Kummer Theory and Algebraic Groups

The project at a glance

Start date:

01 Aug 2017
Duration in months:

7
Funding:

University of Luxembourg / AFR Grant / FNR PRIDE grants / FRQ Québec
Principal Investigator(s):

Antonella PERUCCA

About

Kummer theory is a classical and foundational mathematical theory that has been initiated in the 19th century. It involves extensions of fields that are obtained by adding radicals, provided that certain roots of unity are present in the base field. Perucca’s research group has been investigating Kummer extensions over a cyclotomic extension (in a nutshell, one first adds the necessary roots of unity). With the help of divisibility parameters, Perucca and her team have described the degree and Galois group structure of such extensions. The fields that have been investigated in detail are number fields (especially quadratic and multiquadratic fields), p-adic fields, and function fields. It is also to be mentioned that the PhD students Sebastiano Tronto and Flavio Perissinotto (co-supervised by Peter Bruin and Peter Stevenhagen respectively, and supported by Davide Lombardo) made progress in their theses for Kummer theory of abelian varieties. One application of Kummer theory is Artin’s conjecture on primitive roots. In a foundational work with Olli Järviniemi, and later works joint also with Igor Shparlinski or Pietro Sgobba, Perucca made a far-reaching generalization of Hooley’s method and discovered a uniform lower bound for the non-zero Artin’s densities. Another research direction of Perucca are algebraic groups: in the recent years she has collaborated with Davide Lombardo, Peter Bruin, Francesc Fité, and with her PhD student Antigona Pajaziti. The foundational work with Lombardo achieves the full level of generality for the problem under consideration. It is also to be mentioned that Félix Baril Boudreau (a postdoctoral research with mentors Antonella Perucca, Gabor Wiese from Luxembourg and Chantal David and Henri Darmon from Montreal) is working on a variety of projects in number theory. Perucca is also active in undergraduate research. For example, with the Master student Alexandre Benoist she wrote an article on cyclotomic polynomials. Benoist will join the team as PhD student, and will carry on a project on isogeny graphs that has been developed with Jean Kieffer. Note that isogeny graphs are the core of isogeny-based cryptography.