STUDYING THE CONNECTION BETWEEN ALGEBRAIC GEOMETRY AND NUMBER THEORY

Abstract

This article is dedicated to studying the deep and multifaceted connection between algebraic geometry and number theory. These two areas of mathematics, though initially appearing separate, have shown throughout history an ever-closer and mutually enriching relationship. The article examines the historical evolution from Diophantine equations to modern concepts such as elliptic curves, category theory, motives, the Langlands program, and Arakelov geometry. Furthermore, the significant role of these connections in modern mathematics is analyzed, including their importance in applications such as cryptography and future research directions. The aim of the research is to provide an overview of how geometric methods help solve arithmetic problems and, conversely, how ideas from number theory contribute to the understanding of geometric structures.