Number Theory with Emphasis on Algorithms and Algebraic Number Theory

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Lenstra0100485The proposed research belongs to the interface between number theory andalgebra. It is inspired by problems that come up in an algorithmiccontext and in arithmetic algebraic geometry. Altogether, the proposalcontains 19 problem sets: five from Algorithmic Number Theory, threefrom Algebraic Number Theory, five from Commutative and HomologicalAlgebra, three from the Geometry of Numbers, and three from Group Theory.The collection has been composed with a view towards assisting theinvestigator's many current and future graduate students in choosingsuitable thesis subjects. The problems have the appealing features ofappearing to be feasible without being trivial, and of being specificwithout being narrow. They belong to mainstream areas that will alsoserve the students after obtaining their degrees. Of the nineteenproblem sets, the following two are both easy to formulate andattractive. The first is the development of an algorithmic theory ofquadratic forms over rings and fields of arithmetic interest. A typicalquestion is how quickly one can find a representation of a positiveinteger as a sum of four squares. Or: it is known that any odd unimodularindefinite inner product space over the ring of integers isdiagonalizable; given the symmetric matrix that defines the innerproduct, how quickly can one find the change of basis that diagonalizesthe form? A first investigation shows that one may expect a wide spectrumof answers to the algorithmic questions in this area, displaying all theriches of number-theoretic algorithms. The second is giving class numberestimates for orders in number fields. What is a good upper bound for thenumber 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 thetheory of abelian varieties, and one will need to apply techniques comingfrom commutative algebra, abelian group theory, combinatorics, andelementary analytic number theory.In order to place the project in perspective one may consider the recentdevelopment of number theory. Present day number theory differs in twoimportant respects from number theory twenty five years ago, namely inthe roles played by algorithms and computers, and by algebraic geometry.It has been found that algorithmic number theory has importantapplications, notably in cryptography, and in addition number theoristshave learned how to use computers for their research. Inventing goodcomputational methods for number-theoretic problems has thus become ofcentral importance. One of the principal investigator's strengths is inthe interaction between theory and practice, on the one hand using recenttheoretical advances for algorithmic purposes and on the other handderiving purely mathematical inspiration from the problems suggested bythe applications. At the other end of the spectrum, knowledge ofalgebraic geometry has become a standard requirement for aspiring numbertheorists. Virtually every breakthrough in number theory over the pastfew decades, including Andrew Wiles's work on Fermat's Last Theorem, hasinvolved arithmetic algebraic geometry. Algebraic geometry depends on abroad spectrum of techniques from algebra and algebraic number theory,and gives rise to an unending array of tantalizing questions in thoseareas, of which the project studies a sample. What is maybe the mostexciting of all, is the way in which arithmetic algebraic geometry andalgorithmic number theory are presently being tied together, both in theapplication of geometric objects to cryptography and in the applicationof algorithmic techniques to investigate geometric objects in numbertheory. The project will be carried out by the investigator's graduatestudents, many of whom will, as experience shows, acquire combinedexpertise in these two areas, which is a very precious but fairly rarecommodity.