inf(x,y) ∈Q[x,y]

The most basic question one can ask about a variety *X* over **Q** is to
determine the set *X*(**Q**)
of points with rational coordinates which lie on *X*.
For example, for fixed *n* > 2,
Fermat's last theorem
(proven by
Andrew Wiles in 1994)
states that the only rational points on the *Fermat curve* defined by

are (1,0) and (0,1) (forx+^{n}y= 1^{n}

Although in principle we would like to study arbitrary algebraic
varieties, the only varieties we presently
know much of anything about are curves.
If *X* is a smooth projective curve, the complex points *X*(**C**)
topologically form a compact orientable surface. The
classification of
surfaces thus tells us that *X*(**C**) is homeomorphic to a
torus with *g* holes
for some *g* ≥ 0.
This topological invariant *g* is called the *genus* of
*X*.

It is a remarkable fact that the genus of a curve tells us a great deal
about its set of rational points. It has long been known that a curve
of genus zero has either zero or
infinitely many rational points, and
by the
Hasse-Minkowski theorem we have an effective algorithm for
determining which occurs for a given curve of genus zero. At the other
extreme, the
Mordell Conjecture,
proven by
Gerd Faltings in 1983, states that
the set *X*(**Q**) of rational points on *X* is finite
if *X* has genus at least two.