Gauss sums of finite quadratic modules #

The Gauss sum of a finite quadratic module (A, q) is

G(q) = ∑_{a ∈ A} e^{2πi q(a)},

the sum of the standard character TauCeti.expCircle of ℚ/ℤ over the values of q. In the half-norm convention of discriminant forms, q(x) = B(x, x) / 2 modulo ℤ, the factor in the exponent is 2πi.

This file proves the properties of G(q) that hold for every finite quadratic module, before any classification is available:

it is an isometry invariant, it is multiplicative over orthogonal sums, and negating q conjugates it; the scaled Gauss sums ∑_{a ∈ A} e^{2πi n q(a)} of the forms n • q are isometry invariants too, and multiplicative over orthogonal sums;
restricting to the orthogonal complement of a quadratic-isotropic subgroup leaves G(q) unchanged;
quotienting by a subgroup K in the quadratic radical gives G(q) = #K · G(q/K);
for nondegenerate q, G(q) · conj G(q) = #A, so |G(q)| = √#A. Expanding the product and substituting a = b + c turns it into ∑_c e(q(c)) ∑_b e(b(c, b)), and nondegeneracy kills every inner sum except the one at c = 0;
for a Lagrangian subgroup H (isotropic for q itself, and equal to its orthogonal complement) G(q) = #H, so a nondegenerate metabolic module has G(q) = √#A.

The normalized Gauss sum G(q) / √#A of a nondegenerate module is therefore a complex number of absolute value one, equal to 1 on metabolic modules. It is moreover an eighth root of unity, and this needs no classification of finite quadratic modules. Write 2#A - 1 = a² + b² + c² + d² by Lagrange's four-square theorem. Multiplication by the quaternion a + bi + cj + dk is a bijection of A⁴, because composing it with multiplication by the conjugate quaternion is multiplication by 2#A - 1, which is -1 on A. By Euler's four-square identity for quadratic maps it carries q(x₁) + ⋯ + q(x₄) to (2#A - 1)(q(x₁) + ⋯ + q(x₄)) = -(q(x₁) + ⋯ + q(x₄)), since 2#A kills every value of q. Reindexing the sum that defines G(q)⁴ along it gives G(q)⁴ = conj (G(q)⁴), so G(q)⁸ = (G(q) · conj G(q))⁴ = #A⁴.

The exponent of that root of unity is the Gauss-sum invariant sign q ∈ ℤ/8, defined by

G(q) = √#A · e^{2πi sign(q) / 8},

the invariant that Milgram's theorem compares with the signature of an even lattice. It is an isometry invariant, additive over orthogonal sums, negated by negating q, and zero on metabolic modules; the discriminant form of A₁ has invariant 1.

Quadratic isotropy is needed in the Lagrangian statement, not merely isotropy for the polar pairing: the discriminant form of A₁ ⊕ A₁, the orthogonal sum of two copies of q(x) = x² / 4 on ℤ/2, has a subgroup equal to its own orthogonal complement on which the pairing vanishes, while its Gauss sum is (1 + i)² = 2i ≠ 2, so it is not metabolic.

Main declarations #

TauCeti.FiniteQuadraticModule.gaussSum: the Gauss sum ∑_{a ∈ A} e^{2πi q(a)}.
TauCeti.FiniteQuadraticModule.gaussSum_prod and TauCeti.FiniteQuadraticModule.gaussSum_pi: multiplicativity over orthogonal sums.
TauCeti.FiniteQuadraticModule.scaledGaussSum: the Gauss sum ∑_{a ∈ A} e^{2πi n q(a)} of the scaled form n • q, with its isometry invariance TauCeti.FiniteQuadraticModule.Isometry.scaledGaussSum_eq and multiplicativity TauCeti.FiniteQuadraticModule.scaledGaussSum_prod and TauCeti.FiniteQuadraticModule.scaledGaussSum_pi.
TauCeti.FiniteQuadraticModule.IsNondegenerate.gaussSum_mul_conj and TauCeti.FiniteQuadraticModule.IsNondegenerate.norm_gaussSum: |G(q)|² = #A.
TauCeti.FiniteQuadraticModule.gaussSum_eq_natCard_of_isLagrangian and TauCeti.FiniteQuadraticModule.gaussSum_eq_sqrt_natCard_of_isMetabolic: the value on modules with a Lagrangian subgroup.
TauCeti.FiniteQuadraticModule.gaussSum_zmodStandard_two and TauCeti.FiniteQuadraticModule.isLagrangian_zmultiples_and_not_isMetabolic_zmodStandard_two_prod: the discriminant forms of A₁ and of A₁ ⊕ A₁.
TauCeti.FiniteQuadraticModule.conj_gaussSum_pow_four and TauCeti.FiniteQuadraticModule.IsNondegenerate.gaussSum_pow_eight: G(q)⁴ is real, and G(q)⁸ = #A⁴ for nondegenerate q.
TauCeti.FiniteQuadraticModule.gaussSign: the Gauss-sum invariant sign q ∈ ℤ/8, with its defining property TauCeti.FiniteQuadraticModule.IsNondegenerate.gaussSum_eq and its uniqueness TauCeti.FiniteQuadraticModule.gaussSign_eq_of_gaussSum_eq.
TauCeti.FiniteQuadraticModule.gaussSign_eq_of_gaussSum_eq_mul: nondegenerate modules whose Gauss sums differ by a nonnegative real factor have the same invariant.
TauCeti.FiniteQuadraticModule.gaussSign_prod, TauCeti.FiniteQuadraticModule.gaussSign_pi, TauCeti.FiniteQuadraticModule.gaussSign_neg and TauCeti.FiniteQuadraticModule.gaussSign_eq_zero_of_isMetabolic: additivity over orthogonal sums, behaviour under negation, and vanishing on metabolic modules.

References #

C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298.
J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Appendix 4.
V. V. Nikulin, Integral symmetric bilinear forms and some of their applications, §1.11.

Gauss sums of finite quadratic modules #

Main declarations #

References #

The discriminant forms of A₁ and A₁ ⊕ A₁ #

The eighth power of the Gauss sum #

The Gauss-sum invariant #

The discriminant forms of `A₁` and `A₁ ⊕ A₁` #