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Research Programme 1.1: Number Theory and Algebra

Programme leader: H.W. Lenstra

Staff

(situation at November 1, 2008)

permanent staff

emeritus

PhD students

guest researchers

Description of the project

The main focus of the research programme is number theory. Number theory studies the properties of integers, with a historically strong emphasis on the study of diophantine equations, that is, systems of equations that are to be solved in integers. The methods of number theory are taken from several other branches of mathematics. Traditionally, these include algebra and analysis, and in recent times algebraic geometry has become increasingly important. Another recent development is the discovery that number theory has significant implications in more applied areas, such as cryptography, theoretical computer science, the theory of dynamical systems, and numerical mathematics. This discovery led to the rise of algorithmic and computational number theory, which occupies itself with the design, analysis, and efficient implementation of arithmetical algorithms. The overall result has been a unification rather than a diversification of number theory. For example, the applications in cryptography depend heavily on algebraic geometry, and algebraic number theory, which used to stand on itself, is now pervading virtually all of number theory. Themes of the programme reflect the research areas mentioned. They include finding points on algebraic curves, applications of group theory and algebraic number theory, the theory of finite fields, diophantine approximation, words and sequences, discrete tomography, primality tests and factorization methods, and the development of efficient computer algorithms.

The algebra portion of the programme is strongly oriented towards the applications of algebra in number theory and arithmetic geometry and towards algorithmic aspects. Themes include Galois theory and various aspects of group theory and ring theory.

The research programme also includes cryptology and the history of mathematics. Main themes in cryptology are the applications of number theory and algebra to the design of cryptographic schemes, and foundational issues are considered as well. In the history of mathematics, the emphasis is on the edition and translation of early Islamic mathematical and astronomical texts.

Research results in 2007

Evertse and Bugeaud finished a paper on the approximation of complex algebraic numbers by algebraic numbers of bounded degree. They further investigated the structure of the expansion of an algebraic number with respect to a given base b. Evertse and Ferretti worked out an improvement of the quantitative Subspace Theorem. Bérczes, Evertse and Győry obtained new effective results on equations ax+by=1 with unknowns x,y from a multiplicative group of finite rank. The new feature of this result is that the unknowns x,y are taken from a group which is not contained in a prescribed algebraic number field. Evertse and Győry are writing a book on unit equations and discriminant equations. This is work in progress.

Tijdeman obtained with Hancl results on the irrationality of values of polynomial Cantor series. Saradha and Tijdeman studied arithmetic progressions with common difference divisible by a small prime. Hajdu, Tengely and Tijdeman proved that cubes cannot occur as products of the terms in arithmetic progressions of certain lengths. Tijdeman and Zamboni deduced new results on words with any periods.

Van Dalen started a Ph.D.-project on discrete tomography, Ekkelkamp continued her study of estimates of the expected run time of factorization algorithms, Rosema obtained new results on the connection of substitutions and number systems, and Smeets transformed the LLL-algorithm into a multi-dimensional continued fraction algorithm with similar properties, and with Kraaikamp corrected and sharpened a result on symmetric and asymmetric Diophantine approximation by Tong.

Dassen continued his research in the theory and structure of layered lattices. He completed the embedding theorem and gave a description of the subgroups of a layered Euclidean space which are layered lattices. Streng continued his research on the construction of curves of genus 2 via CM methods.

Daems continued her research on the history of mathematical crystallography, in particular the several classifications of crystal structures in the 19th century.

The connections between secure computation and algebraic function fields with many rational places that were found by Chen and Cramer in 2006 were extended by Chen (Shanghai), Cramer, Goldwasser (MIT), de Haan (CWI) and Vaikuntanathan (MIT) in their work on ramp schemes and secure computation from random error correcting codes. Cramer also concluded his work with Damgaard (Aarhus) and de Haan (CWI) on low communication secure multiplication, and his work with Kiltz, Padró, on secure linear algebra based on new results concerning Moore-Pernrose pseudoinverses. Cramer also studied blackbox construction of chosen ciphertext secure encryption with Hofheinz (CWI), Kiltz (CWI), Pass (Cornell), Shelat (U Virginia) and Vaikuntanathan (MIT).

Brakenhoff has continued his research on subrings of maximal orders and started research on the class numbers of general orders. Palenstijn continued his research on radical field extensions, extending certain results to division points of rank one tori over number fields together with de Smit

The distributed computation project abcathome.com which is run by Lenstra, Palenstijn and de Smit, made substantial progress collecting data related to the ABC conjecture.

De Smit collaborated with Thomas and Florence of the EPFL Lausanne on a valuative criterion for normal basis generators in a wildly ramified extension of local fields. Together with Sutton (Dartmouth) and Gornet (U. Texas, Arlington) he analyzed the behavior of the covering spectrum of isospectral Riemannian manifolds. He supervised the Master’s Thesis of M. Perone (Padova) on a duality theory of commutative monoids and applications to units in group rings. De Smit continued his collaboration with Hanzon (Cork) which aims to apply Galois theory to mathematical finance.

Lenstra worked on several subjects in algebra and number theory, in many cases as part of a collaboration with others or a supervision of a PhD student. These subjects include numerical aspects of the abc conjectures (with de Smit and Palenstijn), algorithms for norm residue symbols (with Bouw), layered lattices (with Dassen), Mersenne primes and class field theory (with Jansen and de Smit), primality testing in polynomial time (with C. Pomerance), the history of the LLL algorithm (with A. Lenstra, L. Lovász and P. van Emde Boas), an apparent error in the Cohen-Lenstra heuristics (with G. Malle), certain aspects of crystallographic groups (with Daems), odd perfect numbers (with M. Roelands) and algorithms for ordered fields.

Stevenhagen and Bröker improved their construction of elliptic curves with point groups of prime order. Stevenhagen, Streng and Freeman (UC Berkeley) created an algorithm for finding pairing friendly abelian varieties in low dimension. With Howe (CCR San Diego) and Lauter (Microsoft), Stevenhagen investigated the construction of cryptographic abelian surfaces over finite fields by CM-methods. He also completed his survey paper on the arithmetic of general number rings.