Weber's cubic: the level-two Siegel functions and the modular invariant j

The twelfth powers of the three level-two Siegel functions g₀ = g_{(0, 1/2)}, g₁ = g_{(1/2, 0)} and g₂ = g_{(1/2, 1/2)} are the three roots of

(X + 16)³ - j X = X³ + 48 X² + (768 - j) X + 4096,

so that (g¹² + 16)³ = j g¹² for each of them. In Weber's notation g₀¹² = 𝔣₂²⁴, g₁¹² = 𝔣₁²⁴ and g₂¹² = -𝔣²⁴.

The proof compares elementary symmetric functions. The product g₀¹² g₁¹² g₂¹² = -4096 is the twelfth power of g₀ g₁ g₂ = 2 ζ₈. The cubic with roots g₀¹², g₁¹², g₂¹² is invariant under SL₂(ℤ) (prod_X_sub_C_siegelFunction_pow_twelve_smul), so g₀¹² + g₁¹² + g₂¹² and g₀¹² g₁¹² + g₀¹² g₂¹² + g₁¹² g₂¹² + j are holomorphic SL₂(ℤ)-invariant functions. Both have limits at i∞, so they are constant (TauCeti.ModularForm.levelOne_eq_of_tendsto_atImInfty), and the constants are the limits -48 and 768.

The limits come from the product expansions. With s = e^{π i τ}, q = s² and Φ(s) = ∏_{n ≥ 0} (1 - s^{2n+1})²⁴, one has g₁¹² = s⁻¹ Φ(s) and g₂¹² = -s⁻¹ Φ(-s), and Φ(s) Φ(-s) = Φ(q). Since Φ(s) = 1 - 24 s + O(s²), at i∞ g₁¹² + g₂¹² = (Φ(s) - Φ(-s)) / s → -48 and g₁¹² g₂¹² + q⁻¹ = (1 - Φ(q)) / q → 24, while g₀¹² = 4096 q / Φ(q) → 0. Together with j - q⁻¹ → 744 this gives the two constants.

This is the input for the cube root γ₂ = (g¹² + 16) / g⁴ of j.

Main results

• TauCeti.ModularForm.prod_X_sub_C_siegelFunction_pow_twelve: Weber's cubic (X - g₀¹²)(X - g₁¹²)(X - g₂¹²) = (X + 16)³ - j X.
• TauCeti.ModularForm.siegelFunction_zero_half_pow_twelve_add_sixteen_pow_three, TauCeti.ModularForm.siegelFunction_half_zero_pow_twelve_add_sixteen_pow_three, TauCeti.ModularForm.siegelFunction_half_half_pow_twelve_add_sixteen_pow_three: (g¹² + 16)³ = j g¹² for g = g₀, g₁, g₂.

References

• H. Weber, Lehrbuch der Algebra, Band III (the functions 𝔣, 𝔣₁, 𝔣₂ and the cubic satisfied by their 24th powers).
• D. A. Cox, Primes of the Form x² + ny², §12 (Weber's functions and γ₂).
• D. H. Shin, Generation of class fields by Siegel–Ramachandra invariants, §5.

The twelfth powers as products in s = e^{π i τ}

Limits at i∞

The symmetric functions of the twelfth powers

Weber's cubic

theorem TauCeti.ModularForm.prod_X_sub_C_siegelFunction_pow_twelve (τ : UpperHalfPlane) : (Polynomial.X - Polynomial.C (siegelFunction ![0, 1 / 2] ↑τ ^ 12)) * (Polynomial.X - Polynomial.C (siegelFunction ![1 / 2, 0] ↑τ ^ 12)) * (Polynomial.X - Polynomial.C (siegelFunction ![1 / 2, 1 / 2] ↑τ ^ 12)) = (Polynomial.X + 16) ^ 3 - Polynomial.C (j τ) * Polynomial.X

Weber's cubic. The twelfth powers of the level-two Siegel functions g₀ = g_{(0, 1/2)}, g₁ = g_{(1/2, 0)} and g₂ = g_{(1/2, 1/2)} are the three roots of (X + 16)³ - j X.

theorem TauCeti.ModularForm.siegelFunction_zero_half_pow_twelve_add_sixteen_pow_three (τ : UpperHalfPlane) : (siegelFunction ![0, 1 / 2] ↑τ ^ 12 + 16) ^ 3 = j τ * siegelFunction ![0, 1 / 2] ↑τ ^ 12

(g₀¹² + 16)³ = j g₀¹² for g₀ = g_{(0, 1/2)}.

theorem TauCeti.ModularForm.siegelFunction_half_zero_pow_twelve_add_sixteen_pow_three (τ : UpperHalfPlane) : (siegelFunction ![1 / 2, 0] ↑τ ^ 12 + 16) ^ 3 = j τ * siegelFunction ![1 / 2, 0] ↑τ ^ 12

(g₁¹² + 16)³ = j g₁¹² for g₁ = g_{(1/2, 0)}.

theorem TauCeti.ModularForm.siegelFunction_half_half_pow_twelve_add_sixteen_pow_three (τ : UpperHalfPlane) : (siegelFunction ![1 / 2, 1 / 2] ↑τ ^ 12 + 16) ^ 3 = j τ * siegelFunction ![1 / 2, 1 / 2] ↑τ ^ 12

(g₂¹² + 16)³ = j g₂¹² for g₂ = g_{(1/2, 1/2)}.