On the divisibility of the class number of imaginary quadratic fields

The theorem presented in this paper provides a sufficient condition for the divisibility of the class number of an imaginary quadratic field by an odd prime. Two corollaries to this theorem are also included. They represent special cases of the theorem which in general use are somewhat easier to apply.     

On the 2-part of the class number of imaginary quadratic number fields

We employ a type number formula from the theory of quaternion algebras to gain information on the 2-part of the class numbers of imaginary quadratic number fields whose discriminants are divisible by three or fewer prime numbers.     

On the average of the number of imaginary quadratic fields with a given class number

Let \(\mathcal {F}(h)\) be the number of imaginary quadratic fields with class number h. In this note, we improve the error term in Soundararajan’s asymptotic formula for the average of \(\mathcal {F}(h)\). Our argument leads to a similar refinement of the asymptotic for the average of \(\mathcal {F}(h)\) over odd h, which was recently obtained by Holmin, Jones, Kurlberg, McLeman and Petersen.     

On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅¹⁰⁹. We determine all these K's with dK?¹⁰⁶. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅¹⁰¹¹.     

Ono invariants of imaginary quadratic fields with class number three

Let Ed(x) denote the “Euler polynomial” x²+x+(1−d)/4 if and x²−d if . Set Ω(n) to be the number of prime factors (counting multiplicity) of the positive integer n. The Ono invariantOnod of is defined to be except when d=−1,−3 in which case Onod is defined to be 1. Finally, let hd=hk denote the class number of K. In 2002 J. Cohen and J. Sonn conjectured that hd=3⇔Onod=3 and is a prime. They verified that the conjecture is true for p<1.5×¹⁰⁷. Moreover, they proved that the conjecture holds for p>¹⁰¹⁷ assuming the extended Riemann Hypothesis. In this paper, we show that the conjecture holds for p?2.5×¹⁰¹³ by the aid of computer. And using a result of Bach, we also proved that the conjecture holds for p>2.5×¹⁰¹³ assuming the extended Riemann Hypothesis. In conclusion, we proved the conjecture is true assuming the extended Riemann Hypothesis.     

Ideal class groups of imaginary quadratic fields

The Ramanujan Journal - Let d be a square-free positive integer and $$\mathrm{CL}(-d)$$ the ideal class group of the imaginary quadratic field $${\mathbb {Q}}(\sqrt{-d})$$ . In this paper, we show...     

On 2-ciass field towers of some imaginary quadratic number fields

We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2, 2, 2) whose Hilbert 2-class fields are finite.     

The tame kernel of imaginary quadratic fields with class number 2 or 3

This paper presents improved bounds for the norms of exceptional finite places of the group , where is an imaginary quadratic field of class number 2 or 3. As an application we show that .

     

On a class of elliptic functions associated with imaginary quadratic fields

Let be the field discriminant of an imaginary quadratic field. We construct a class of elliptic functions associated naturally with the quadratic field which, combined with the general theory of elliptic functions, allows us to provide a unified theory for two fundamental results (one classical and one due to Ramanujan) about the elliptic functions.

     

Normal bases of ray class fields over imaginary quadratic fields

We develop a criterion for a normal basis (Theorem 2.4), and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than ${\mathbb{Q}(\sqrt{-1})}$ and ${\mathbb{Q}(\sqrt{-3})}$ (Theorem 4.2). This result would be an answer for the Lang-Schertz conjecture on a ray class field with modulus generated by an integer (≥2) (Remark 4.3).     

Tame and wild kernels of quadratic imaginary number fields

For all quadratic imaginary number fields of discriminant
we give the conjectural value of the order of Milnor's group (the tame kernel) where is the ring of integers of Assuming that the order is correct, we determine the structure of the group and of its subgroup (the wild kernel). It turns out that the odd part of the tame kernel is cyclic (with one exception, ).

     

Divisibility of class numbers of imaginary quadratic function fields by a fixed odd number

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field $\mathbb{F}_{q}(x)$ whose class groups have elements of a fixed odd order. More precisely, for q, a power of an odd prime, and g a fixed odd positive integer ≥?3, we show that for every ε?>?0, there are $\gg q^{L(\frac{1}{2}+\frac{3}{2(g+1)}-\epsilon)}$ polynomials $f \in \mathbb{F}_{q}[x]$ with $\deg f=L$ , for which the class group of the quadratic extension $\mathbb{F}_{q}(x, \sqrt{f})$ has an element of order g. This sharpens the previous lower bound $q^{L(\frac{1}{2}+\frac{1}{g})}$ of Ram Murty. Our result is a function field analogue which is similar to a result of Soundararajan for number fields.     

A new family of imaginary quadratic fields whose class number is divisible by five

In this paper, we prove that the class number of the imaginary quadratic field (s?0) is divisible by 5, where Fn is the nth number in the Fibonacci sequence. Moreover we give a polynomial with integer coefficients whose splitting field over Q is an unramified cyclic quintic extension of .