Department of

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Tuesday, January 15, 2008

**Abstract:** An elliptic curve $E$ possessing a rational point is an arithmetic-algebraic object: It is simultaneously a nonsingular projective curve with an affine equation $Y^2 = X^3 + AX + B$, which allows one to perform arithmetic on its points; and a finitely generated abelian group $E(\Q) \simeq E(\Q)_{tors} \times \Z^r$, which allows one to apply results from abstract algebra. The abstract nature of its rank $r$ can be made explicit by searching for rational points $(X,Y)$. In this talk, we give some history on the problem of determining properties of $r$, explain its importance by discussing the conjecture of Birch and Swinnerton-Dyer, and analyze various approaches to finding curves of large rank.

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