# Introduction to Elliptic Curves

MATH 900

In no mathematical field do diverse disciplines—algebra, number theory, and geometry—come together so elegantly as they do in the study of elliptic curves. The most fascinating of these is the so-called “group law,” which associates two known solutions on the curve with a third. Elliptic curves have proved useful in cyptography and in the solution to Fermat’s Last Theorem.

In no mathematical field do diverse disciplines—algebra, number theory, and geometry—come together so elegantly as they do in the study of elliptic curves. In simple terms, with a few little subleties, an elliptic curve is one whose defining equation can be put in the form y2 = x3 + ax + b. In the case of rational coefficients, the search for “rational points” on the curve—that is, points whose coordinates are rational numbers—opens up a panoply of investigative techniques drawn from the “toolboxes” of the three disciplines. The most fascinating of these is the so-called “group law,” which associates two known solutions on the curve with a third. Elliptic curves have proved useful in cyptography and in the solution to Fermat’s Last Theorem.
Prerequisites
Familiarity with the basic tenets of number number and group theory.

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