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4 Introductiongiving more general background on algebraic curves in Section 5.1, an alternate point of viewis presented in Section 5.2 for the analytic case. Over the complex numbers, elliptic curves canindeed be described as complex tori. Such considerations lead to the main theorems of complexmultiplication and the definition of the Hilbert class polynomial in Section 5.3. Some applicationsof such curves in the context of asymmetric cryptography are given in Section 5.4. The purposeof Chapter 6 is then to extend the results of Chapter 5 to higher dimension with an emphasison the case of non-maximal orders which is usually dismissed for simplicity or concision. Thestructure of this chapter is essentially the same as that of the previous one. Section 6.1 providessuperficial background on the algebraic and analytic theories of abelian varieties. The theory offractional ideals in orders of number fields is presented in Section 6.2 where it is shown how classgroups and units of non-maximal orders can be explicitly computed. Section 6.3 is devoted to thegeneral theory of complex multiplication whereas Section 6.4 is restricted to dimension 2 andgives algorithms to compute class polynomials in that case — the Igusa class polynomials — fora potentially non-maximal order.