Diophantine Geometry

Diophantine equations are systems of poiynomial equations to be solved in integers or rational numbers, and Diophantine geometry is the study of Diophantine equations using ideas and techniques from algebraic geometry. This is a very natural approach, since the basic objects studied in algebraic geometry, namely algebraic varieties, are themselves defined by systems of polynomial equations. The difference is that a (classical) algebraic geometer studies solutions in complex numbers or in some other algebraically closed field, while a number theorist studies solutions in a ring or field of arithmetic interest.