Type:

Doctoral thesis 
Title:

Complex multiplication of abelian surfaces

Author:

Streng, Theodorus Cornelis

Publisher:

Mathematical Institute, Faculty of Science, Leiden University 
Issue Date:

20100601 
Keywords:

Abelian surface Abelian variety Algorithm Class field Class polynomial CM Complex multiplication Curve Embedding degree Finite field Geometry Igusa Number theory Pairing 
Abstract:

The theory of complex multiplication makes it possible to construct certain class fields and abelian varieties. The main theme of this thesis is making these constructions explicit for the case where the abelian varieties have dimension 2.
Chapter I is an introduction to complex multiplication, and shows that a general result of Shimura can be improved for degree4 CMfields.
Chapter II gives an algorithm for computing class polynomials for quartic CMfields,
based on an algorithm of Spallek. We make the algorithm more explicit, and use Goren and Lauterâ€™s recent bounds on the denominators of the coefficients, which yields the first running time bound and proof of correctness of an algorithm computing these polynomials.
Chapter III studies and computes the irreducible components of the modular variety of abelian surfaces with CM by a given primitive quartic CMfield. We adapt the algorithm of Chapter II to compute these components.
Chapters IV and V construct certain `Weil numbers'. They have properties that are number theoretic in nature and are motivated by cryptography. Chapter IV is joint work with David Freeman and Peter Stevenhagen. Chapter V is joint work with Laura Hitt O'Connor, Gary McGuire, and Michael Naehrig. 
Description:

Promotor: Peter Stevenhagen With summary in Dutch 
Faculty:

Faculteit der Wiskunde en Natuurwetenschappen

Citation:

Streng, T.C., 2010, Doctoral thesis, Leiden University 
ISBN:

9789053352915 
Handle:

http://hdl.handle.net/1887/15572 

