Programme leader: R. Tijdeman
The research in the programme Number Theory ranges from purely theoretical to applied computational, and from algebraic number theory and group theory to analytic number theory and ergodic theory. In algebraic number theory the focus is on algebraic units, class groups, Galois module structures and computational methods. Methods from class field theory, homological algebra and analytic number theory play a prominent role. Methods from algebraic number theory and numerical analysis are applied in factorization methods for large integers. Computational methods are also applied to determine all solutions of diophantine equations and to determine all integer or even rational points on algebraic curves. Here methods from algebraic number theory, algebraic geometry and diophantine approximations are being combined. Effective and ineffective diophantine approximation methods are investigated and applied to diophantine equations, recurrence sequences and matrices. Metrical and ergodic properties of (multi-dimensional) continued fractions algorithms are further subject of research. There is some link with an investigation of Sturmian sequences. Finally, group varieties are studied in connection to representation theory of groups as well as Dade correspondence theory and extension theory of characters.
The coherence, size and intensity of the number theory research has been stimulated by the Number Theory Seminar which has had biweekly meetings since October 1993 and the national SWON-NWO (groot) project Number Theory which has started in September 1995 and will last for five years. The latter project made it possible to hire De Weger, De Smit and Bosma to do number theoretical research for two years and Roskam for writing a thesis. Additional Stieltjes support enabled members of the number theory group to invite some guests and to visit some conferences.
In the past two years some remarkable results have been obtained by Stieltjes number theorists. We mention some highlights.
- The visit of P.L. Montgomery as a Stieltjes guest eventually led to active participation of Elkenbracht-Huizing in a world-wide project to factor the record number RSA 130. Not only did she produce considerable part of the required relations, but the last two and critical steps of the computations were carried out at the CWI at Amsterdam using methods of Montgomery and Elkenbracht-Huizing. This was the first time that a factorization record was established by the general number field sieve.
- Evertse and Tijdeman solved a diophantine problem of Pollington on matrices. Their result enabled Brown, Moran and Pollington to solve an old problem of Schmidt on normality of matrices for dimension 2.
- Evertse improved an upper bound of Bombieri for the number of solutions of Thue-Mahler equations.
- De Weger produced some elliptic curves with very large Tate-Shafarevich groups using good examples for the abc-conjecture.
- Stevenhagen worked on the determination of densities for the sets of prime divisors of recurrent sequences. For most classical sequences, which are of torsion type, unconditional explicit density results were obtained. For more general sequences the problem is related to Artin's conjecture on primitive roots and could only be attacked under the assumption of suitable Riemann hypotheses.
- De Smit worked on the application of commutative algebra in Wiles's proof of Fermat's Last Theorem. In joint publications with H.W. Lenstra, R. Schoof (Rome) and K. Rubin (Columbus, Ohio) Wiles' proof was simplified.
- Bosma and De Smit investigated class number relations and arithmetically equivalent fields. This problem has both theoretical and computational aspects. They proved, for instance, that the zeta function of a number field of degree at most 10 determines its class number up to a factor dividing 4.