Formal Group Laws

Power series generalizing group structure: from elliptic curves to chromatic homotopy theory

Formal Group Law

F(x,y) = x + y
(additive group Ĝ_a)

The additive formal group: F(x,y)=x+y. Identity element 0. Inverse: ι(x)=-x. This is the simplest FGL, associated to ordinary cohomology.

Height & Classification

Lazard: L ≅ ℤ[a_{ij}] (poly ring)
Height h: [p](x) = p·x + ... + x^{pʰ}
Lubin-Tate: height-h FGLs over 𝔽̄_p

The height of a formal group over 𝔽_p measures complexity. Height ∞ = additive, height 1 = multiplicative (Ĝ_m), height h for elliptic curves h=1 or 2. Chromatic filtration in homotopy theory.

Visualization

N = 3