On congruent primes and class numbers of imaginary quadratic fields

• Autorzy: Nils Bruin , Brett Hemenway
• Języki publikacji: EN

Abstrakty

EN

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 ℚ(√-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.

Kategorie tematyczne

• 11G05: Elliptic curves over global fields
• 11R29: Class numbers, class groups, discriminants

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 159
• Numer: 1
• Strony: 63-87

Daty

wydano

2013

Twórcy

autor

Nils Bruin

• Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada

autor

Brett Hemenway

• Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, U.S.A.

Identyfikatory

DOI

10.4064/aa159-1-4