I like to call my area "arithmetic geometry". Roughly speaking algebraic geometry studies the geometric objects (algebraic varieties) defined by polynomial equations over an algebraically closed field, and arithmetic geometry studies the same objects over arithmetically interesting fields such as the rational numbers. Thus it is a fascinating mixture of algebraic number theory and algebraic geometry.
My own work, and that of my students, has varied from almost straight algebraic number theory to almost straight algebraic geometry.
To give a little of the flavor of the subject, consider an elliptic curve
Y2=X3+aX+b
with a,b in Q. It is known that the points on this curve with coordinates in Q form a finitely generated group, but no algorithm is known for finding the rank of the group. There are many surprisingly deep problems concerning these simple curves. A key to the proof of Fermat's Last Theorem was the realization that a point on a Fermat curve would give rise to an elliptic curve over Q with properties so weird that it couldn't possibly exist.
A projective algebraic variety with a group structure is called an abelian variety---elliptic curves are the abelian varieties of dimension one. Most of my research has concerned abelian varieties, or more general objects called motives. My early work mainly studied the arithmetic properties of the abelian varieties themselves, but more recently I have been studying the varieties, called Shimura varieties, that parametrize abelian varieties (or motives). Initially these are defined as quotients of certain complex domains by discrete groups, but a difficult theorem shows that they have a natural structure as algebraic varieties over the complex numbers, and an even more difficult theorem shows that these algebraic varieties are defined over finite extensions of Q. Thus their study falls into arithmetic geometry. Their arithmetic properties have interested mathematicians for 150 years. For example, it is known that one can obtain interesting Galois extensions of Q by adjoining the coordinates of certain special points. The simplest Shimura varieties are the elliptic modular curves.
My course notes on Elliptic Curves can be viewed as an introduction to arithmetic geometry (they are available from my homepage).