Divisibility of the class numbers of imaginary quadratic fields

For a given odd integer $n>1$, we provide some families of imaginary quadratic number fields of the form $\mathbb{Q}(\sqrt{x^2-t^n})$ whose ideal class group has a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$.Comment: 10 pages, accepted for publication in Journal of Number Theory (2017

Averages and moments associated to class numbers of imaginary quadratic fields

For any odd prime $\ell$, let $h_\ell(-d)$ denote the $\ell$-part of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Nontrivial pointwise upper bounds are known only for $\ell =3$; nontrivial upper bounds for averages of $h_\ell(-d)$ have previously been known only for $\ell =3,5$. In this paper we prove nontrivial upper bounds for the average of $h_\ell(-d)$ for all primes $\ell \geq 7$, as well as nontrivial upper bounds for certain higher moments for all primes $\ell \geq 3$.Comment: 26 pages; minor edits to exposition and notation, to agree with published versio

On congruent primes and class numbers of imaginary quadratic fields

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.Comment: 21 pages; changes w.r.t. version 1: some added references and minor correction

Indivisibility of class numbers of imaginary quadratic fields

We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to -X whose class numbers are indivisible by a given prime and whose imaginary quadratic fields satisfy any given set of local conditions. This estimate matches the best results in the direction of the Cohen-Lenstra heuristics for the number of imaginary quadratic fields with class number indivisible by a given prime. This general result is applied to study rank 0 twists of certain elliptic curves.Comment: 11 pages, revised version based on reviewer's comment

Hilbert modular forms and class numbers

In 1975, Goldfeld gave an effective solution to Gauss's conjecture on the class numbers of imaginary quadratic fields. In this paper, we generalize Goldfeld's theorem to the setting of totally real number fields.Comment: 35 page

Divisibility of class numbers of imaginary quadratic fields whose discriminant has only three prime factors

We prove the existence of infinitely many imaginary quadratic fields whose discriminant has exactly three distinct prime factors and whose class group has an element of a fixed large order. The main tool we use is solving an additive problem via the circle method. © 2012 Akadémiai Kiadó, Budapest, Hungary

Divisibility Criteria for Class Numbers of Imaginary Quadratic Fields Whose Discriminant Has Only Two Prime Factors

We will prove a theorem providing sufficient condition for the divisibility of class numbers of certain imaginary quadratic fields by 2g, where g>1 is an integer and the discriminant of such fields has only two prime divisors