The Eichler integral of a cusp form #
For a function f on ℍ with q-expansion f = ∑ aₘ qᵐ at width h (where
q = exp (2πiτ / h)), the n-fold Eichler integral is the termwise antiderivative
E_n f = ∑ (h / m)ⁿ aₘ qᵐ
for the normalized derivative D = (2πi)⁻¹ d/dτ of Derivative.normalizedDerivOfComplex, which
acts on qᵐ as multiplication by m / h. For n ≥ 1 the constant term is dropped (the
coefficient (h / 0)ⁿ is 0), so E_n f vanishes at i∞, and differentiating n times returns
f minus its constant term. For a cusp form f of weight k ≥ 2 the function E_{k-1} f is the
classical Eichler integral of f, a (k - 1)-fold antiderivative: D^{k-1} E_{k-1} f = f.
Classically, the failure of the Eichler integral to transform in weight 2 - k is a polynomial
whose coefficients are periods of f, which is how vanishing periods force a cusp form to vanish
(the injectivity of the Eichler–Shimura period map); that transformation law is not part of this
file.
Main definitions #
TauCeti.eichlerIntegral h n f: then-fold Eichler integral∑ (h / m)ⁿ aₘ qᵐ.
Main results #
TauCeti.eichlerIntegral_zero,TauCeti.eichlerIntegral_add,TauCeti.eichlerIntegral_smul,TauCeti.eichlerIntegral_neg: linearity under the appropriate analytic hypotheses.TauCeti.hasSum_eichlerIntegral: the defining series converges toeichlerIntegral h n fat every point ofℍ.TauCeti.qExpansion_eichlerIntegral_coeff: theq-expansion coefficients ofeichlerIntegral h n fare(h / m)ⁿ aₘ.TauCeti.mdifferentiable_eichlerIntegral: the Eichler integral is holomorphic onℍ.TauCeti.isZeroAtImInfty_eichlerIntegral: forn ≥ 1it vanishes ati∞.TauCeti.normalizedDerivOfComplex_eichlerIntegral_add_two,TauCeti.normalizedDerivOfComplex_eichlerIntegral_one:D E_{n+2} f = E_{n+1} fandD E_1 f = f - a₀.TauCeti.iterate_normalizedDerivOfComplex_eichlerIntegral:D^{n+1} E_{n+1} f = f - a₀.TauCeti.CuspFormClass.iterate_normalizedDerivOfComplex_eichlerIntegral: for a cusp form,D^{n+1} E_{n+1} f = f.
References #
- G. Shimura, Introduction to the arithmetic theory of automorphic functions, §8.2.
- M. Eichler, Eine Verallgemeinerung der Abelschen Integrale, Math. Z. 67 (1957), 267–298.
- The AINTLIB
LeanModularFormsproject,ModularSymbols/EichlerInjective.lean(eichlerCoeff,bol_iterated_eichler), whose coefficient normalizationaₘ / m^{k-1}is the width-one case of the one used here.
The n-fold Eichler integral of f : ℍ → ℂ at width h: the series
∑ (h / m)ⁿ aₘ qᵐ, where aₘ are the coefficients of qExpansion h f and q = 𝕢 h τ.
It is the termwise n-fold antiderivative of the q-expansion of f for the normalized
derivative D = (2πi)⁻¹ d/dτ, with the constant term dropped when n ≥ 1. For a cusp form of
weight k ≥ 2, eichlerIntegral h (k - 1) f is the classical Eichler integral of f.
Equations
- TauCeti.eichlerIntegral h n f τ = ∑' (m : ℕ), (↑h / ↑m) ^ n * (PowerSeries.coeff m) (UpperHalfPlane.qExpansion h f) * Function.Periodic.qParam h ↑τ ^ m
Instances For
The defining series of the Eichler integral converges at every point of ℍ.
The Eichler integral of the zero function is zero.
The Eichler integral commutes with complex scalar multiplication when the cusp function is analytic at zero.
The Eichler integral commutes with negation when the cusp function is analytic at zero.
The Eichler integral is additive on holomorphic periodic functions bounded at i∞.
The Eichler integral commutes with subtraction on holomorphic periodic functions bounded
at i∞.
The 0-fold Eichler integral is the function itself.
The Eichler integral has the same period h as the q-parameter.
The Eichler integral is holomorphic on ℍ.
The Eichler integral vanishes at i∞ once at least one antiderivative is taken: its
q-expansion has no constant term.
The q-expansion of the Eichler integral: its m-th coefficient is (h / m)ⁿ aₘ.
Differentiating an Eichler integral of order at least two lowers the order by one:
D E_{n+2} f = E_{n+1} f.
Differentiating the first Eichler integral returns the function minus its constant term:
D E_1 f = f - a₀.
The Eichler integral is an iterated antiderivative: differentiating E_{n+1} f exactly
n + 1 times with the normalized derivative D returns f minus its constant term.
The Eichler integral of a cusp form is an iterated antiderivative:
D^{n+1} E_{n+1} f = f. For a cusp form of weight k ≥ 2 and n + 1 = k - 1, this recovers f
from its Eichler integral.