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The graded ring of level-one modular forms

Statement

The graded $C$ -algebra $M_{*} = ⨁_{k e v e n} M_{k}$ is freely generated by $E_{4}$ and $E_{6}$ : the substitution $X \mapsto E_{4}$ , $Y \mapsto E_{6}$ defines an isomorphism of graded $C$ -algebras $C [X, Y] \to M_{*}$ , where $\deg X = 4$ and $\deg Y = 6$ . Equivalently, for every even $k \geq 0$ the forms $E_{4}^{a} E_{6}^{b}$ with $4 a + 6 b = k$ , $a, b \geq 0$ , form a basis of $M_{k}$ , and the cusp forms form the principal ideal $S_{*} = Δ M_{*}$ ; explicitly $S_{k} = Δ M_{k - 12}$ for all $k \geq 0$ (with $M_{k - 12} = 0$ for $k < 12$ ).

Facts & Assumptions

Given: The graded spaces $M_{k}, S_{k}$ and the Eisenstein forms $E_{4}, E_{6}$ with $E_{4} (\infty) = E_{6} (\infty) = 1$ , together with $Δ = (E_{4}^{3} - E_{6}^{2}) / 1728 \in S_{12}$ of order one at the cusp and without zeros on $H$ (Level-one modular forms and cusp forms, Eisenstein series are modular forms; their Fourier coefficients, The discriminant is a nonvanishing cusp form of weight 12, The modular discriminant and the j-invariant).

[F1]

For even $k$ the monomials $E_{4}^{a} E_{6}^{b}$ with $4 a + 6 b = k$ are linearly independent in $M_{k}$ , and their number equals $\dim M_{k}$ from the dimension formula; every such monomial has value $1$ at the cusp (The dimension of the space of level-one modular forms, Eisenstein series are modular forms; their Fourier coefficients).

[F2]

A nonzero weight- $k$ form has a well-defined constant term $f (\infty)$ at the cusp, and $S_{k}$ is exactly the kernel of $f \mapsto f (\infty)$ (Level-one modular forms and cusp forms).

[F3]

$Δ$ has vanishing order $1$ at the cusp and no zeros on $H$ ; therefore $f \mapsto f / Δ$ maps $S_{k}$ bijectively onto $M_{k - 12}$ for every even $k$ : it is holomorphic on $H$ , has the correct weight, and at the cusp the order drops by exactly $1$ ; conversely $Δ h \in S_{k}$ for $h \in M_{k - 12}$ (The discriminant is a nonvanishing cusp form of weight 12, The modular discriminant and the j-invariant, The level-one valence formula, The zeros of E4 and E6 at the elliptic points).

[F4]

For every even $k \geq 4$ there is a monomial of weight $k$ (and for $k = 0$ the empty monomial $1$ ); $M_{2} = 0$ and $M_{k}$ is one-dimensional for $k = 0, 4, 6, 8, 10$ (The dimension of the space of level-one modular forms).

Proof

technique · induction on the weight $k$ in steps of $12$

1.1baseF1F2F4givenalgebra

Base cases: for $k = 0$ , $M_{0} = C$ consists of constants and the empty monomial $1$ has weight $0$ ; for $k = 2$ , $M_{2} = 0$ ; for $k = 4, 6, 8, 10$ the space $M_{k}$ is one-dimensional and contains the monomial $E_{4}, E_{6}, E_{4}^{2}, E_{4} E_{6}$ respectively, whose value at the cusp is $1$ , so every $f \in M_{k}$ equals $f (\infty)$ times that monomial. Hence in these degrees the monomials span $M_{k}$ .

1.2ihgiven

Assume, as induction hypothesis, that for some even $k \geq 12$ the weight- $(k - 12)$ monomials span $M_{k - 12}$ ; equivalently every element of $M_{k - 12}$ is a polynomial in $E_{4}, E_{6}$ .

2.1F2F3F4step 1.1step 1.2givenalgebra

Let $f \in M_{k}$ and choose a monomial $P = E_{4}^{a} E_{6}^{b}$ of weight $k$ [F4]; then $P (\infty) = 1$ and $f - f (\infty) P \in M_{k}$ has zero constant term at the cusp, so $f - f (\infty) P \in S_{k}$ [F2]. By [F3] there is $h \in M_{k - 12}$ with $f - f (\infty) P = Δ h$ ; by the induction hypothesis 1.2 the form $h$ is a polynomial in $E_{4}, E_{6}$ , so $f = f (\infty) P + Δ h$ is a polynomial in $E_{4}, E_{6}$ (using $Δ = \frac{1}{1728} (E_{4}^{3} - E_{6}^{2})$ ). Hence the monomials span $M_{k}$ for every even $k \geq 0$ .

3.1F1F3step 2.1discharge-induction∎

The graded algebra map $φ : C [X, Y] \to M_{*}$ with $φ (X) = E_{4}$ , $φ (Y) = E_{6}$ is well defined because $E_{4}^{a} E_{6}^{b} \in M_{4 a + 6 b}$ and multiplication of modular forms adds weights. In degree $k$ the source is spanned by the monomials of weight $k$ , whose images are linearly independent and as numerous as $\dim M_{k}$ [F1]; since they also span $M_{k}$ by the induction result, $φ$ is an isomorphism in each degree and hence an isomorphism of graded algebras. Therefore the monomials form a basis and $E_{4}, E_{6}$ are algebraically independent. Finally [F3] gives $S_{k} = Δ M_{k - 12}$ for every $k$ , that is $S_{*} = Δ M_{*}$ .

Depends on

Used by

The j-invariant classifies complex tori Theorem
The j-invariant uniformizes X(1) Theorem

Dependency tree · two levels

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Sources

D. Zagier, Elliptic Modular Forms and Their Applications, in The 1-2-3 of Modular Forms (Universitext, Springer, 2008) (standard reference, not scraped)
J. S. Milne, Modular Functions and Modular Forms (v1.31, 2017) (standard reference, not scraped)
C. T. McMullen, Advanced Complex Analysis, Math 213a course notes (Harvard, 2010) (standard reference, not scraped)