On congruent primes and class numbers of imaginary quadratic fields

Nils Bruin, and Brett Hemenway. "On congruent primes and class numbers of imaginary quadratic fields." Acta Arithmetica 159.1 (2013): 63-87. <http://eudml.org/doc/279280>.

@article{NilsBruin2013,
abstract = {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.},
author = {Nils Bruin, Brett Hemenway},
journal = {Acta Arithmetica},
keywords = {congruent numbers; Tate-Shafarevich groups; class numbers; descent},
language = {eng},
number = {1},
pages = {63-87},
title = {On congruent primes and class numbers of imaginary quadratic fields},
url = {http://eudml.org/doc/279280},
volume = {159},
year = {2013},
}

TY - JOUR
AU - Nils Bruin
AU - Brett Hemenway
TI - On congruent primes and class numbers of imaginary quadratic fields
JO - Acta Arithmetica
PY - 2013
VL - 159
IS - 1
SP - 63
EP - 87
AB - 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.
LA - eng
KW - congruent numbers; Tate-Shafarevich groups; class numbers; descent
UR - http://eudml.org/doc/279280
ER -