Number Theory with Emphasis on Algorithms and Algebraic Number Theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Lenstra 0100485 The proposed research belongs to the interface between number theory and algebra. It is inspired by problems that come up in an algorithmic context and in arithmetic algebraic geometry. Altogether, the proposal contains 19 problem sets: five from Algorithmic Number Theory, three from Algebraic Number Theory, five from Commutative and Homological Algebra, three from the Geometry of Numbers, and three from Group Theory. The collection has been composed with a view towards assisting the investigator's many current and future graduate students in choosing suitable thesis subjects. The problems have the appealing features of appearing to be feasible without being trivial, and of being specific without being narrow. They belong to mainstream areas that will also serve the students after obtaining their degrees. Of the nineteen problem sets, the following two are both easy to formulate and attractive. The first is the development of an algorithmic theory of quadratic forms over rings and fields of arithmetic interest. A typical question is how quickly one can find a representation of a positive integer as a sum of four squares. Or: it is known that any odd unimodular indefinite inner product space over the ring of integers is diagonalizable; given the symmetric matrix that defines the inner product, how quickly can one find the change of basis that diagonalizes the form? A first investigation shows that one may expect a wide spectrum of answers to the algorithmic questions in this area, displaying all the riches of number-theoretic algorithms. The second is giving class number estimates for orders in number fields. What is a good upper bound for the number of equivalence classes of fractional ideals of a giving order, expressed as a function of the degree and the discriminant of the order? And can one find better estimates for orders that have nice properties, such as being Gorenstein? This type of question is of importance in the theory of abelian varieties, and one will need to apply techniques coming from commutative algebra, abelian group theory, combinatorics, and elementary analytic number theory. In order to place the project in perspective one may consider the recent development of number theory. Present day number theory differs in two important respects from number theory twenty five years ago, namely in the roles played by algorithms and computers, and by algebraic geometry. It has been found that algorithmic number theory has important applications, notably in cryptography, and in addition number theorists have learned how to use computers for their research. Inventing good computational methods for number-theoretic problems has thus become of central importance. One of the principal investigator's strengths is in the interaction between theory and practice, on the one hand using recent theoretical advances for algorithmic purposes and on the other hand deriving purely mathematical inspiration from the problems suggested by the applications. At the other end of the spectrum, knowledge of algebraic geometry has become a standard requirement for aspiring number theorists. Virtually every breakthrough in number theory over the past few decades, including Andrew Wiles's work on Fermat's Last Theorem, has involved arithmetic algebraic geometry. Algebraic geometry depends on a broad spectrum of techniques from algebra and algebraic number theory, and gives rise to an unending array of tantalizing questions in those areas, of which the project studies a sample. What is maybe the most exciting of all, is the way in which arithmetic algebraic geometry and algorithmic number theory are presently being tied together, both in the application of geometric objects to cryptography and in the application of algorithmic techniques to investigate geometric objects in number theory. The project will be carried out by the investigator's graduate students, many of whom will, as experience shows, acquire combined expertise in these two areas, which is a very precious but fairly rare commodity.