Siegel functions #
For r = (r₁, r₂) ∈ ℝ² the Siegel function g_r is the holomorphic function on the upper
half-plane given by the product
g_r(τ) = -e^{π i B₂(r₁) τ} e^{π i r₂ (r₁ - 1)} (1 - w) ∏_{n ≥ 1} (1 - qⁿ w) (1 - qⁿ w⁻¹),
where q = e^{2π i τ}, w = e^{2π i (r₁ τ + r₂)} and B₂(x) = x² - x + 1/6 is the second
Bernoulli polynomial. For rational r these functions are the basic building blocks of modular
units of level N: suitable powers and products of them are modular functions whose values at
imaginary quadratic points generate ray class fields, and they control the small-power class field
inclusions in the proof of the class number one theorem.
This file constructs g_r and develops its elementary analytic and algebraic properties:
the defining product converges locally uniformly on the upper half-plane, so g_r is
holomorphic there; g_r vanishes nowhere on the upper half-plane when r ∉ ℤ²; and g_r
transforms by an explicit root of unity under integral translations of r and under τ ↦ τ + 1.
The phases are exact: no root of unity is lost to a choice of branch. Finally it computes the three
Siegel functions of level two, g₀ = g_{(0, 1/2)}, g₁ = g_{(1/2, 0)} and g₂ = g_{(1/2, 1/2)},
as explicit products (up to constant factors they are the squares 𝔣₂², 𝔣₁² and 𝔣² of
Weber's modular functions), proves g₀ g₁ g₂ = 2 ζ₈ with ζ₈ = e^{2π i / 8}, expands their
fourth powers in q = e^{2π i τ} and s = e^{π i τ}, and records how τ ↦ τ + 1 permutes them
up to roots of unity.
Main definitions and results #
TauCeti.ModularForm.siegelFunction r τ: the Siegel functiong_r(τ).TauCeti.ModularForm.siegelFunction_eq_qTheta,TauCeti.ModularForm.siegelFunction_eq_mul_tprod_mul_tprod:g_ras a phase times the theta productθ_q(w), and as a phase times1 - wand its two half products.TauCeti.ModularForm.hasProdLocallyUniformlyOn_siegelFunction: the defining product∏_{n ≥ 1} (1 - qⁿ w) (1 - qⁿ w⁻¹)converges locally uniformly on the upper half-plane.TauCeti.ModularForm.differentiableOn_siegelFunction,TauCeti.ModularForm.mdifferentiable_siegelFunction:g_ris holomorphic on the upper half-plane, as a function onupperHalfPlaneSetand as a function onℍ.TauCeti.ModularForm.siegelFunction_ne_zero: forr ∉ ℤ²,g_rhas no zero on the upper half-plane.TauCeti.ModularForm.siegelFunction_add_intCast: fors ∈ ℤ²,g_{r + s} = (-1)^{s₁ s₂ + s₁ + s₂} e^{-π i (s₁ r₂ - s₂ r₁)} g_r.TauCeti.ModularForm.siegelFunction_apply_add_one:g_r(τ + 1) = e^{π i / 6} g_{(r₁, r₁ + r₂)}(τ), the transformation under the translationT = [[1, 1], [0, 1]]acting on the row vectorron the right.TauCeti.ModularForm.siegelFunction_zero_half,TauCeti.ModularForm.siegelFunction_half_zero,TauCeti.ModularForm.siegelFunction_half_half: the product expansions ofg₀,g₁andg₂, andTauCeti.ModularForm.siegelFunction_zero_half_pow_twelve:g₀¹² = 4096 q ∏_{n ≥ 1} (1 + qⁿ)²⁴.TauCeti.ModularForm.siegelFunction_zero_half_mul_half_zero_mul_half_half:g₀ g₁ g₂ = 2 ζ₈on the upper half-plane.TauCeti.ModularForm.siegelFunction_zero_half_pow_four,TauCeti.ModularForm.siegelFunction_half_zero_pow_four,TauCeti.ModularForm.siegelFunction_half_half_pow_four: the product expansionsg₀⁴ = 16 q^{1/3} / P(q)⁸,g₁⁴ = s^{-1/3} P(s)⁸andg₂⁴ = -s^{-1/3} P(-s)⁸fors = e^{π i τ},q = s²andP(s) = ∏_{n ≥ 0} (1 - s^{2n+1}).TauCeti.ModularForm.siegelFunction_pow_four_mul_mul:g₀⁴ g₁⁴ g₂⁴ = -16.TauCeti.ModularForm.siegelFunction_zero_half_apply_add_one,TauCeti.ModularForm.siegelFunction_half_zero_apply_add_one,TauCeti.ModularForm.siegelFunction_half_half_apply_add_one:τ ↦ τ + 1sendsg₀, g₁, g₂toe^{π i / 6} g₀,e^{π i / 6} g₂ande^{-π i / 3} g₁.
References #
- D. S. Kubert, S. Lang, Modular Units, Grundlehren der mathematischen Wissenschaften 244, Chapter 2, §1 (the Klein forms and Siegel functions and their transformation properties).
- D. H. Shin, Generation of class fields by Siegel–Ramachandra invariants, §2.
- D. A. Cox, Primes of the Form x² + ny², §12 (Weber's functions
𝔣, 𝔣₁, 𝔣₂and the identity𝔣 𝔣₁ 𝔣₂ = √2, equivalent tog₀ g₁ g₂ = 2 ζ₈).
The Siegel function g_r(τ) attached to r = (r₁, r₂) ∈ ℝ²:
g_r(τ) = -e^{π i B₂(r₁) τ} e^{π i r₂ (r₁ - 1)} (1 - w) ∏_{n ≥ 1} (1 - qⁿ w) (1 - qⁿ w⁻¹)
with q = e^{2π i τ}, w = e^{2π i (r₁ τ + r₂)} and B₂ = bernoulliFun 2 the second Bernoulli
polynomial x² - x + 1/6. The product converges for τ in the upper half-plane, where g_r is
holomorphic; elsewhere the value is a junk value.
Instances For
The defining product of the Siegel function.
The Siegel function as a phase times the theta product θ_q(w):
g_r(τ) = -e^{π i B₂(r₁) τ} e^{π i r₂ (r₁ - 1)} θ_q(w) with q = e^{2π i τ} and
w = e^{2π i (r₁ τ + r₂)}.
Convergence and holomorphy on the upper half-plane #
On the upper half-plane the Siegel function splits into its two half products,
g_r(τ) = -e^{π i B₂(r₁) τ} e^{π i r₂ (r₁ - 1)} (1 - w) P(w) P(w⁻¹) where
P(v) = ∏_{n ≥ 1} (1 - qⁿ v), with q = e^{2π i τ} and w = e^{2π i (r₁ τ + r₂)}.
The infinite product ∏_{n ≥ 1} (1 - qⁿ w) (1 - qⁿ w⁻¹) defining the Siegel function, with
q = e^{2π i τ} and w = e^{2π i (r₁ τ + r₂)}, converges locally uniformly on the upper
half-plane.
The Siegel function is holomorphic on the upper half-plane.
The Siegel function is holomorphic as a function on ℍ.
Translations of r and of τ #
Translating r by an integral vector. For s = (s₁, s₂) ∈ ℤ²,
g_{r + s} = (-1)^{s₁ s₂ + s₁ + s₂} e^{-π i (s₁ r₂ - s₂ r₁)} g_r on the upper half-plane. In
particular g_r depends on r modulo ℤ² only up to a root of unity when r is rational.
The translation τ ↦ τ + 1: g_r(τ + 1) = e^{π i / 6} g_{r T}(τ), where
r T = (r₁, r₁ + r₂) is the row vector r multiplied on the right by T = [[1, 1], [0, 1]].
The factor e^{π i / 6} in siegelFunction_apply_add_one is a twelfth root of unity.
The Siegel functions of level two #
With s = e^{π i τ}, so that q = s², the three Siegel functions g_r with r ∈ (½ ℤ)² \ ℤ²
modulo ℤ² are the products below. Multiplying them, Euler's identity
∏_{n ≥ 1} (1 + qⁿ) ∏_{n ≥ 0} (1 - q^{2n+1}) = 1 collapses every infinite product, leaving the
constant 2 ζ₈.
g_{(0, 1/2)}(τ) = 2 i e^{π i τ / 6} (∏_{n ≥ 1} (1 + qⁿ))² on the upper half-plane, where
q = e^{2π i τ}.
g_{(0, 1/2)}(τ)¹² = 4096 q (∏_{n ≥ 1} (1 + qⁿ))²⁴ on the upper half-plane, where
q = e^{2π i τ}.
g_{(1/2, 0)}(τ) = -e^{-π i τ / 12} (∏_{n ≥ 0} (1 - s^{2n+1}))² on the upper half-plane,
where s = e^{π i τ}.
g_{(1/2, 1/2)}(τ) = -e^{-π i / 4} e^{-π i τ / 12} (∏_{n ≥ 0} (1 + s^{2n+1}))² on the upper
half-plane, where s = e^{π i τ}.
The product of the level-two Siegel functions: g₀ g₁ g₂ = 2 ζ₈ on the upper half-plane,
where g₀ = g_{(0, 1/2)}, g₁ = g_{(1/2, 0)}, g₂ = g_{(1/2, 1/2)} and ζ₈ = e^{2π i / 8}.
g₀⁴ = 16 q^{1/3} / (∏_{n ≥ 0} (1 - q^{2n+1}))⁸ for q = e^{2π i τ}, where
q^{1/3} = e^{2π i τ / 3}.
g₁⁴ = s^{-1/3} (∏_{n ≥ 0} (1 - s^{2n+1}))⁸ for s = e^{π i τ}, where
s^{-1/3} = e^{-π i τ / 3}.
g₂⁴ = -s^{-1/3} (∏_{n ≥ 0} (1 - (-s)^{2n+1}))⁸ for s = e^{π i τ}, where
s^{-1/3} = e^{-π i τ / 3}.
g₀⁴ g₁⁴ g₂⁴ = -16, the fourth power of g₀ g₁ g₂ = 2 ζ₈.
g₀(τ + 1) = e^{π i / 6} g₀(τ) for g₀ = g_{(0, 1/2)}.
g₁(τ + 1) = e^{π i / 6} g₂(τ) for g₁ = g_{(1/2, 0)} and g₂ = g_{(1/2, 1/2)}.
g₂(τ + 1) = e^{-π i / 3} g₁(τ) for g₁ = g_{(1/2, 0)} and g₂ = g_{(1/2, 1/2)}.