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The discriminant is a nonvanishing cusp form of weight 12

Statement

Define $Δ : = (E_{4}^{3} - E_{6}^{2}) / 1728$ . Then $Δ$ is a cusp form of weight $12$ whose $q$ -expansion is $Δ (τ) = q - 24 q^{2} + O (q^{3}), q = e^{2 π i τ};$ in particular $Δ \neq 0$ and ${ord}_{\infty} (Δ) = 1$ , and the valence formula gives $Δ (τ) \neq 0$ for every $τ \in H$ . Moreover $E_{4}^{3}$ and $E_{6}^{2}$ are linearly independent in $M_{12}$ , this space being two-dimensional with basis ${E_{4}^{3}, E_{6}^{2}}$ .

Facts & Assumptions

Given: $E_{4} \in M_{4}$ and $E_{6} \in M_{6}$ with $E_{4} = 1 + 240 q + 2160 q^{2} + O (q^{3})$ and $E_{6} = 1 - 504 q - 16632 q^{2} + O (q^{3})$ (Eisenstein series are modular forms; their Fourier coefficients, The level-one Eisenstein series E_k and the weight-two series E_2, The divisor power sums $σ_{k}$ ).

[F1]

$M_{12}$ is a two-dimensional complex vector space. Multiplication adds weights, with $M_{4} M_{4} \subseteq M_{8}$ , $M_{8} M_{4} \subseteq M_{12}$ and $M_{6} M_{6} \subseteq M_{12}$ ; the cusp forms are the kernel of the constant-term functional $f \mapsto f (\infty)$ (The dimension of the space of level-one modular forms, Level-one modular forms and cusp forms).

[F2]

Valence formula: for even $k$ and $0 \neq f \in M_{k}$ , $\sum_{P \neq \infty} {ord}_{P} (f) / ν_{P} = k / 12 - {ord}_{\infty} (f)$ (The level-one valence formula).

Proof

1.1F1givenalgebra

$E_{4}^{3} \in M_{12}$ and $E_{6}^{2} \in M_{12}$ by [F1], hence $Δ = (E_{4}^{3} - E_{6}^{2}) / 1728 \in M_{12}$ . From the displayed expansions, $E_{4}^{3} = 1 + 720 q + (3 \cdot 2160 + 3 \cdot 240^{2}) q^{2} + O (q^{3}) = 1 + 720 q + 179280 q^{2} + O (q^{3})$ and $E_{6}^{2} = 1 - 1008 q + (504^{2} - 2 \cdot 16632) q^{2} + O (q^{3}) = 1 - 1008 q + 220752 q^{2} + O (q^{3})$ . Therefore $Δ = \frac{1}{1728} (1728 q - 41472 q^{2} + O (q^{3})) = q - 24 q^{2} + O (q^{3})$ . In particular $Δ \neq 0$ , $Δ (\infty) = 0$ , so $Δ$ is a cusp form, and ${ord}_{\infty} (Δ) = 1$ .

2.1F1F2step 1.1givenalgebra∎

Applying the valence formula [F2] to $Δ$ of weight $12$ gives $\sum_{P \neq \infty} {ord}_{P} (Δ) / ν_{P} = 12 / 12 - 1 = 0$ ; every summand is a nonnegative multiple of $1 / ν_{P}$ , so all vanish and $Δ$ has no zeros on $H$ . Finally, if $a E_{4}^{3} + b E_{6}^{2} = 0$ then comparing constant terms gives $a + b = 0$ and comparing $q$ -coefficients gives $720 a - 1008 b = 0$ , whence $(720 + 1008) a = 0$ and $a = b = 0$ ; so $E_{4}^{3}, E_{6}^{2}$ are linearly independent in the two-dimensional space $M_{12}$ and therefore form a basis.

Depends on

Used by

The modular discriminant and the j-invariant Definition
The first Fourier coefficients of E4, E6, Delta and j Example
The j-invariant of the Legendre normal form Lemma
The Jacobi product formula for the discriminant Lemma
The graded ring of level-one modular forms Theorem
The j-invariant classifies complex tori Theorem

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Sources

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