Guide to Graduate Study in Number Theory

Introduction

The object of number theory is to study intrinsic properties of integers, and, more generally of numbers. Here we shall discuss some of the main areas of number theory and some of the important problems in each area.

Elementary Number Theory

Included in elementary number theory are divisibility and prime factorization, residue classes, congruences, the quadratic reciprocity law, representation of numbers by forms, diophantine equations, continued fraction approximations and sieves. Because of its charm and general accessibility, this is one of the best known areas of number theory. The description "elementary" refers more to the nature of the methods employed than to the level of difficulty of the subject.

Analytic Number Theory

In analytic number theory an arithmetical phenomenon is represented by a related function, generally an analytic function of a complex variable. Information about the arithmetical problem, generally of an asymptotic nature, is then extracted by analysis of the associated function. It is remarkable that study of continuous quantities yields information in discrete problems.

The first famous result in this area is Dirichlet's theorem that any arithmetic progression, a, a + q, a + 2q, ... contains an infinite number of primes provided only that a and q are relatively prime. Corresponding results for non-linear polynomial sequences (n^2 + 1, for example) are almost certainly true but remain unproved. Sieves are combinatorial devices for counting, in a given integer sequence, elements having very few prime factors. In combination with analytic means, these devices have had considerable success in recent years in a variety of contexts ranging from measuring gaps between consecutive primes to Fermat's Last Theorem. Outstanding unsolved problems include the Riemann hypothesis on the location of the zeros of the Riemann zeta function and the conjecture that there exist an infinite number of "twin primes" p and p + 2.

Algebraic Number Theory

Algebraic number theory deals with fields of algebraic numbers, that is with numbers which are roots of a polynomial equation with rational coefficients. Among areas of study are decomposition laws for primes in a number field, extensions of the famous quadratic reciprocity law and of Gauss and Jacobi sums, class numbers and units, and the connection between Stickelberger elements and p-adic L-functions.

Many different techniques from algebra and analysis are used to attack these problems, despite the sometimes expressed view that algebraic number theory is the algebraic aspect of number theory

Abelian class field theory, the description of all the extensions of a number field with abelian Galois group, represents an important solved problem. Major unsolved problems are the development of a nonabelian class field theory and its connection with modular forms, Vandiver's conjecture on class numbers of totally real cyclotomic fields, and the holomorphy of Artin L-functions.

Probabilistic Number Theory

In probabilistic number theory statistical limit theorems are established in problems involving "almost independent" random variables. Methods used include a combination of probabilistic, elementary and analytic ideas.

One of the first achievements in this area was the Erdos-Kac theorem, which asserts that properly normalized values of a rather general additive arithmetical function have a Gaussian limit distribution. The determination of necessary and sufficient conditions for such functions to have a limit distribution is an outstanding problem.

Another famous probabilistic result, due to Khintchine and Levy states that for almost all real numbers x between 0 and 1, the geometric mean of the first n denominators of the continued fraction expansion of x converges to the unique number exp(r2/12 log 2).

Diophantine Approximation and the Geometry of Numbers

How well a given real number can be approximated by a rational number with a relatively small denominator is a typical problem of Diophantine Approximation. The Geometry of Numbers, a technique developed by Minkowski, often transforms such questions into problems about lattice points in convex bodies, or about packing of spheres in n-space. This leads, for example, to the Roth-Schmidt theorem on simultaneous approximation. Completely different is the analytic approach that has led to Baker's theorems on the linear independence of logarithms of algebraic numbers. Outstanding unsolved problems include the transcendence of Euler's constant, Schanuel's conjecture on the algebraic independence of numbers and their logarithms, and the determination of the densest sphere packing in n-dimensions.

Combinatorial Number Theory

Many problems in number theory involve combinatorial ideas in their formulation or solution. Abstraction into combinatorial terms sometimes provides the insight and simplification needed to handle a difficult problem. A noteworthy achievement of combinatorial methods is the following problem on progressions.

Say that an increasing sequence A = {a1,a2, . . . } of natural numbers has the "progression property" if for every k there are k members of A in arithmetic progression. An important problem in combinatorial number theory is to find natural conditions on A guaranteeing that it will have the progression property.

Over 50 years ago it was shown by van der Waerden that if successive members of A differ by at most a fixed (but arbitrary) bound, then A has the progression property. Recently it was shown by Szemereldi that if n/a does not have the limit zero as n -> ° , then A has the progression property. It is easily seen that this theorem implies van der Waerden's. It is not known whether the divergence of the sum of the reciprocals of the elements of A implies that A must have the progression property. The well known combinatorial number theorist Erdos offered a $3000 prize for the solution of this problem!

Elliptic Curves and Modular Forms

Elliptic curves are geometric objects like tori, and modular forms are special functions such as Ramanujan's q \prod (1-q^n)^{24} (q = exp(2\pi iz)). They are, however, intimately related to each other and vital to the solution of various Diophantine problems, such as Fermat's Last Theorem. Arithmetic invariants of these curves and forms produce L-functions and representations of Galois groups, the study of which (by analysis, algebra, ...) sheds light on the original problems. Great progress has been made recently in resolving conjectures in the area, but one famous unsolved problem is the conjecture of Birch and Swinnerton-Dyer, which roughly says that the rank of the group of rational points of an elliptic curve equals the order of the zero of its L-function at s = 1.

Arithmetic Geometry

Arithmetic Geometry studies the solutions of systems of polynomial equations in several variables, but one is mainly interested in solutions which are integers or rational numbers. The techniques applied are often geometric in nature, meaning that one pays attention to the possible existence of singularities, or that one uses geometric constructions such as projective space and tangent bundles. The techniques are also arithmetic in nature, depending heavily on the geometry of numbers or the prime factorization of ideals. Powerful machinery developed over the years in this area has led to the solution of important problems, such as Faltings' solution of Mordell's conjecture about certain polynomial equations in two variables having only a finite number of solutions.

Number Fields and Function Fields

Algebraic number fields are obtained by adjoining to the rationals the root(s) of a polynomial with rational coefficients. Such a field K contains a ring R of all elements satisfying a monic polynomial with integer coefficients, e.g. the Gaussian integers in the field Q(i). In this ring unique factorization of elements into irreducibles is replaced by unique factorization of ideals into prime ideals. The study of the action of Galois automorphisms on R leads to some surprisingly deep connections with the functional equation of the zeta function of K, a generalization of the Riemann zeta function. Class field theory deals with generalizations of the quadratic reciprocity law for quadratic fields to other Galois extensions with abelian Galois group. Other topics are class groups, unit groups, Gauss sums, Stickelberger relations, and p-adic L-functions. Throughout there is a parallel theory of function fields over finite fields, in which some of the questions have an easier solution than in the number field case.

Courses in Number Theory

The following three courses are offered every year: elementary number theory (Math 353), algebraic number theory (Math 405), and analytic number theory (Math 453). Topics courses are offered on specialized subjects, typically two each semester. In recent years the following topics have been offered (some more than once): modular forms, elementary number theory from an advanced viewpoint, Gauss and Jacobi sums, quadratic fields, distribution of prime numbers, sieve methods, distribution of additive and multiplicative arithmetic functions, Abelian varieties, probabilistic number theory, combinatorial number theory, cyclotomic fields, zeta- and L functions, geometry of numbers, diophantine approximation, P -V numbers, Ramanujan's lost notebooks, theory of partitions, elliptic functions with applications to number theory, q-series, and continued fractions.

The department also supports an active program of seminars and colloquium talks in number theory. There are three or four seminars weekly, each focusing on some area of number theory. Most of these are accessible and of interest to graduate students.

Reading courses and thesis research direction are provided for advanced students.

Last modified January 23, 2002