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6.4. Class polynomials for genus 2 1816.4.1 Jacobian varietyDefinition 6.4.1 (Jacobian variety). Let C be an algebraic curve 18 of genus g defined over k;There exists an abelian variety of dimension g called the Jacobian variety 19 of C and denoted byJac(C) such that Jac(C)(l) ≃ Pic 0 l (C) for every extension l of k such that C(l) ≠ ∅.The Jacobian variety is equipped with a canonical principal polarization.Over the complex numbers, the Jacobian variety can be defined analytically as [204, III.2]Jac(C) = H 0 (C, Ω 1 ) ∗ /H 1 (C, Z)where H 1 (C, Z) is embedded into H 0 (C, Ω 1 ) ∗ through the map γ ↦→ ∫ γ ·.If P ∈ C(l), then there is a well defined mapC(l) → Jac(C)(l)Q ↦→ [Q − P ]and the Riemann–Roch theorem implies that the g-fold product of C is a cover of Jac(C) 20 .A theorem of Torelli states that classifying curves is equivalent to classifying their Jacobianvarieties.Theorem 6.4.2 (Torelli’s theorem [204, Theorem III.12.1]). Let C and C ′ be two algebraic curvesof genus g. Then C and C ′ are isomorphic if and only if their Jacobian varieties, together withtheir canonical polarizations, are.We say that a curve has complex multiplication by a CM algebra E or an order R within it ifits Jacobian variety has.Finally, it should be noted that over the complex numbers:• in dimension 2, every simple p.p.a.v. is the Jacobian variety of a curve which is necessarilyhyperelliptic;• in dimension 3, every simple p.p.a.v. is the Jacobian of a curve, but it is not necessarily ahyperelliptic curve anymore;• in dimension greater than or equal to 4, consideration of dimension shows that there existsimple p.p.a.v. which are not Jacobian varieties.Classifying the simple p.p.a.v. which arise as Jacobian varieties is an unsolved question known asthe Schottky problem. However, it is possible to characterize period matrices of hyperellipticcurves among those of p.p.a.v., but for a general number field there will not be any such matricesin the set of isomorphism classes of complex abelian varieties with complex multiplication byK [56, 18.3.1].6.4.2 Igusa invariantsThe moduli space of hyperelliptic curves of genus 2 is of dimension 3, so three invariants areneeded to classify them up to isomorphism. Igusa defined such invariants which are valid in anycharacteristic [139], based on invariants defined by Clebsch [51]. In a field k of characteristicdifferent from 2, every hyperelliptic curve of genus 2 can be given by an equation of the form18 Here a curve is a projective geometrically irreducible smooth scheme of dimension 1 over a perfect field.19 This is obviously a simplified definition which will be more than enough for our purposes.20 This can in fact be used to actually construct the Jacobian variety [204, III.7].