The discriminant function and q-expansions

The expression (E₄³−E₆²)/1728 transforms with weight 12. Its constant term vanishes because both Eisenstein constant terms are one, and its coefficient of q is one because 720−(−1008)=1728. Jacobi's product identity states that this form equals q times the product of (1−qⁿ)²⁴ over positive n. Establishing the product's modular transformation is a theorem; expanding its first few factors does not independently establish the identity on the half-plane.

For |q|<1, the series of |q|ⁿ converges, so the product converges locally uniformly to a nonzero value away from q=0 after its initial q factor is removed. None of its factors vanishes there. Consequently Δ has no zero in the upper half-plane and has a simple zero at the cusp. These facts distinguish a complex lattice's nonzero discriminant from degeneration at the boundary. The function is nonvanishing inside the parameter space even though its Fourier coefficients alternate in sign and may be large.

To compute through qᴺ in Δ, only finitely many product factors matter: after removing the initial q, factors with index greater than N−1 contribute only terms too high to affect the requested coefficients. This is an exact formal-power-series statement, not a numerical approximation near one chosen τ. Binomial coefficients then give integer Fourier coefficients directly. Evaluation at a complex q, by contrast, introduces a tail and needs an analytic error estimate if a certified numerical value is requested.

The leading factor is important when choosing a truncation order. To find the coefficient of q³ in Δ, compute its remaining product through q². The n=1 factor contributes 1−24q+276q², the n=2 factor contributes 1−24q², and higher factors contribute nothing at those orders. Multiplication gives 1−24q+252q² before restoring q. Confusing the product's internal degree with the final series degree creates an off-by-one error that can survive otherwise correct arithmetic.

A q-expansion determines a holomorphic modular form uniquely by the identity theorem, because equality of all coefficients gives equality near the cusp and hence throughout the connected half-plane. A finite list needs a stronger input. For full level, the later valence bound limits a nonzero weight-k form's vanishing order to k/12. Thus sufficiently many equal initial coefficients can certify equality once both functions are already known modular forms of the same weight. Arbitrary analytic series have no such finite certificate.

Dividing by Δ shifts weight by minus twelve and decreases cusp order by one, without introducing any interior poles because Δ is nonvanishing there. This makes the discriminant a bridge between cusp forms and lower-weight holomorphic forms. Dividing a noncusp form may instead create a pole at infinity. The invariant quotient j provides a useful example: its weight is zero and it is holomorphic on the half-plane, but its cusp pole prevents it from being a holomorphic weight-zero modular form on the compactified curve.