Arithmetic algebraic geometry is a branch of mathematics that combines the concepts and techniques of algebraic geometry with number theory. It aims to study the algebraic structures and properties of geometric objects defined by algebraic equations, but with a particular focus on the arithmetic aspects of these objects.
In traditional algebraic geometry, the underlying field of study is typically the field of complex numbers. However, in arithmetic algebraic geometry, the field of study is usually the field of rational numbers or other number fields. This is because the arithmetic properties of geometric objects are closely related to the properties of their solutions in fields other than the complex numbers.
Arithmetic algebraic geometry investigates the behavior of algebraic curves, surfaces, and higher-dimensional varieties over number fields. It studies questions related to the existence of rational points on these varieties, their arithmetic properties, and the connections between the algebraic and arithmetic structures.
The subject has found extensive applications in number theory, where it has been used to prove various deep results, such as Fermat's Last Theorem and the Birch and Swinnerton-Dyer conjecture. It has also provided insights into the distribution of prime numbers and the study of Diophantine equations.
Overall, arithmetic algebraic geometry provides a powerful framework for studying the geometry of algebraic objects with a focus on their arithmetic properties, thereby deepening our understanding of both algebraic and number-theoretic phenomena.
