On a theorem of Coleman (Q1911185)

The paper is motivated by Coleman's effective version of Chabauty's theorem which yields in this context the following: Let \(C/\mathbb{Q}\) be a curve of genus 2 (plus some extra conditions on the equation defining \(C)\) with Jacobian \(J\), such that the rank of \(J(\mathbb{Q})\) is at most 1. Then \(C(\mathbb{Q})\) is finite and can be computed effectively. In order to apply this theorem one must compute the rank of \(J(\mathbb{Q})\). In earlier papers the author presented an algorithm to compute this rank by descent via isogeny. This method involves the construction of certain homogeneous spaces described by 72 quadratic forms in \(\mathbb{P}^{15}\). Since these calculations can not be verified by a reader without a computer algebra system, the author presents here an improvement of his earlier algorithm which avoids these homogeneous spaces. He applies his method to an example which really can be checked without the use of a computer.