  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 20,
21, 22, 23, and 24. For the known curve of rank at least 28, we get a bound of
30.
