#### Vol. 1, No. 1, 2013

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Conditionally bounding analytic ranks of elliptic curves

### Jonathan W. Bober

Vol. 1 (2013), No. 1, 135–144
##### 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 $\mathfrak{3}\mathfrak{0}$.

##### Keywords
elliptic curve, rank, L-function, explicit formula
Primary: 11M41
Secondary: 14G10
##### Milestones
Published: 14 November 2013
##### Authors
 Jonathan W. Bober Department of Mathematics University of Washington Seattle, WA 98195-4350 United States Howard House University of Bristol Queens Avenue Bristol BS8 1SN United Kingdom