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23 Weierstrass Elliptic and Modular FunctionsWeierstrass Elliptic Functions

§23.3 Differential Equations

  1. §23.3(i) Invariants, Roots, and Discriminant
  2. §23.3(ii) Differential Equations and Derivatives

§23.3(i) Invariants, Roots, and Discriminant

The lattice invariants are defined by

23.3.1 g2 =60w𝕃{0}w4,
23.3.2 g3 =140w𝕃{0}w6.

The lattice roots satisfy the cubic equation

23.3.3 4z3g2zg3=0,

and are denoted by e1,e2,e3. The discriminant1.11(ii)) is given by

23.3.4 Δ=g2327g32=16(e2e3)2(e3e1)2(e1e2)2.

In consequence,

23.3.5 e1+e2+e3=0,
23.3.6 g2=2(e12+e22+e32)=4(e2e3+e3e1+e1e2),
23.3.7 g3=4e1e2e3=43(e13+e23+e33).

Let g2327g32, or equivalently Δ be nonzero, or e1,e2,e3 be distinct. Given g2 and g3 there is a unique lattice 𝕃 such that (23.3.1) and (23.3.2) are satisfied. We may therefore define

23.3.8 (z;g2,g3)=(z|𝕃).

Similarly for ζ(z;g2,g3) and σ(z;g2,g3). As functions of g2 and g3, (z;g2,g3) and ζ(z;g2,g3) are meromorphic and σ(z;g2,g3) is entire.

Conversely, g2, g3, and the set {e1,e2,e3} are determined uniquely by the lattice 𝕃 independently of the choice of generators. However, given any pair of generators 2ω1, 2ω3 of 𝕃, and with ω2 defined by (23.2.1), we can identify the ej individually, via

23.3.9 ej=(ωj|𝕃),

In what follows, it will be assumed that (23.3.9) always applies.

§23.3(ii) Differential Equations and Derivatives

23.3.10 2(z)=43(z)g2(z)g3,
23.3.11 2(z)=4((z)e1)((z)e2)((z)e3),
23.3.12 ′′(z)=62(z)12g2,
23.3.13 ′′′(z)=12(z)(z).

See also (23.2.7) and (23.2.8).