The period integral of a cusp form against a binary form #
Let f be a cusp form of weight k = w + 2 on an arithmetic subgroup and P a binary form of
degree w with coefficients in a commutative semiring R mapping to ℂ. The period integrand
is the
one-form f(z) P(z, 1) dz, and its integrals
∫_β^α f(z) P(z, 1) dz
along the geodesics between cusps α, β ∈ ℙ¹(ℚ) are the periods of f. They are the values
of the period pairing between cusp forms and modular symbols, under which the symbol
{α, β} ⊗ P is sent to the integral above; the pairing itself, its descent to the coinvariants
𝕄_w(Γ; R) and its Hecke equivariance are built on the results of this file.
Two facts about the integrand are established. First, its transformation law under a matrix
g ∈ GL(2, ℚ) of positive determinant: substituting z ↦ g • z in f(z) P(z, 1) dz produces
(det g)⁻ʷ · (f ∣[k] g)(z) · P(az + b, cz + d) dz, and for γ ∈ SL(2, ℤ) this is the one-form
of f ∣[k] γ against P ∣ γ, the right action of TauCeti.binaryFormRep. This is what makes
the periods compatible with the relation {γα, γβ} ⊗ P = {α, β} ⊗ (P ∣ γ) defining modular
symbols. Second, absolute convergence: both endpoints of the geodesic are cusps, and the
integrand is integrable along the whole geodesic. Moving each endpoint to i∞ reduces this to
the exponential decay of a cusp form there, which beats the polynomial growth of P. The same
estimate, uniform on vertical strips, gives the hypotheses of TauCeti.cuspIntegral_add_adjacent,
so the periods are additive: ∫_γ^β + ∫_β^α = ∫_γ^α, the analytic form of the relation
{α, β} + {β, γ} = {α, γ} of modular symbols. Only the SL(2, ℤ) form of the transformation
law, stated through TauCeti.binaryFormRep, needs R to be a ring.
Main definitions #
TauCeti.ModularSymbols.periodIntegrand f P: the functionz ↦ f(z) · P(z, 1)onℍ.
Main results #
TauCeti.ModularSymbols.periodIntegrand_add_left,TauCeti.ModularSymbols.periodIntegrand_sum_left,TauCeti.ModularSymbols.periodIntegrand_smul_left: the integrand isℂ-linear in the function, so that the periods define a linear map on cusp forms.TauCeti.ModularSymbols.periodIntegrand_add_right,TauCeti.ModularSymbols.periodIntegrand_smul_right: the integrand isR-linear in the binary form, so that the periods extend linearly to the module of modular symbols.TauCeti.ModularSymbols.periodIntegrand_slash_apply: the weight-2slash of the integrand by a rational matrixgof positive determinant is(det g)⁻ʷ · (f ∣[k] g)(τ) · P(aτ + b, cτ + d).TauCeti.ModularSymbols.mdifferentiable_periodIntegrand: the integrand of a holomorphicfis holomorphic.TauCeti.ModularSymbols.integrableOn_resToImagAxis_periodIntegrand_slash: absolute convergence of the period integral of a cusp form of weightw + 2against a binary form of degreewalong the geodesic between any two distinct cusps.TauCeti.ModularSymbols.tendsto_periodIntegrand_slash: the slashed integrand tends to0ati∞uniformly on vertical strips.TauCeti.ModularSymbols.cuspIntegral_periodIntegrand_add_adjacent: additivity of the periods,∫_α^β f(z) P(z, 1) dz + ∫_β^γ f(z) P(z, 1) dz = ∫_α^γ f(z) P(z, 1) dz.TauCeti.ModularSymbols.cuspIntegral_mul_sub_pow: the period∫_α^β f(z) (z - τ)ʷ dzis a polynomial inτwhose coefficients are the periods∫_α^β f(z) zʲ dz.TauCeti.ModularSymbols.periodIntegrand_slash_mapGL: forγ ∈ SL(2, ℤ), slashing the integrand offagainstPbyγgives the integrand off ∣[k] γagainstP ∣ γ.TauCeti.ModularSymbols.geodesicIntegral_mapGL_mul_periodIntegrand: the transformation law∫_{γβ}^{γα} f(z) P(z, 1) dz = ∫_β^α (f ∣[k] γ)(z) (P ∣ γ)(z, 1) dzforγ ∈ SL(2, ℤ), andTauCeti.ModularSymbols.geodesicIntegral_mapGL_mul_periodIntegrand_of_mem: its form forγin the level off, andTauCeti.ModularSymbols.cuspIntegral_periodIntegrand_mapGL_smul_of_mem: the same between arbitrary cusps,∫_{γβ}^{γα} f(z) P(z, 1) dz = ∫_β^α f(z) (P ∣ γ)(z, 1) dz.TauCeti.ModularSymbols.periodIntegrand_adjugate_slash: for an integral matrixδof positive determinant, slashing the integrand offagainstP ∣ adj δbyδgives the integrand off ∣[k] δagainstP, andTauCeti.ModularSymbols.cuspIntegral_periodIntegrand_slash: the resulting substitution∫_β^α (f ∣[k] δ)(z) P(z, 1) dz = ∫_{δβ}^{δα} f(z) (P ∣ adj δ)(z, 1) dz, the adjunction between the slash action on forms and the action of integral matrices on modular symbols that underlies the Hecke equivariance of the period pairing.
References #
- G. Shimura, Introduction to the arithmetic theory of automorphic functions, §8.2, (8.2.15)–(8.2.16).
- Y. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66, §1.
The period integrand z ↦ f(z) · P(z, 1) of a function f : ℍ → ℂ against a binary form
P of degree w: the one-form f(z) P(z, 1) dz whose integrals along geodesics between cusps
are the periods of f.
Equations
- TauCeti.ModularSymbols.periodIntegrand f P z = f z * (MvPolynomial.aeval ![↑z, 1]) ↑P
Instances For
The period integrand of the zero function vanishes.
The period integrand is additive in the function.
The period integrand commutes with finite sums of functions.
The period integrand is ℂ-linear in the function.
The period integrand against the zero form vanishes.
The period integrand is additive in the binary form.
The period integrand is R-linear in the binary form.
The transformation law #
The transformation law of the period integrand. Slashing f(z) P(z, 1) in weight 2 by
a rational matrix g of positive determinant gives (det g)⁻ʷ · (f ∣[k] g)(τ) · P(aτ + b, cτ + d),
where k = w + 2 and aτ + b, cτ + d are the numerator and denominator of the Möbius action
of g: the factors (cτ + d) of the two slashes combine with the homogeneity of P in degree
w.
Absolute convergence #
The period integrand of a holomorphic function is holomorphic.
Convergence at i∞: for a cusp form f of weight w + 2 and a binary form P of degree
w, the slashed integrand (f(z) P(z, 1)) ∣[2] g is integrable along the imaginary axis away
from 0, by exponential decay of f ∣[k] g against polynomial growth of P.
Absolute convergence of the period integral. For a cusp form f of weight w + 2 on an
arithmetic subgroup, a binary form P of degree w, and a rational matrix g of positive
determinant, the integrand of ∫_{g • 0}^{g • ∞} f(z) P(z, 1) dz is integrable along the whole
geodesic between the two distinct cusps g • 0 and g • ∞: each endpoint is moved to i∞, the
finite one by the reflection S.
Decay on vertical strips: for a cusp form f of weight w + 2 on an arithmetic
subgroup, a binary form P of degree w, and a rational matrix g of positive determinant, the
slashed integrand (f(z) P(z, 1)) ∣[2] g tends to 0 at i∞ uniformly on every vertical strip:
the exponential decay of f ∣[k] g beats the polynomial growth of P, which on a strip is
polynomial in the imaginary part.
The periods are additive: for a cusp form f of weight w + 2 on an arithmetic subgroup
and a binary form P of degree w, the periods satisfy
∫_α^β f(z) P(z, 1) dz + ∫_β^γ f(z) P(z, 1) dz = ∫_α^γ f(z) P(z, 1) dz for all cusps α, β,
γ. Since the symbol {α, β} ⊗ P pairs to ∫_β^α f(z) P(z, 1) dz, this is the analytic
counterpart of the relation {α, β} + {β, γ} = {α, γ} of modular symbols, and is what lets the
periods define a linear functional on degree-zero divisors on the cusps.
The period polynomial is a polynomial in τ with periods as coefficients. For a cusp form
f of weight w + 2 on an arithmetic subgroup, τ ∈ ℂ and cusps α, β, the binomial
expansion of (z - τ)ʷ gives
∫_α^β f(z) (z - τ)ʷ dz = ∑_{j ≤ w} (w choose j) (-τ)ʷ⁻ʲ ∫_α^β f(z) zʲ dz,
where f(z) zʲ is the period integrand of f against the monomial X₀ʲ X₁ʷ⁻ʲ.
The transformation law under SL(2, ℤ) #
The transformation law under SL(2, ℤ): slashing the integrand of f against P by
γ ∈ SL(2, ℤ) gives the integrand of f ∣[k] γ against P ∣ γ, the right action of γ on
binary forms (TauCeti.binaryFormRep).
The transformation law of the periods under SL(2, ℤ):
∫_{γ g • 0}^{γ g • ∞} f(z) P(z, 1) dz = ∫_{g • 0}^{g • ∞} (f ∣[k] γ)(z) (P ∣ γ)(z, 1) dz,
the substitution z ↦ γ • z in the period integral. The identity holds for every rational g
(it is Matrix.GeneralLinearGroup.geodesicIntegral_mul and the transformation law of the
integrand); both sides are period integrals along geodesics when 0 < det g.
The periods of a form respect the modular-symbol relation: for γ in the level Γ of a
slash-invariant f,
∫_{γ g • 0}^{γ g • ∞} f(z) P(z, 1) dz = ∫_{g • 0}^{g • ∞} f(z) (P ∣ γ)(z, 1) dz, the analytic
counterpart of {γα, γβ} ⊗ P = {α, β} ⊗ (P ∣ γ) in 𝕄_w(Γ; R). As for
geodesicIntegral_mapGL_mul_periodIntegrand, the identity holds for every rational g, and both
sides are period integrals along geodesics when 0 < det g.
The periods between cusps respect the modular-symbol relation: for γ in the level Γ
of a slash-invariant f and cusps α, β,
∫_{γβ}^{γα} f(z) P(z, 1) dz = ∫_β^α f(z) (P ∣ γ)(z, 1) dz, the analytic counterpart of
{γα, γβ} ⊗ P = {α, β} ⊗ (P ∣ γ) in 𝕄_w(Γ; R) (TauCeti.ModularSymbols.symbol_mapGL_smul).
The transformation law under integral matrices #
The transformation law under integral matrices of positive determinant. For a rational
matrix g with integral entries A and 0 < det g, slashing the integrand of f against
P ∣ adj A in weight 2 by g gives the integrand of f ∣[k] g against P: the adjugate sends
(aτ + b, cτ + d) to det g • (τ, 1), and the resulting factor (det g)ʷ cancels the
(det g)⁻ʷ of periodIntegrand_slash_apply. On SL(2, ℤ) the adjugate is the inverse, and this
is periodIntegrand_slash_mapGL read backwards.
The substitution z ↦ g • z for integral matrices of positive determinant: for a rational
matrix g with integral entries A and 0 < det g,
∫_β^α (f ∣[k] g)(z) P(z, 1) dz = ∫_{gβ}^{gα} f(z) (P ∣ adj A)(z, 1) dz. Since the symbol
{α, β} ⊗ P pairs to ∫_β^α f(z) P(z, 1) dz, this says that slashing a form by g is adjoint to
the action {α, β} ⊗ P ↦ {gα, gβ} ⊗ (P ∣ adj A) of g on modular symbols.