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It is said that $$ \mathcal{M}(\text{SL}_2(\mathbb{Z}))=\mathbb{C}[E_4,E_6] $$

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.

I'm new to modular form and not quite familiar about the Eisenstein series so I fail to prove it.

Any one give some ideas?

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 I'm sorry, but I don't think this is appropriate for MO (which is about research level mathematics). This is a result found in any undergraduate textbook on modular forms. – Dan Petersen Jan 11 at 16:09
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 I agree with D.P. and point you to Serre's "Course in arithmetic". – Julien Puydt Jan 11 at 16:41
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 Isn't "undergraduate textbook on modular forms" an oxymoron? Which universities are teaching modular forms to undergraduates? – wccanard Jan 11 at 20:01

closed as off topic by Dan Petersen, Felipe Voloch, David Loeffler, Angelo, Will Sawin Jan 11 at 17:30

1 Answer

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See Proposition 1.3.4 of Bump's book "Automorphic forms and representations".

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OK, Thank you.. – hx Jan 12 at 1:07

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