Computational methods for abelian varieties over number fields with complex multiplication

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Computational arithmetic geometry and its related areas generalize considerably both importance and techniques of classical computational number theory. The introduction of algebraic curves to cryptography emphasized that arithmetic geometry possesses not only an extremely interesting theoretical value that is rapidly growing; it also provides us with an exciting computational side. Many questions are still unsolved and need more investigation. This project is concerned with explicit algorithmic problems in the arithmetic of abelian varieties over number fields with complex multiplication. There are a multitude of recent numerical results for dimension one abelian varieties with complex multiplication, also motivated from applications to cryptography. We wish to solve various problems for low dimensional abelian varieties with complex multiplication, both algorithmically and theoretically. This includes the following topics: Torsion points of abelian varieties with CM over finite fields, small invariants and explicit class field theory, construction of curves for pairing-based cryptography, investigation of the Igusa-invariants, efficient algorithms and explicit implementation of new methods.