Class group computation in large fields

• Applicant: Professor Dr. Claus Fieker
• Subject Area: Mathematics
• Term: from 2013 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 239413268

The class group is one of the most important invariants of number fields; it underpins all problems regarding the multiplicative structure of number fields. While class groups have been at the centre of attention for a long time now, their behavior is still mysterious: for example in randomly chosen fields, class groups tend to be trivial, yet it is not known if there are infinitely many fields with trivial class group! On the other hand, in specific families class groups are known to be large. Class groups and their computation are also intimately linked to problems rearding the distribution of so called smooth numbers: algebraic integers that have no large prime divisors. This project has two core aims: mathematically, we want to study the distribution of smooth numbers in number fields in order to gain a better understanding of the performance of various class group related algorithms. In particular, the project will develop new methods to generate and analyses smooth numbers, to work with large very sparse matrices over the integers and study Brauer relations with the aim of verifying class groups. On the computer algebra side, this projects aims to provide the foundations of algebraic number theory, namely algorithms for class and unit group computations, smoothness tests and practical linear algebra in low dimensions to the SPP 1489. Those algorithms will form the foundation of an independent new computer algebra system that is tightly integrated with both Singular and Gap; hence will open the gate towards interdisciplinary applications. In this sense, the project aims to be a successor to the no longer active systems Kant/KaSH, LiDIA and Simath.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1489: Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory
• International Connection: Australia, France, United Kingdom
• Participating Persons: Professor Dr. John J. Cannon; Dr. Pierrick Gaudry; Dr. Max Neunhöffer; Dr. Allan Steel; Professor Dr. Damien Stehle; Dr. Emmanuel Thome