in A Course in Arithmetic by Jean-Pierre Serre, he states a theorem in 3 parts:
(i) $M_{k}=0$ for k<0 and k=1.
(ii) For k=0,,2,3,4,5, $M_{k} is a vector space of dimension 1 with basis 1.
(iii) Multiplication by $\Delta$ defines an isomorphism of $M_{k-6}$ onto $M_{k}^{0}$
First assuming (i), (iii) are true, I need someone to confirm whether I understand the proof of (ii). For $M_{k}$ where k=0,2,3,4,5, we have that $M_{k}=M_{k}^{0} \bigoplus C.G_{k}$ so by part (iii), $M_{k}=M_{k-6} \bigoplus C.G_{k}$. Since $k\leq 5$, by part (i), $M_{k-6}=0$ so we have $M_{k}=C.G_{k}$ which is a non-zero element so $M_{k}$ is dimension 1. Is this correct?
Next, I understand why the map from part (iii) is surjective but to prove injectivity, I need to show given f,g $\in M_{k-6}$ such that $\Phi(f)=\Phi(g)$, then $f=g$ where $\Phi$ is the multiplication by $\Delta$ map. So $\Delta f=\Delta g$ so from the proof by Serre, $\Delta$ does not vanish on H so we can divide by $\Delta$ and get f=g so we get injectivity. Is this correct?
Now I have questions on the Corollary which states that the space $M_{k}$ is has a basis $G_{2}^{\alpha}G_{3}^{\beta}$. I understand Serre's proof that these family of monomials span the space but I don't get his proof for linear independence. If we assume the monomials were not linearly independent, then one of these monomials can be expressed as a linear combination of the others. I don't see how he goes from there to the idea that $G_{2}^{3}/G_{3}^{2} satisfies a polynomial in C (and also, I don't know what polynomial he is referring to).