If M is a manifold with an action of a group G, then the homology group H_1(M,**Q**) is naturally a **Q**[G]-module, where **Q**[G] denotes the rational group ring. We prove that for every finite group G, and for every **Q**[G]-module V, there exists a closed hyperbolic 3-manifold M with a free G-action such that the **Q**[G]-module H_1(M,**Q**) is isomorphic to V. We give an application to spectral geometry: for every finite set P of prime numbers, there exist hyperbolic 3-manifolds N and N' that are strongly isospectral such that for all p in P, the p-power torsion subgroups of H_1(N,**Z**) and of H_1(N',**Z**) have different orders. We also show that, in a certain precise sense, the rational homology of oriented Riemannian 3-manifolds with a G-action "knows" nothing about the fixed point structure under G, in contrast to the 2-dimensional case. The main geometric techniques are Dehn surgery and, for the spectral application, the Cheeger-Mueller formula, but we also make use of tools from different branches of algebra, most notably of regulator constants, a representation theoretic tool that was originally developed in the context of elliptic curves.

Given a finite group G, a G-covering of closed Riemannian manifolds, and a so-called G-relation, a construction of Sunada produces a pair of manifolds M_1 and M_2 that are strongly isospectral. Such manifolds have the same dimension and the same volume, and their rational homology groups are isomorphic. We investigate the relationship between their integral homology. The Cheeger-Mueller Theorem implies that a certain product of orders of torsion homology and of regulators for M_1 agrees with that for M_2. We exhibit a connection between the torsion in the integral homology of M_1 and M_2 on the one hand, and the G-module structure of integral homology of the covering manifold on the other, by interpreting the quotients Reg_i(M_1)/Reg_i(M_2) representation theoretically. Further, we prove that the p-primary torsion in the homology of M_1 is isomorphic to that of M_2 for all primes p not dividing #G. For p ≤ 71, we give examples of pairs of strongly isospectral arithmetic hyperbolic 3-manifolds for which the p-torsion homology differs.

Deciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. For indefinite quaternion algebras, the decision problem reduces to that in the underlying number field. Finding a generator is hard, and we present a heuristically subexponential algorithm.

Arithmetic Kleinian groups are arithmetic lattices in PSL_2(**C**). We present an algorithm which, given such a group Γ, returns a fundamental domain and a finite presentation for Γ with a computable isomorphism. It is an improvement of the algorithm of my master's thesis. The Magma package is available here.

Méthodes explicites pour les groupes arithmétiques. [ thesis ]

This thesis was supervised by Karim Belabas and Andreas Enge. Here is the abstract of the thesis.

Computing fundamental domains for arithmetic Kleinian groups. [ HAL | thesis | defending ]

This thesis was supervised by John Voight.

L'équation de Pell-Fermat non commutative. [ text (french) | presentation (french) ]

Computing good covers of compact arithmetic manifolds, at King's College London on 2016-09-8 at the workshop Automorphic forms: theory and computation. [ slides ]

Torsion homology of hyperbolic 3-manifolds in Jacquet-Langlands pairs and isospectral pairs, in the University of Luxembourg on 2016-07-12, at the Bianchi Modularity workshop. [ slides ]

Torsion in the homology of isospectral manifolds, in the AUB (Beirut, Lebanon) on 2016-04-27. [ slides ]

Torsion dans l'homologie et régulateurs de variétés isospectrales, in the IMJ (Paris, France) on 2016-02-10. [ slides (french) ]

Torsion dans l'homologie des groupes kleinéens arithmétiques, in the IRMAR (Rennes, France) on 2015-12-11, for the effective algebra and geometry seminar. [ slides (french) ]

Torsion homology of arithmetic Kleinian groups, in Amherst College (MA, US) on 2015-11-17, for the Five College Number Theory Seminar. [ slides ]

Isospectrality, regulators and special value formulas, in Brown University, Providence (RI, US) on 2015-11-16, for the ICERM peer-to-peer seminar. [ slides ]

Aspects algorithmiques des groupes d'unités, in Luminy on 2015-03-19, for the ATI seminar. [ slides (french) ]

Tutorial: associative and central simple algebra, in the IMB (Bordeaux, France) on 2015-01-13, PARI/GP workshop. [ gp script (updated version) ]

Calcul de formes modulaires de Klein, in Clermont-Ferrand (France) on 2014-12-9, for the number theory seminar. [ slides (french) ]

Computing Klein modular forms, in Bristol (UK) on 2014-11-12, for the Heilbronn seminar. [ slides ]

Computing Klein modular forms, in the University of Warwick (Coventry, UK) on 2014-10-13, in the number theory seminar. [ slides ]

Computing Klein modular forms, in Sheffield (UK) on 2014-09-15, for the WANDS seminar. [ slides ]

An algorithm for the principal ideal problem in indefinite quaternion algebras in GyeongJu (Korea) on 2014-08-11, Algorithmic Number Theory Symposium XI. [ slides ]

Computing Klein modular forms, in the Université Paris 13 on 2014-07-24, Workshop on Analytic Number Theory and Geometry. [ slides ]

Computing Kleinian modular forms, in the University of Warwick (Coventry, UK) on 2014-06-4, LMFDB Workshop. [ slides ]

The principal ideal problem in quaternion algebras, in the CIRM (Luminy, France) on 2014-03-11, workshop Number Theory and Applications. [ slides ]

Central simple algebras for PARI, in the LMB (Besançon, France) on 2014-01-7, 2014 PARI/GP workshop.

Groupes kleinéens arithmétiques et formes automorphes pour GL(2), in the IRMAR (Rennes, France) on 2013-12-3, meeting of the ANR Peace. [ slides (french) | Bianchi -14 | Bianchi -223 | quartic -275 (open with geomview) ]

Computing Kleinian modular forms, in the LMB (Besançon, France) on 2013-09-24, conference Algebraic and explicit methods in number theory. [ slides ]

Le théorème de Pólya, in the IMB (Bordeaux, France) on 2013-05-29, TNT day of the number theory PhD students. [ text on the same topic ]

Calcul explicite de formes automorphes, in the Grands Moulins (Paris, France) on 2013-02-7, Jussieu PhD students seminar. [ slides (french) ]

La correspondance de Jacquet-Langlands, in the IMB (Bordeaux, France) on 2012-10-23, TNT day of the number theory PhD students.

Cryptologie ? (with Nicolas Mascot), in the IMB (Bordeaux, France) on 2012-10-9, French Science Festival.

Algorithms for arithmetic Kleinian groups, in the BIRS (Banff, Canada) on 2012-07-5, workshop Torsion in the homology of arithmetic groups : geometry, arithmetic, and computation. [ video ]

Calcul de groupes kleinéens arithmétiques, in the IRMAR (Rennes, France) on 2012-05-4, cryptography seminar.

L'équation de Pell-Fermat non-commutative, in the IMB (Bordeaux, France) on 2012-03-20, number theory PhD students seminar.

Quaternion algebras, in the IMB (Bordeaux, France) on 2012-01-27, 2012 PARI/GP workshop.

Algorithms for arithmetic Kleinian groups, in the Universität Heidelberg (Germany) on 2011-09-6, conference Computations with Modular Forms 2011. [ slides ]

Calculs de domaines fondamentaux de groupes arithmétiques, in the IMB (Bordeaux, France) on 2010-12-8, AlgoL workshop. [ part I (french) | part II (french) ]