The valence formula and dimension calculations

For a nonzero meromorphic full-level form, let v∞ be its order in q at the cusp, and let vₚ be its ordinary local analytic order in τ at an interior point. Poles count negatively. The valence formula says that v∞ plus half the order at i, one third the order at ρ, and the orders at one representative of every other orbit equals k/12. Boundary representatives must not be counted twice. The halves and thirds concern stabilizer orders in PSL₂(ℤ), not the doubled SL₂ matrix stabilizers.

One proof applies the argument principle to a truncated fundamental domain, then compares paired edge integrals using the weight transformation law. Small indentations around elliptic points account for their angular fractions. We use the resulting theorem rather than claiming that a drawing proves it. For a holomorphic form every term is nonnegative, so its cusp order cannot exceed k/12. Negative weight consequently admits no nonzero holomorphic form, and at weight zero a holomorphic form must be constant by compactness.

For E₄ the total budget is one third. The fixed-point transformation at ρ forces a zero there, so it is exactly simple and there are no other zeros. Similarly E₆ has budget one half and a simple zero at i. A weight-two form would be forced to vanish at both elliptic points: the contribution would be at least one half plus one third, larger than its budget one sixth. Thus M₂ is zero. Omitting the stabilizer weights either miscounts these examples or obscures the contradiction.

The full graded ring is the polynomial ring in E₄ and E₆, whose weights are four and six. This structure theorem can be proved by successively canceling cusp coefficients and dividing cusp forms by Δ. Since Δ has a single cusp zero and no interior zeros, multiplication gives Sₖ=ΔMₖ₋₁₂, with negative-weight spaces interpreted as zero. Counting nonnegative pairs a,b satisfying 4a+6b=k gives the dimension of Mₖ. This ring statement is specific to full level and cannot be transferred unchanged to congruence subgroups.

Suppose f and g are known holomorphic forms of the same full-level weight. If their coefficients agree for all indices from zero through floor(k/12), their difference would have cusp order greater than k/12 unless it were zero. The valence bound therefore proves equality. The constant term is included because otherwise two different noncusp forms could escape the test. This exact complex equality argument is separate from arithmetic congruence versions of Sturm's theorem, which require coefficient-ring hypotheses.

A basis obtained by counting monomials makes calculation concrete, but a basis adapted to successive leading q powers often makes certification easier. Linear combinations can cancel initial coefficients without changing weight, and Δ provides a controlled way to raise vanishing. The finite-dimensional perspective is useful in reverse as well: a proposed Fourier series that vanishes beyond the budget cannot represent a nonzero holomorphic form of that weight, even if its coefficients look arithmetically plausible. Local power-series regularity alone supplies no existence theorem.