The cusps of Γ₁(N) #
Away from the levels 1, 2 and 4, the number ε∞ of cusps of Γ₁(N) satisfies
2 ε∞ = ∑_{g ∣ N} φ(g) φ(N / g). This is the cusp count in the dimension formulas for Γ₁(N).
Cusps are counted as the cusp orbits of the effective image of Γ₁(N) in PSL(2, ℝ), the
boundary orbits adjoined to its analytic quotient.
Every cusp is γ • ∞ for some γ ∈ SL(2, ℤ). Take γ and γ' with bottom rows (c, d) and
(c', d'). The cusps γ • ∞ and γ' • ∞ are Γ₁(N)-equivalent exactly when
(c', d') ≡ ±(c, d + j c) modulo N for some integer j. Without the sign, this congruence is
detected by the label of γ. The label is the divisor g = gcd(c, N) of N, the unit c / g
modulo N / g, and the unit d modulo g. Every label occurs. When N ∉ {1, 2, 4}, the
matrices γ and -γ have different labels. So each cusp then has exactly two labels.
At the three exceptional levels the criterion is a finite check modulo N. Every cusp of Γ₁(1)
is ∞, and every cusp of Γ₁(2) is ∞ or 0. Every cusp of Γ₁(4) is ∞, 0, or one with
c ≡ 2 modulo 4, and the last ones are all equivalent (to 1/2). That cusp is irregular:
Γ₁(4) contains γ (-T) γ⁻¹, the matrix -T being the negative of the translation T.
Main definitions #
TauCeti.gamma1CuspLabel: the label(g, u, v)ofγ ∈ SL(2, ℤ).
Main results #
TauCeti.cuspOrbitOfSL2Z_Gamma1_eq_iff: the congruence criterion for equivalence of cusps.TauCeti.cuspOrbitOfSL2Z_Gamma1_eq_iff_gamma1CuspLabel: two matrices represent the same cusp exactly when one has the label of the other or of its negative.TauCeti.card_cuspOrbit_Gamma1: the cusp count2 ε∞ = ∑_{g ∣ N} φ(g) φ(N / g)forN ∉ {1, 2, 4}.TauCeti.card_cuspOrbit_Gamma1_of_prime:Γ₁(p)hasp - 1cusps for an odd primep.TauCeti.cuspOrbitOfSL2Z_Gamma1_one,TauCeti.cuspOrbitOfSL2Z_Gamma1_two,TauCeti.cuspOrbitOfSL2Z_Gamma1_four: cusp representatives at the levels1,2and4.TauCeti.cuspOrbitOfSL2Z_Gamma1_four_eq_of_intCast_eq_two,TauCeti.mul_neg_T_mul_inv_mem_Gamma1_four: the cusps ofΓ₁(4)withc ≡ 2modulo4form one irregular cusp.
References #
- F. Diamond and J. Shurman, A First Course in Modular Forms, §3.8.
Equivalence of cusps #
The criterion for equivalence of cusps of Γ₁(N). For γ and γ' with bottom rows
(c, d) and (c', d'), the cusps γ • ∞ and γ' • ∞ are Γ₁(N)-equivalent exactly when
(c', d') ≡ ±(c, d + j c) modulo N for some integer j.
Labels of cusps #
The label of the cusp γ • ∞ of Γ₁(N), up to sign. For γ ∈ SL(2, ℤ) with bottom
row (c, d), it is the divisor g = gcd(c, N) of N together with the unit c / g modulo
N / g and the unit d modulo g. Two matrices have the same label exactly when their bottom
rows are congruent modulo N up to adding a multiple of c to d
(gamma1CuspLabel_eq_iff); the cusps of Γ₁(N) are the labels up to the sign of γ
(cuspOrbitOfSL2Z_Gamma1_eq_iff_gamma1CuspLabel).
Instances For
The divisor in the label of γ is gcd(c, N), for c the lower-left entry of γ.
The first unit in the label of γ is c / g modulo N / g, for c the lower-left entry
of γ and g = gcd(c, N).
The second unit in the label of γ is d modulo gcd(c, N), for (c, d) the bottom row
of γ.
The label of γ is (g, u, v) as soon as gcd(c, N) = g, c / g reduces to u and d
reduces to v.
Equality of labels. Two matrices with bottom rows (c, d) and (c', d') have the same
label exactly when c' ≡ c and d' ≡ d + j c modulo N for some integer j.
Every divisor g of N, unit modulo N / g and unit modulo g form a label.
The criterion for equivalence of cusps of Γ₁(N) in terms of labels. The cusps γ • ∞
and γ' • ∞ of Γ₁(N) agree exactly when γ' has the label of γ or of -γ.
The label of -γ. It has the divisor of the label of γ and the negatives of its two
units.
Away from the levels 1, 2 and 4, the matrices γ and -γ have different labels.
The cusp count #
The cusp count ε∞ of Γ₁(N): away from the levels 1, 2 and 4, twice the number
of cusps of Γ₁(N) is ∑_{g ∣ N} φ(g) φ(N / g).
Γ₁(p) has p - 1 cusps for an odd prime p.
The cusps at the levels 1, 2 and 4 #
Γ₁(1) = SL(2, ℤ) has the single cusp ∞.
Every cusp of Γ₁(2) is ∞ = 1 • ∞ or 0 = S • ∞.
Every cusp of Γ₁(4) is ∞ = 1 • ∞, or 0 = S • ∞, or γ • ∞ with c ≡ 2 modulo 4
for the lower-left entry c of γ.
The cusps γ • ∞ of Γ₁(4) whose lower-left entry is 2 modulo 4 are all equivalent.
The cusp of Γ₁(4) with c ≡ 2 modulo 4 is irregular: if the lower-left entry of γ
is 2 modulo 4, then Γ₁(4) contains the conjugate γ (-T) γ⁻¹ of the negative translation.