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MORDELL-WEIL THEOREM AND RANK OF ELLIPTIC CURVES

ABSTRACT

The Mordell-Weil theorem is a cornerstone in the study of elliptic curves, playing a crucial role in modern number theory and arithmetic geometry. It asserts that the set of rational points on an elliptic curve defined over a number field forms a finitely generated abelian group. This result has far-reaching implications, providing a foundational framework for understanding the arithmetic properties of elliptic curves and their associated Diophantine equations. In this paper, we present a comprehensive exploration of the Mordell-Weil theorem, delving into its proof, key techniques involved, and its broader significance. We outline the necessary background on elliptic curves, including their group structure, the Weierstrass equation, and the concept of rational points. A critical component of this discussion is the rank of an elliptic curve, which quantifies the number of independent infinite-order rational points in the group structure. The determination of rank remains one of the central challenges in the field, and we provide an overview of established methods for computing it, such as descent techniques, height functions, and computational algorithms. To illustrate these ideas, we include concrete examples of elliptic curves with varying ranks and examine how these examples align with theoretical predictions. Moreover, we discuss open problems and conjectures concerning the rank of elliptic curves, particularly the Birch and Swinnerton-Dyer (BSD) conjecture. This deep and influential conjecture suggests a direct relationship between the rank of an elliptic curve and the order of vanishing of its associated L-function at s=1. Despite significant progress in recent decades, the full proof of the BSD conjecture remains an open problem and is one of the Clay Millennium Prize Problems. Finally, we highlight recent developments in the study of elliptic curves, including advances in computational number theory, heuristic approaches to rank distribution, and new insights into the behavior of rational points. This discussion underscores the enduring significance of the Mordell-Weil theorem and its central role in ongoing research at the intersection of algebraic geometry, arithmetic, and analytic number theory.

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