. That's an old saying. Some of you understand it. Crazy is a good word. We are cuckoo. How do you handle that? And so this person came in and was an irregular person and spoke to me for a while and walks away. And my music minister said something I've never forgotten. Obviously it just comes to mind as I'm teaching you. It just came to mind. And he says to me, how do you do that? And I said, do what? Do what? How did, he said, because she was kind of different because he said, you stood there and you didn't leave. You didn't, you listened. How'd you do that? Because he couldn't do that at that time. And I said, how did I do that? He goes, yeah. I said, you love him. You love him. That's how you do that. That's not a new idea, is it? You love him. Because you know what I think? I think this. And this may sound bad. Maybe I shouldn't say it, but I will. I don't think it's that bad. Because this poor person wakes up with himself every day. Can you imagine how painful their life can be?  That's their life. And I'm gonna make it worse by not giving them my time. Why would I make their life worse? When maybe just by loving them for a moment it might help them in their future. People have done that for me. I've been the irregular person. I've been that person that nobody wants to be around. I've been that person that was gonna be the failure. I was that person that, I'll give you an example. I was in 10 years old. I was 10 years old and a friend of mine named Bill and I decided to cut school. And we got caught. We weren't the best. I got better later. And so the principal had us in the office. And he called my friend's mom. And he said, I wanna make a recommendation. Keep your son away from this boy, David. He's a bad influence. And I was sitting right across from him. The principal's right there. And I'm sitting here and he's looking at me as he's speaking. You keep your son away from him. He's a bad influence. And I'm just sitting there looking at him. That was my best friend.  You know, the principal was right. I was a bad influence, I confess. But guess what? My friend Bill comes and meets with me every month so I can disciple him in the ways of Jesus Christ. I'm thankful for God, not keeping June, his mama, keeping him from being near me because to this day at our age, I'm still dear friends with him and ministering to him the things of the kingdom of God. So when people say things like that, they don't know what they're talking about. And we have to be careful the way that we speak of others. You know, one of the good things that you can learn to do if I may is learn to speak of others as if they were standing next to you while you're speaking. How would you speak if they were standing there at that moment? That helps me. I keep that in mind. If they were listening to me right now, how would I be speaking of them? Would I be saying or would I be speaking with charity? It's an easy thing to do but that's what you do. You learn to do that. You have to be careful. You have to be careful not to speak evil of one another. He says he who speaks evil of a brother and judges his brother speaks evil of the law. If you judge the law, you're not a doer of the law, but a judge. You've put yourself above that. He's saying, who gives you the right to judge? You have usurped God's right. He's saying, God is the ultimate judge and God judges fairly. And so finally, verse 12, there's one lawgiver who is able to save and to destroy who are you to judge another? You can only see the outside. You never know the thoughts, the motives. You never know the intense of the heart. We're not the final judge. Only God is. And constant criticism and nitpicking destroys the body of Christ. So instead of destroying, God help us to uplift and encourage. God help us to speak truth but to always do it with love. To honor him and care for somebody else, to learn to do that. And it is something God will teach you. It is. God help us to learn that.


Hi, y'all, Liz here. And today we are talking about the ABCs of cleaning. And today I want to talk about saturate, agitate, and remove. S-A-R. The three basics of cleaning, of cleaning any surface. Now, we should actually start with R, remove, and then go to S-A-R because most of the time you will want to look and see if something can be removed on its own before having to saturate. For example, you're dusting. You're in a fine living room or a living room with fine items. You're not going to have to saturate anything. You're probably going to be able to use your microfiber and just be able to remove. Now, if you do remove, try to remove and stuff doesn't come up. Now you're going to hit S-A-R. You're going to have to saturate, agitate, and then remove again. All right. I know you guys know all of this stuff. I'm just trying to give you some little pointers, tips to remember. Okay, so R-S-A-R. Saturate. First off, whatever it is that you're using, you need to let whatever it is do its job. And it doesn't really matter what it is. Maybe you're using steam. If you're using steam, you need to make sure that there's enough steam to saturate the area. If you're using a chemical, is there enough chemical, enough product to saturate the area? Let whatever it is that cleaning compound do its work. Sometimes it's steam, water, maybe a chemical of some type, top job, or whatever it is that you all use. We use a lot of buckeye products, so we might be dealing with a buckeye product down there. It needs to saturate the stuff that needs to be removed. That should be edited. It's probably not going to be, because you know how my editor is, so go with it, okay? Saturate the stuff that needs to be removed. And after it does its work, the product chemical cleaning solution does its work, then agitate it. Move it around. Give it a little scrub. See what it is that you need to add? There's some more scrubbing, less scrubbing, with a brush maybe, or maybe just with a cloth or a sponge. But some type of agitation usually is necessary. Sometimes you don't need to do a lot of agitation. The simple removing third step is enough agitation to remove it, especially if it was something light. You've got a very, very light smattering of dust that's stuck with maybe a little bit of grease, just the tiniest bit of grease, and you've used a little bit of ammonia. You've let it saturate, and now agitation can simply be the act of wiping it off. Maybe not. You might have to wipe it twice.  Should have just agitated. Agitate, agitate, agitate, and then remove. So, S-A-R, saturate, agitate, and remove. All right. Now remember, keep in mind what we talked about yesterday, direction. So when you're using the S-A-R method, make sure that you're working top to bottom, right to left, and I forgot yesterday, back to front, right? Don't clean from the front to the back. Back, bringing it forward. All right. That's it for today. But my question for you is, what is the area that you know you need to remember S-A-R? That you need to remember to saturate, agitate, and remove? Because sometimes you accidentally go to remove before you're ready to go there. Is it showers? A lot of people showers. That's a biggie. Another one, though, that's very, very common is stove tops.  It might look so, so clean, especially a clear, clean, clear glass top.  But then you're going to get that weird haziness that actually needs to have some saturation, and it's probably going to need a little bit of agitation. So what's your area? What's the one thing that you always have? The reason why I want you guys to answer these questions is so that tomorrow, or today, when you're out cleaning, you'll remember, and you'll go, oh, that's right, I have a problem with that one, and you'll save yourselves time and energy and frustration. Okay, so figure out your answer, and I'll talk to you tomorrow. Bye.