number theory II

Number Theory

Number theory, according to Gauß the queen of mathematics, is concerned with properties of whole numbers. One often encounters a situation where a quastion can easily be asked, but an answer is seamingly impossible to achieve. Nowadays of course some knowledge of mathematics will be requires in order to understand the more subtle questions in number theory. Here are some typical questions from number theory:

1. How many prime numbers are there below some given real number?

2. Which natural numbers can be written (and in how many ways) as $x^2+y^2$ ( $x$ and
$y$ being integers)?

3. How many types of quadratic forms $ax^2+2bxy + cy^2$ are there with integers
$a,b,c$ and given discriminant $b^2-ac$ ?

In number theory one has some profit from cooperating with other mathematical disciplines. This leads to different sub-areas of number theory.

Algebraic number theory

Here the properties of integral numbers are being explored systematically. These are complex roots of normed polynomials with coefficients in $\mathbb Z$ . Studying the zeros of such a polynomial $f$ often means to study its splitting field $K$ , the smallest subfield of the complex numbers which contains all roots of $f$ . This is a field of some finite dimension $n$ over the field of rationals. The set of of all integral numbers inside $K$
is a subring ${\cal O}$ of $K$ , a Dedekindring, which shares some proerties with
${\mathbb Z}$ . A main difference which often occures is that there will not be a unique factorization into prime elements in the ring ${\cal O}$ . This defect leads to the introduction of the class group of ${\cal O}$ , which measures how far ${\cal O}$ is away from unique factorisation, and which turns out to be a finite group. Question 3 above is closely related to this question for specific choices of number fields $K$ .

An important feature is the possibility to embed $K$ into certain fields carry a metric defined by using $K$ , and which are complete wrt this metric and locally compact wrt the topology defined by the metric. It often is possible to separate phenomena which are cuased by different prime ideals in ${\cal O}$ , as there is one such completion for every prime ideal.

Geometry of numbers

The ring ${\cal O}$ can be embedded naturally as a lattice (=discrete subgroup) in an $n$ -dimensional real vectorspace . A prominent example for this is the ring ${\mathbb Z}[{\rm i}]$ of Gaussian numbers which is a lattice in the field of complex numbers. This paves the way in order to use geometric arguments in proving arithmetic facts. For instance, the finiteness of the class group alluded to above is a consequence of Minkowski's famous lattice point theorem. An answer on question 2 also can be derived using this feature (but there are also some other arguments).

A generelazition of this process leads to the introduction of certain discrete subgroups of Lie-Groups, which play an important role in differential geometry, algebraic geometry and the authors life. Here is a place where information goes from arithmetic to geometry and vice versa.

analytic number theory

The ring ${\cal O}$ has a Dedekind zeta-function $\zeta_K$ . This is a generating function encoding the number of ideals in ${\cal O}$ of given index. In the special case ${\cal O}= \mathbb Z$ we find Riemann's zeta-function. Every Dedekind zeta-function can be continued analytically to a meromorphic function on $\mathbb C$ with a single pole at $s=1$ . Analytic properties of $\zeta_K$ reflect arithmetic properties of ${\cal O}$ . Riemann's zeta-function answers question 1 above: there are approximately $x/\log x$ primes below $x$ . The famous Riemann conjecture is equivalent to the best possible error estimation of this so-called prime number theorem.

A more recent variant of zeta-functions is given by the theore of $p$ -adic L-series. Many problems in number theory also lead to other types of analytic functions, e.g. modular forms. These in turn also have L-series attached to them. The analytic properties of these "automorphic" L-series are easier to understand than those of "arithmetic" L-series.

Arithmetic Geometry

Using methods from algebraic geometry one treats systems of polynomial equations with coefficients in ${\cal O}$ . Besides purely algebraic methods one can sometimes use the completions of $K$ (local-global-princple). Moreover, there will be some L-series corresponding to every Well-behaved" system of polynomial equations, which converges in some right half-plane. With some luck, this arithmetic L-series will be equal to some automorphic L-series and this would lead to new arithmetic information to be got out of the L-series. This miracle is being expected to hold in a systematic way (Langlands programme). A prominent example where it did work out was Wiles' proof of Fermat's last theorem.

Many of the applications of number theory (cryptography, coding theory) are situated in the realm of arithmetic geometry.