Van der Blij's characteristic-norm congruence

In a unimodular integral lattice, every characteristic vector has integral norm congruent to the signature difference modulo eight. The statement applies to all signatures and to both even and odd lattices.

For the canonical even sublattice N, the original lattice determines a bilinear Lagrangian H = L/N in the discriminant form of N. A characteristic vector w gives a dual vector w/2; the quadratic form is constant on its class plus H, with value w²/8. Character averaging therefore evaluates the normalized Gauss sum as expCircle (w²/8). Milgram's theorem identifies the same phase with the signature of N, which is the signature of L. When L is even and w = 0, the congruence gives divisibility of the signature by eight.

References

• F. van der Blij, An invariant of quadratic forms mod 8, Indag. Math. 21 (1959), 291–293.
• J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Appendix 4.

theorem TauCeti.IntegralLattice.vanDerBlij {V : Type u} [AddCommGroup V] [Module ℚ V] (L : IntegralLattice V) (hL : L.IsUnimodular) {w : ↥L.carrier} (hw : L.IsCharacteristicVector w) : ↑L.sigPos - ↑L.sigNeg ≡ L.integralNorm w [ZMOD 8]

Van der Blij's congruence: the signature difference of a unimodular integral lattice is congruent modulo eight to the norm of any characteristic vector.