It is well known that the graded algebra $\mathcal{M}(1)$ of Modular forms for $\Gamma = PSL_2(\mathbb{Z})$ is the polynomial algebra $$ \mathcal{M}(1) = \mathbb{C}[E_4, E_6] $$ where $E_4$ and $E_6$ are the Eisenstein series of weights 4 and 6, respectively. It is also true that, while $E_2 = -\frac{1}{24} + \sum_{k=1}^\infty \sigma(k) q^k$, the Eisenstein series of weight 2 is not modular, it is quasi-modular, and satisfies a similar transformation law. Moreover, the graded algebra $\mathcal{QM}(1)$ of quasi-modular forms is given by $$ \mathcal{QM}(1) = \mathcal{M}(1)[E_2] = \mathbb{C}[E_2, E_4, E_6]. $$

What can be said about quasi-modular forms for congruence subgroups of $\Gamma$? Is it still the case that, if we denote by $\mathcal{M}(N)$ and $\mathcal{QM}(N)$ the algebras of modular (resp. quasi-modular) forms for, say $\Gamma(N)$ (or perhaps $\Gamma_0(N)$, etc), that we can write $$ \mathcal{QM}(N) = \mathcal{M}(N)[E_2]? $$ If not, is there some way of determining generators for $\mathcal{QM}(N)$ over $\mathcal{M}(N)$, say for even small values of $N$ ($N = 2, 4$ are of interest to me)?

If this is not known, are there at least dimension formulae?


1 Answer 1


Dear Simon: Your assertion is right. Let $\Gamma$ be a subgroup of finite index in $SL_2(Z)$. Then any quasi-modular form for $\Gamma$ can be written uniquely as a polynomial in $E_2$ with coefficients which are modular forms for $\Gamma$. This is proved in a paper by Kaneko and Zagier, A generalized Jacobi theta function and quasi-modular forms in The Moduli Space of Curves, Progress in Math., Vol. 129, Birkhauser, 1995, pp. 165-172. Hope this helps.

Ram Murty


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.