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Elliptic Tales by Robert Gross, Avner Ash

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Chapter 15

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THE CONJECTURE OF BIRCH AND SWINNERTON-DYER

Road Map

We are finally at our journey’s end. Armed with our tools, we can describe the mysterious conjecture we have aimed at all along. It relates properties of the L-function to properties of the elliptic curve used to construct the L-function.

The conjecture says that the order of the zero of the L-function of an elliptic curve E at s = 1 is equal to the rank of the abelian group E(Q) of rational points on the elliptic curve. In this way, the number of rational solutions to a cubic equations with integer coefficients can be connected to analytic properties of generating functions coming ...

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