section_path: Main subdivisions/Diophantine geometry
section_name: Diophantine geometry
paragraph_text: Diophantine geometry
  
  The central problem of Diophantine geometry is to determine when a Diophantine equation has 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 curve, a surface or some other such object in 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 if there are finitelyor infinitely many rational points on a given curve (or surface).
  
  In the Pythagorean equation x 2 + y 2 = 1, {\displaystyle x^{2}+y^{2}=1,} we would like to study its rational solutions, that is, its solutions (x, y) {\displaystyle (x,y)} such that x and y are both rational. This is the same as asking for all integer solutionsto a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}}; any solution to the latter equation gives us a solution x = a / c {\displaystyle x=a/c}, y = b / c {\displaystyle y=b/c} to the former. It is also thesame as asking for all points with rational coordinates on the curve described by x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1}. (This curve happens to be a circle of radius 1 around the origin.)
  
  The rephrasing of questions on equations in terms of points on curves turns out to be 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 f (x, y) = 0 {\displaystyle f(x,y)=0}, where f {\displaystyle f} is a polynomial in two variables—turns out to depend crucially on the genus of the curve. The genus can be defined as follows: allow the variables in f (x, y) = 0 {\displaystyle f(x,y)=0} to be complex numbers; then f (x, y) = 0 {\displaystyle f(x,y)=0} 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). If we count the number of (doughnut) holes in the surface; we call this number the genus of f (x, y) = 0 {\displaystyle f(x,y)=0}. Other geometrical notions turn out to be just as crucial.
  
  There is also the closely linked area of Diophantine approximations: given a number x {\displaystyle x}, then finding how well can it be approximated by rationals. (We are looking for approximations that are good relative to the amount of space that it takes to write the rational: call a / q {\displaystyle a/q} (with gcd (a, q) = 1 {\displaystyle \gcd(a,q)=1}) a good approximation to x {\displaystyle x} if | x − a / q | < 1 q c {\displaystyle |x-a/q|<{\frac {1}{q^{c}}}}, where c {\displaystyle c} is large.) This question is of special interest if x {\displaystyle x} is an algebraic number. If x {\displaystyle x} cannot be well approximated, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of height) turn out to be 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 better approximated than any algebraic number, then it is a transcendental number. It is by this argument that π and 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, however, is a contemporary term for much the same domain as that covered by the term Diophantine geometry. The term arithmetic geometry is arguably used most often when one wishes to emphasise the connections to modern algebraic geometry (as in, for instance, Faltings's theorem) rather than to techniques in Diophantine approximations.