here - Sites personnels de TELECOM ParisTech - Télécom ParisTech

166 Chapter 6. Complex multiplication in higher generawhere ˜x is any element of V over x and H(·, ˜x) is indeed a C-antilinear map [65, VI.4.3], [213,II.9], [16, Lemma 2.4.5]. The kernel of φ L is K(L) = Λ(L)/Λ whereΛ(L) = {x ∈ V | ω(Λ, x) ⊂ Z} .If L is non degenerate, then the matrix of ω in a symplectic basis is( 0) ∆−∆ 0,and#K(L) = deg(ψ L ) = pf(ω) 2 = (d 1 · · · d g ) 2 .To conclude this section let us mention that, as was pointed out by Weil [282], the correctgeneralization of elliptic curves to higher dimensions are polarized abelian varieties. They haveindeed finite automorphism groups and are well suited for moduli problems.Theorem 6.1.18 ([213, Theorem IV.21.5], [204, Proposition I.14.4]). Let A be an abelian varietyand ψ a polarization. Then the automorphism group of (A, ψ) is finite.Over the complex numbers, isomorphism classes of polarized abelian variety of a given typecan be classified using the fact that an element Ω ∈ H g can be associated with every polarizedabelian variety of a given type.Definition 6.1.19 (Symplectic group). Let J be the 2g × 2g matrixJ =( 0)Ig−I g 0The symplectic group SP 2g (Q) is the set of 2g × 2g matrices M such that MJ t M = J:SP 2g (Q) = { M ∈ GL 2g (Q) | MJ t M = J } ..As was the(case in)dimension 1, there is an action of the symplectic group SP 2g (Q) on H ga bgiven for M = ∈ SPc d 2g (Q) and Ω ∈ H g by [65, Proposition VII.1.1]M · Ω = (aΩ + b)(cΩ + d) −1 .Proposition 6.1.20 ([65, VII.1], [16, Proposition 8.1.3]). Let A and B be two complex polarizedabelian varieties of type ∆. Then A and B are isomorphic if and only if there exists M ∈ G ∆such that M · A = B where G ∆ is the following subgroup of SP 2g (Q):{( ) a bG ∆ =c d}∈ SP 2g (Q) | a, b∆ −1 , ∆c, ∆d∆ −1 ∈ Mat g (Z)In particular the moduli space of complex polarized abelian varieties of a given type is ofdimension g(g+1)2..