[SOUND] [MUSIC] So now we have to analyze the modular behavior of the Eisenstein series, G2. But first of all, I would like to formulate a nice fact, as an exercise. You can find the proof of this fact in the paper, in the famous paper of Richard Borcherds from 95 about automorphic product. But, you can prove this fact using, The description of the kernel of modular operator. So I can write this fact that delta belong to the kernel of D12. So maybe I can put the first question. This is an elementary question to analyze the kernel of this modular differential operator for any k. Any k, and then the fact, very nice lemma from the Borcherds paper. Then, The following map from the space of modular function, with respect to S iota zed, into the space of weak modular form of weight 2, with respect to S iota zed. Then this map is surjective. I repeat, then you can prove this fact using the definition of modular differential operator. Moreover, the corollary from the surjectivity of this operator, if f in tau is a weak modular form of weight 2. Then, the constant term is always 0. Certainly, this is direct corollary of the factor. Now, we have to analyze, Modular equation, Of the Eisenstein series, G2. This is a modular equation of the logarithmic derivative. This is, I can derive definition once more. This is- 1/2 pi i n prime in tau over eta in tau. So, modular equation. First of all, let me write down the modular equation of the Dedekind eta function. Multiply system in M, so or for the 24, (C tao + d) to the power 1/2 eta tau, for any M = (a, b, c, d) and S. What do we have for M prime? So, let's calculate the duration of the left and the right-hand side. So, what do we get? (C tau + d) to the power -2 n prime (a tau + b/c tau + d) is equal now without any changes, v eta (M), this automorphic factor, n prime (tau) + v eta and M, c/2 (C tau + d) to the power 1/2 eta (tau). And now, we have to divide. The first term by the second term, and we have to add the constant term- 1/2 pi i. What we get, we get (c tau + d) to the power -2 G2 (tau) is equal, I'm sorry. Here, I made a mistake. G2 (a tau + b / c tau + d) is equal, The first, we have to divide the first term by, The right hand side. So we get G2 (tau). Then, we'll have to consider the second term. And we get, with this additional k function, that -c / 4 pi i, or maybe I change a little the form of this factor. 1 / 4 pi i, c / c tau + d. And this is true for any M in the modular group. We see that the Eisenstein series, G2, is not modular. So G2, is a quasi modular form of weight 2. This is simply by definition, because without this linear, fractional linear term, in c tau + d, well we'd have a modular form of weight 2, but this additional term certainly changed the behavior of our function. So this is the difference between modular and quasi modular form, but now we have to put the question. So we have to calculate [SOUND] this differential operator, Modular differential operator, in Eisenstein series E6, E8 and so on. But what is about [SOUND] G2? So the question which we have to put, what do we have about D(G2)? If I calculate the derivation of this modular equation, we get a term of order 2. So you can check, and if we add here the coefficient 2 G2, multiplied by G2, G2 to the square, we get a function which transforms like modular form of weight 4. Please check this. So it means that this sum has very good modular behavior with respect to the full modular group. But now you can calculate this function because to calculate it we'll have to analyze the first, the constant term. The constant term, we have only here, yes? And calculating this constant term, you can find that this function is equal up to some constant of Eisenstein series, E4. I leave to find this question to you. Certainly, the coefficient here, this is 2 times 1/24 to the square. So we'll have something here, like 1 /2 times 12 to the square. Please find this constant. But now we can make the conclusion from our calculation. [SOUND] In fact, we have proved a nice theorem about the space of quasi modular form, but I have to define this space. And I can make this definition inside the theorem, but I would like sometimes we use two notation for our quasi modular form. We use, from time to time, the following notation. E2, this is 1-24 the sum sigma 1(n) q to the power n, n greater or equal to 1. This is our Eisenstein series, G2, with coefficient (-24) G2 (tau). Please compare the function of, the Fourier expansion, so, [SOUND] I would like to analyze once more the Eisenstein series. So the Eisenstein series, E, has the first Fourier coefficient 1. So E series, E Eisenstein series, all of them start by 1. And I would like to define the same Eisenstein series for weight 2. [SOUND] Up to the constant -24. This is the quasi modular Eisenstein, G2. And now I can formulate the theorem, and to define the space of quasi modular form theorem. The gradient ring of all modular form with respect to S iota zed, Is generated by two Eisenstein series, E4 and E6. Its extension by E2 means the gradient ring generated by three Eisenstein series, the quasi modular Eisenstein series E2, E4 and E6. This ring is called, The ring of quasi modular form. And we prove that differential operator D X on the space. Means the rank of quasi modular form is invariant with respect to action to its differential operator, D. This is the first fact. The second was our proof, but you can do it by yourself. This is not complicated. E2, E4, and E6 are algebraically, Independent. That's exciting. Now we can consider the method of a demotic correction. [SOUND] [MUSIC]
