On congruent primes and class numbers of imaginary quadratic fields

Nils Bruin ¹   ; Brett Hemenway ²

¹ Department of Mathematics Simon Fraser University Burnaby, BC V5A 1S6, Canada
² Department of Mathematics University of Michigan Ann Arbor, MI 48109-1043, U.S.A.

Acta Arithmetica, Tome 159 (2013) no. 1, pp. 63-87

Résumé

We consider the problem of determining whether a given prime $p$ is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate–Shafarevich groups of the associated elliptic curves. We also present a related criterion for primes $p$ such that $16$ divides the class number of the imaginary quadratic field $\mathbb {Q}(\sqrt {-p})$. Both results are based on descent methods. While we cannot show for either criterion individually that there are infinitely many primes that satisfy it nor that there are infinitely many that do not, we do exploit a slight difference between the two to conclude that at least one of the criteria is satisfied by infinitely many primes.

Détail

DOI : 10.4064/aa159-1-4

Keywords: consider problem determining whether given prime congruent number present easily computed criterion allows conclude certain primes which congruency previously undecided congruent result get additional information possible sizes tate shafarevich groups associated elliptic curves present related criterion primes divides class number imaginary quadratic field mathbb sqrt p results based descent methods while cannot either criterion individually there infinitely many primes satisfy nor there infinitely many exploit slight difference between conclude least criteria satisfied infinitely many primes