Integrals along geodesics between cusps, and their additivity #
For F : ℍ → ℂ and cusps a, b ∈ ℙ¹(ℚ), this file defines the integral
TauCeti.cuspIntegral F a b = ∫_a^b F(z) dz of the one-form F(z) dz along the hyperbolic
geodesic from a to b: it is the geodesic integral g.geodesicIntegral F of
TauCeti.NumberTheory.ModularForms.GeodesicIntegral.Basic for any rational matrix g of positive
determinant with g • 0 = a and g • ∞ = b, which only depends on the endpoints.
The main result is additivity, ∫_a^b F(z) dz + ∫_b^c F(z) dz = ∫_a^c F(z) dz: Cauchy's
theorem for the ideal triangle with vertices a, b, c, all three of which lie on the boundary.
It holds for holomorphic F whose weight-2 slashes F ∣[2] g by rational matrices of positive
determinant are integrable near i∞ along the imaginary axis and tend to 0 at i∞ uniformly on
vertical strips; the period integrand f(z) P(z, 1) of a cusp form satisfies both conditions
(TauCeti.NumberTheory.ModularForms.ModularSymbols.Period.Integral). The proof uses a primitive Φ
of F on ℍ (TauCeti.Analysis.Complex.UpperHalfPlane.Primitive), whose composite with g is a
primitive of F ∣[2] g. The first condition gives Φ a limit at each cusp along each geodesic
ending there; the second shows that the limit at a cusp does not depend on the geodesic, since
after moving the cusp to i∞ two such geodesics become vertical lines, joined by horizontal
segments on which F ∣[2] g is uniformly small. Each geodesic integral is then the difference of
the values of Φ at its endpoints, and additivity follows.
Additivity is what makes the periods of a cusp form a function of degree-zero divisors on the cusps, the first step of the period pairing between cusp forms and modular symbols.
The same primitive also computes integrals from a point τ ∈ ℍ to i∞ along the vertical ray
z = τ + i t, as its limit at i∞ minus its value at τ. Comparing this with its values at the
cusps gives the substitution z ↦ g • z in such an integral: the image of the vertical ray from
τ ends at the cusp g • ∞, and is replaced by the vertical ray from g • τ followed by the
geodesic from i∞ to g • ∞. This is how the Eichler integral ∫_τ^{i∞} f(z) (z - τ)ⁿ dz of a
cusp form transforms under SL(2, ℤ) up to periods.
Main definitions #
TauCeti.cuspIntegral F a b: the integral∫_a^b F(z) dzalong the geodesic from the cuspato the cuspb.
Main results #
TauCeti.cuspIntegral_smul_zero_smul_infty:∫_{g • 0}^{g • ∞} F(z) dz = g.geodesicIntegral F.TauCeti.cuspIntegral_same,TauCeti.cuspIntegral_symm:∫_a^a = 0and∫_b^a = -∫_a^b.TauCeti.cuspIntegral_slash: the substitutionz ↦ γ • z,∫_a^b (F ∣[2] γ)(z) dz = ∫_{γ • a}^{γ • b} F(z) dz.TauCeti.cuspIntegral_zero,TauCeti.cuspIntegral_smul: the integral of0vanishes, and the integral is homogeneous under scalar multiplication of the integrand.TauCeti.cuspIntegral_add,TauCeti.cuspIntegral_sum: additivity in the integrand, for integrands that are integrable along the geodesic.TauCeti.cuspIntegral_add_adjacent: additivity,∫_a^b + ∫_b^c = ∫_a^c.TauCeti.integral_Ioi_slash_eq_add_cuspIntegral: the substitutionz ↦ g • zin an integral from a point ofℍtoi∞,∫_τ^{i∞} (F ∣[2] g)(z) dz = ∫_{g • τ}^{i∞} F(z) dz + ∫_{i∞}^{g • ∞} F(z) dz.
References #
- G. Shimura, Introduction to the arithmetic theory of automorphic functions, §8.2.
- Y. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66, §1.
Primitives on the upper half-plane #
Limits of a primitive at i∞ #
Values of a primitive at the cusps #
Integrals between cusps #
The integral ∫_a^b F(z) dz of the one-form F(z) dz along the hyperbolic geodesic from the
cusp a to the cusp b of ℙ¹(ℚ). For a ≠ b it is g.geodesicIntegral F for any rational
matrix g of positive determinant with g • 0 = a and g • ∞ = b, the choice not mattering
(cuspIntegral_smul_zero_smul_infty); for a = b it is 0. Both endpoints are improper, and,
as for Matrix.GeneralLinearGroup.geodesicIntegral, the value is 0 when the integrand is not
integrable.
Equations
Instances For
The integral between the cusps g • 0 and g • ∞ is the geodesic integral along g.
The integral from a cusp to itself vanishes.
Reversing the orientation changes the sign: ∫_b^a F(z) dz = -∫_a^b F(z) dz.
The substitution z ↦ γ • z: for γ of positive determinant, the integral of the
pulled-back one-form (F ∣[2] γ)(z) dz from a to b is the integral of F(z) dz from γ • a
to γ • b.
The integral between cusps is homogeneous under scalar multiplication of the integrand:
∫_a^b c F(z) dz = c ∫_a^b F(z) dz. Additivity in the integrand needs integrability, and holds
under the hypotheses of TauCeti.cuspIntegral_add.
The integral of the zero integrand between two cusps vanishes.
The integral between the cusps g • 0 and g • ∞ is additive in the integrand when both
integrands are integrable along the geodesic g.
The integral between the cusps g • 0 and g • ∞ commutes with finite sums of integrands
that are integrable along the geodesic g.
Additivity of integrals between cusps: ∫_a^b F(z) dz + ∫_b^c F(z) dz = ∫_a^c F(z) dz
for a holomorphic F whose weight-2 slashes by rational matrices of positive determinant are
integrable near i∞ along the imaginary axis and tend to 0 at i∞ uniformly on vertical
strips. These are the conditions under which the integrals along the three sides of the ideal
triangle with vertices a, b, c are computed by a primitive Φ of F: each is the difference
of the limits of Φ at its two endpoints, and the limit of Φ at a cusp does not depend on the
geodesic along which the cusp is approached.
Integrals from a point of ℍ to i∞ #
The substitution z ↦ g • z in an integral from a point of ℍ to i∞. For g a
rational matrix of positive determinant and τ ∈ ℍ,
∫_τ^{i∞} (F ∣[2] g)(z) dz = ∫_{g • τ}^{i∞} F(z) dz + ∫_{i∞}^{g • ∞} F(z) dz,
the integrals from τ and from g • τ taken along the vertical rays z = τ + i t and
z = g • τ + i t, t > 0, and the last along the geodesic between the two cusps. The left side
is the integral of F(z) dz along the image under g of the vertical ray from τ, which ends at
the cusp g • ∞, and the identity is Cauchy's theorem for the triangle with vertices g • τ,
g • ∞ and i∞. The hypotheses on F are those of TauCeti.cuspIntegral_add_adjacent, and
the two vertical integrands are assumed integrable.