arithmetic geometry

Arithmetic geometry is a field that combines algebraic geometry and number theory. It focuses on studying integer solutions to polynomial equations. This area of mathematics was central to Andrew Wiles's proof of Fermat's Last Theorem. Algebraic geometry itself studies the geometric properties of solutions to polynomial equations, extending beyond three dimensions. It emerged from analytic geometry and uses concepts from topology, complex analysis, and algebra to analyze algebraic curves, which are graphs of polynomial equations. These curves are classified by their genus, a non-negative integer that can be calculated from their polynomial.1