On congruent primes and class numbers of imaginary quadratic fields

TL;DR

This paper introduces new criteria based on descent methods to determine whether certain primes are congruent or have specific class number properties, providing insights into their distribution and implications for elliptic curves.

Contribution

It presents easily computed criteria for identifying non-congruent primes and primes with class number divisible by 16, advancing understanding of their distribution and properties.

Findings

Certain primes are proven not to be congruent using the new criterion.

Primes with 16 dividing their class number are characterized by the second criterion.

At least one of the criteria applies to infinitely many primes.

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(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.