= Rational structure of level-one modular forms
{title2=$M_*(SL_2(\mathbb Z),\mathbb Q)=\mathbb Q[E_4,E_6]$}
A level-one <modular form> with rational <Fourier coefficients> is a weighted homogeneous polynomial with rational coefficients in the normalized <Eisenstein series> $E_4,E_6$. Subtract its constant coefficient times a same-weight monomial. The remainder is a <cusp form>; divide it by $\Delta_0=(E_4^3-E_6^2)/1728=q+O(q^2)$. The <valence formula for the modular group> shows that $\Delta_0$ has no interior zeros, so this quotient is holomorphic and has rational coefficients. Induction on weight proves the assertion. Euler's <Bernoulli number> formula for even zeta values then gives the corresponding rational weighted polynomial in the unnormalized <lattice Eisenstein sums> $G_4,G_6$.
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