Arithmetic geometry is a branch of mathematics that combines concepts from algebraic geometry and number theory. It studies the solutions of polynomial equations and their properties, focusing on both rational and integer solutions. This field helps mathematicians understand the relationships between geometric shapes and numerical values. One of the key areas of interest in arithmetic geometry is the study of elliptic curves, which are specific types of curves defined by cubic equations. These curves have applications in various areas, including cryptography and modular forms, making arithmetic geometry an important field in modern mathematics.