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180 Chapter 6. Complex multiplication in higher generaFor a polarized principal CM abelian variety (A, i, ψ) of type (E, Φ, a, ξ) defined over the complexnumbers, it is then relatively easy to show that (A σ , i σ , ψ σ ) is of type (E, Φ, N Φ r(p) −1 , N(p)ξ)and that the following diagram is commutative:C g /Φ(a)θ ◦ ΦAcan.C g /Φ(N Φ r(p) −1 a)θ ′ ◦ ΦA σλUsing the same method as in dimension 1 one then shows the main theorem over the reflexfield. This first version of the main theorems of complex multiplication deals with the action of theabsolute Galois group of the reflex field on complex abelian varieties with complex multiplication.Theorem 6.3.24 (Over the reflex field [238, Theorems IV.18.6 and IV.18.8], [159, Theorem 3.6.1],[205, Theorem II.9.17], [57, Theorem 6.3]). Let (A, i, ψ) be a polarized complex CM abelian varietyof type (E, Φ, a, ξ) with respect to θ. Let σ ∈ Aut(C/E r ) and s be an idèle of E r such thatσ = (s, E r ) E r ab. Then there exists a unique uniformization θ ′ such that (A σ , i σ , ψ σ ) is of type(E, Φ, N Φ r(s −1 )a, N Q (s)ξ) with respect to θ ′ and the following diagram commutes:K/aθ ◦ ΦA torN Φ r(s −1 )σK/ N Φ r(s −1 )aθ ′ ◦ ΦA σ torThe previous theorem can then be extended to arbitrary types of conjugations for a CM fieldK. It is much more involved and uses the cyclotomic character χ cyc : Aut C → Ẑ× = A × Q,f and amap f Φ : Aut(C) → A × K,f /K× verifying f Φ (σ)f Φ (σ) = χ cyc (σ)K × extending the map induced bythe reflex norm N Φ : Aut(C/K r ) → A × K,f /K× and called the type transfer [159, 7], [205, II.10].Theorem 6.3.25 (Over the rationals [159, Theorem 7.3.1], [205, Theorem II.10.1]). Let (A, i, ψ)be a polarized complex CM abelian variety of type (K, Φ, a, ξ) with respect to θ. Let σ ∈ Aut(C)and s ∈ f Φ (σ) ⊂ A × K,f . Then there exists a unique uniformization θ′ such that (A σ , i σ , ψ σ ) is oftype (K, σΦ, sa, χ(σ)ss) with respect to θ′ and the following diagram commutes:K/aθ ◦ ΦA torsθK/sa′ ◦ σΦσA σ tor6.4 Class polynomials for genus 2In this section we extend the construction of the Hilbert class polynomial to curves of genus 2.