Why is $\mathcal{M}(\text{SL}_2(\mathbb{Z}))$ spanned by $E_4$ and $E_6$? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-21T16:37:07Z http://mathoverflow.net/feeds/question/118639 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118639/why-is-mathcalm-textsl-2-mathbbz-spanned-by-e-4-and-e-6 Why is $\mathcal{M}(\text{SL}_2(\mathbb{Z}))$ spanned by $E_4$ and $E_6$? hx 2013-01-11T16:07:16Z 2013-01-11T17:27:56Z <p>It is said that $$\mathcal{M}(\text{SL}_2(\mathbb{Z}))=\mathbb{C}[E_4,E_6]$$</p> <p>where $\mathcal{M}(\text{SL}_2(\mathbb{Z}))$ is the graded ring of module forms over $\text{SL}_2(\mathbb{Z})$ and $E_4,E_6$ are normalized Eisenstein series.</p> <p>I'm new to modular form and not quite familiar about the Eisenstein series so I fail to prove it.</p> <p>Any one give some ideas?</p> http://mathoverflow.net/questions/118639/why-is-mathcalm-textsl-2-mathbbz-spanned-by-e-4-and-e-6/118651#118651 Answer by Vahid Shirbisheh for Why is $\mathcal{M}(\text{SL}_2(\mathbb{Z}))$ spanned by $E_4$ and $E_6$? Vahid Shirbisheh 2013-01-11T17:27:56Z 2013-01-11T17:27:56Z <p>See Proposition 1.3.4 of Bump's book "Automorphic forms and representations". </p>