ED Mathématiques et Informatique
Arithmetic group actions: reduction theories and enumeration algorithms
by Anne-Edgar WILKE (IMB - Institut de Mathématiques de Bordeaux)
The defense will take place at 15h30 - Salle de conférences Institut de mathématiques de Bordeaux Université de Bordeaux, bâtiment A33 351 cours de la Libération 33405 Talence
in front of the jury composed of
- Florent JOUVE - Professeur - Université de Bordeaux - Examinateur
- Aurel PAGE - Chargé de recherche - Université de Bordeaux - Examinateur
- Michael STOLL - Full professor - Universität Bayreuth - Rapporteur
- Cécile DARTYGE - Maîtresse de conférences - Université de Lorraine - Examinateur
- Emmanuel BREUILLARD - Full professor - University of Oxford - Rapporteur
- Jean-François QUINT - Directeur de recherche - Université de Montpellier - Examinateur
- Sébastien BOUCKSOM - Directeur de recherche - Sorbonne Université - Examinateur
- Stéphane CHARPENTIER - Maître de conférences - Aix Marseille Université - Examinateur
Given an action of an arithmetic group on a set, this thesis focuses on two problems. The first problem is to construct an explicit fundamental domain, by picking a distinguished point in each orbit: these points are then said to be reduced, and we say that we have a reduction theory for the group action. The second problem is to find algorithms to enumerate the orbits satisfying given conditions. Having a reduction theory enables one to reformulate problems of enumeration of orbits into problems of enumeration of points inside the fundamental domain. The first part of the thesis is devoted to an analytic concept which turns out to be important in reduction theory, namely, plurisubharmonicity. First I make the known analogy between convexity and plurisubharmonicity more precise; then I introduce a notion of strict plurisubharmonicity analogous to strict convexity, and I use this notion to give a new, short, unified proof of a characterisation of L^p direct integrals satisfying the strong maximum modulus principle. The second part deals with the search for reduction theories for arithmetic group actions. I develop an approach which answers this question in a very general setting. The underlying object, which I call the Kempf-Ness covariant, generalises the covariants introduced by Hermite, Julia, and later Stoll and Cremona for reducing binary forms. I show that the Kempf-Ness covariant of a sequence of points in the complex Grassmannian can be seen as a barycenter of these points; this generalises observations due to Stoll and Cremona in the case of the projective line, and, independently, to Kapovich, Leeb and Millson in the case of the projective space. The third part is concerned with effective enumeration of cubic and quartic number fields. Thanks to the parametrisation of cubic fields discovered by Levi and later by Davenport and Heilbronn, and to the parametrisation of quartic fields discovered by Bhargava, these are in fact problems of enumeration of orbits. I show how to solve them by replacing the fundamental domain with an overapproximation of it, defined by simpler, so-called monomial equations. In the cubic case, one essentially recovers Belabas's algorithm; in the quartic case, one obtains a new algorithm, better than the Hunter-Martinet method, but less efficient than the approaches relying on class field theory under the generalised Riemann hypothesis. Finally, the fourth part presents another idea to obtain algorithms for enumerating orbits: cover the fundamental domain with a finite family of balls of constant radius. I carry out this approach in the case of truncated Siegel sets. Since the fundamental domains used for arithmetic group actions are generally made from such sets, one can hope that my work will give rise to many new enumeration algorithms.