The cusp forms have codimension at most the number of cusps #
Let H be a finite-index subgroup of SL(2, ℤ), acting through its image in GL(2, ℝ). The
constant term of a modular form on H at the cusp represented by γ ∈ SL(2, ℤ) depends on γ
only through the cusp orbit of γ • ∞, up to a sign: if γ • ∞ and γ' • ∞ are
H-equivalent, then γ' = h γ (±Tʲ) with h ∈ H, so the two constant-term functionals differ
by the factor (±1)ᵏ, the same for every form.
Consequently the constant terms at one representative of each cusp orbit already have the cusp
forms as their common kernel, and a subspace of M_k(H) meeting the cusp forms trivially
embeds into the functions on the cusp orbits. Its dimension is therefore at most the number
ε∞ of cusps of H. This is the upper bound in Diamond–Shurman's computation of the dimension
of the Eisenstein space: comparing it with the number of linearly independent Eisenstein series
shows that those series span a complement of the cusp forms.
Cusps at which every form of weight k has constant term zero can be left out of the count. In
odd weight these are all cusps when -1 ∈ Γ, and the irregular cusps. This sharper bound is what
the comparison needs at the levels without a cusp count, where the constant terms are read at
explicit representatives.
Main results #
TauCeti.ModularForm.cuspOrbitOfSL2Z_inv_eq_of_cuspTranslationOrbit_mk_eq: integral matrices in the same cusp translation orbit have inverses representing the same cusp.TauCeti.ModularForm.exists_constantTermAt_eq_zpow_smul_of_cuspOrbitOfSL2Z_eq: the constant terms at two representatives of the same cusp orbit agree up to a factor(±1)ᵏ.TauCeti.ModularForm.constantTermAt_eq_of_cuspOrbitOfSL2Z_eq: at even weight, the constant terms at equivalent cusp representatives agree.ModularForm.constantTermAt_mul_constantTermAt_eq_of_cuspOrbitOfSL2Z_eq: when the weights have even sum, the product of two constant terms is the same at equivalent cusp representatives.TauCeti.ModularForm.finrank_le_finrank_range_constantTerms_of_disjoint: a subspace disjoint from cusp forms has dimension at most the image of representative constant terms.TauCeti.ModularForm.finrank_le_card_cuspOrbit_of_disjoint: a subspace ofM_k(H)with zero intersection with the cusp forms has dimension at most the number of cusps ofH.TauCeti.ModularForm.finrank_le_natCard_of_disjoint: the same bound by the number of any family of cusp representatives that misses only cusps without constant terms in weightk.
References #
Integral matrices whose cosets lie in the same cusp translation orbit have inverses representing the same cusp.
Constant terms at equivalent cusps agree up to sign. If γ • ∞ and γ' • ∞ lie in the
same cusp orbit of H, the constant-term functionals at γ and γ' on M_k(H) differ by the
factor uᵏ for a sign u = ±1.
At even weight, constant terms at representatives of the same cusp orbit agree, without
requiring that -I ∈ H.
Reading constant terms at enough cusp representatives bounds the dimension of a subspace
disjoint from cusp forms by the dimension of the resulting constant-term image. Here the r i
need only meet every cusp that carries a constant term in weight k (an irregular cusp in odd
weight, for instance, may be skipped: ModularForm.constantTermAt_eq_zero_of_odd).
Constant terms at enough cusps bound a complement of the cusp forms. Let r i be
finitely many cusp representatives such that every cusp either is equivalent to some r i or
carries no constant term in weight k (for instance an irregular cusp in odd weight,
ModularForm.constantTermAt_eq_zero_of_odd). Then a subspace of M_k(H) whose intersection with
the cusp forms is zero has dimension at most the number of representatives.
The cusp forms have codimension at most the number of cusps. A subspace of M_k(H)
whose intersection with the cusp forms is zero, such as a complement of the cusp forms, has
dimension at most the number of cusps of H.
When the weights of f and g have even sum, the product of their constant terms at a cusp
does not depend on the representative: the signs of the two constant terms cancel.