Conditionally bounding analytic ranks of elliptic curves

TL;DR

This paper introduces a method to bound the rank of elliptic curves assuming the Birch and Swinnerton-Dyer conjecture and the generalized Riemann hypothesis, providing explicit upper bounds for several high-rank curves.

Contribution

It presents a new conditional method for bounding elliptic curve ranks, improving understanding of their possible maximum ranks under key conjectures.

Findings

Bounded ranks of specific elliptic curves with known high ranks

Provided explicit upper bounds under conjectural assumptions

Demonstrated the method on curves with ranks up to at least 28

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