Chapter

Open Problems in Mathematics and Computational Science

pp 37-68

Date:

Isogenies in Theory and Praxis

  • Gerhard FreyAffiliated withInstitute for Experimental Mathematics, University of Duisburg-Essen Email author 

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Abstract

We want to give an overview on arithmetical aspects of abelian varieties and their torsion structures, isogenies, and resulting Galois representations. This is a wide and deep territory with a huge amount of research activity and exciting results ranging from the highlights of pure mathematics like the proof of Fermat’s last theorem to stunning applications to public-key cryptography. Necessarily we have to be rather superficial, and thus specialists in the different aspects of the topics may be disappointed. But I hope that for many, and in particular for young researchers, the chapter may serve as an appetizer and will raise interest for a fascinating area of mathematics with many open problems (some are very hard and worth a Fields Medal but others are rather accessible).

The first section of the chapter gives basic notions, definitions, and properties of abelian varieties. Disguised as examples one will find their theory over the complex numbers \(\mathbb{C}\) and the special case of elliptic curves. The second section discusses the situation over finite fields, in particular the role of the Frobenius endomorphism, and over number fields where the most interesting results and challenging conjectures occur. Finally we discuss algorithmic aspects of isogenies, mostly of elliptic curves, and relations to cryptography.