Editing Number theory (section)

=== Diophantine geometry ===
{{Main|Diophantine geometry}}

The central problem of Diophantine geometry is to determine when a [[Diophantine equation]] has integer or rational solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object.

For example, an equation in two variables defines a curve in the plane. More generally, an equation or system of equations in two or more variables defines a [[algebraic curve|curve]], a [[algebraic surface|surface]], or some other such object in {{math|''n''}}-dimensional space. In Diophantine geometry, one asks whether there are any ''rational points'' (points all of whose coordinates are rationals) or
''integral points'' (points all of whose coordinates are integers) on the curve or surface. If there are any such points, the next step is to ask how many there are and how they are distributed. A basic question in this direction is whether there are finitely
or infinitely many rational points on a given curve or surface.

Consider, for instance, the [[Pythagorean equation]] <math>x^2+y^2 = 1</math>. One would like to know its rational solutions, namely <math>(x,y)</math> such that ''x'' and ''y'' are both rational. This is the same as asking for all integer solutions
to <math>a^2 + b^2 = c^2</math>; any solution to the latter equation gives us a solution <math>x = a/c</math>, <math>y = b/c</math> to the former. It is also the
same as asking for all points with rational coordinates on the curve described by <math>x^2 + y^2 = 1</math> (a circle of radius 1 centered on the origin).

[[File:ECClines-3.svg|thumb|Two examples of [[elliptic curve]]s, that is, curves of genus 1 having at least one rational point.]]

The rephrasing of questions on equations in terms of points on curves is felicitous. The finiteness or not of the number of rational or integer points on an algebraic curve (that is, rational or integer solutions to an equation <math>f(x,y)=0</math>, where <math>f</math> is a polynomial in two variables) depends crucially on the [[genus (mathematics)|genus]] of the curve.<ref group="note">The ''genus'' can be defined as follows: allow the variables in <math>f(x,y)=0</math> to be complex numbers; then <math>f(x,y)=0</math> defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables; that is, four dimensions). The number of doughnut-like holes in the surface is called the ''genus'' of the curve of equation <math>f(x,y)=0</math>.</ref> A major achievement of this approach is [[Wiles's proof of Fermat's Last Theorem]], for which other geometrical notions are just as crucial.

There is also the closely linked area of [[Diophantine approximations]]: given a number <math>x</math>, determine how well it can be approximated by rational numbers. One seeks approximations that are good relative to the amount of space required to write the rational number: call <math>a/q</math> (with <math>\gcd(a,q)=1</math>) a good approximation to <math>x</math> if <math>|x-a/q|<\frac{1}{q^c}</math>, where <math>c</math> is large. This question is of special interest if <math>x</math> is an algebraic number. If <math>x</math> cannot be approximated well, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of [[Glossary of arithmetic and diophantine geometry#H|height]]) are critical both in Diophantine geometry and in the study of Diophantine approximations. This question is also of special interest in [[transcendental number theory]]: if a number can be approximated better than any algebraic number, then it is a [[transcendental number]]. It is by this argument that [[Pi|{{pi}}]] and [[e (mathematical constant)|e]] have been shown to be transcendental.

Diophantine geometry should not be confused with the [[geometry of numbers]], which is a collection of graphical methods for answering certain questions in algebraic number theory. [[Arithmetic geometry]] is a contemporary term for the same domain covered by Diophantine geometry, particularly when one wishes to emphasize the connections to modern algebraic geometry (for example, in [[Faltings's theorem]]) rather than to techniques in Diophantine approximations.