Abstract

We describe a method for bounding the rank of an elliptic curve under the assumptions of the Birch and Swinnerton-Dyer conjecture and the generalized Riemann hypothesis. As an example, we compute, under these conjectures, exact upper bounds for curves which are known to have rank at least as large as $\mathfrak{2}\mathfrak{0},\mathfrak{2}\mathfrak{1},\mathfrak{2}\mathfrak{2},\mathfrak{2}\mathfrak{3}$, and $\mathfrak{2}\mathfrak{4}$. For the known curve of rank at least $\mathfrak{2}\mathfrak{8}$, we get a bound of $\mathfrak{3}\mathfrak{0}$.

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Mathematical Subject Classification 2010

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