An algebraic variety over Q is a subset of some complex n-space Cn defined by the vanishing of a set of polynomial equations with rational coefficients. For example, the zero locus of a polynomial
f(x,y) ∈ Q[x,y]
in C2 defines a curve. (It is of course much easier to visualize a zero locus in R2; historically, as in this case, the terminology is often based on the real rather than the complex picture. Also, we really should be considering the projective closure of these affine varieties; I will do this without further comment below.) Arithmetic geometry is the study of algebraic varieties over Q.

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

xn + yn = 1
are (1,0) and (0,1) (for n odd) or (±1,0) and (0,±1) (for n even).

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.

Next: Elliptic Curves