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.     

On the class number of certain imaginary quadratic fields

Theorem. Let 2$"> denote an integer, the square-free part of and the class number of the field . Then except for the case , divides .

     

Class numbers of imaginary quadratic fields

The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number . The first complete results were for by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any . Indeed, after Oesterlé handled , in 1985 Serre wrote, ``No doubt the same method will work for other small class numbers, up to 100, say.' However, more than ten years later, after doing , Wagner remarked that the case seemed impregnable. We complete the classification for all , an improvement of four powers of 2 (arguably the most difficult case) over the previous best results. The main theoretical technique is a modification of the Goldfeld-Oesterlé work, which used an elliptic curve -function with an order 3 zero at the central critical point, to instead consider Dirichlet -functions with low-height zeros near the real line (though the former is still required in our proof). This is numerically much superior to the previous method, which relied on work of Montgomery-Weinberger. Our method is still quite computer-intensive, but we are able to keep the time needed for the computation down to about seven months. In all cases, we find that there is no abnormally large ``exceptional modulus' of small class number, which agrees with the prediction of the Generalised Riemann Hypothesis.

     

Computation of class numbers of quadratic number fields

We explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields. We therefore end up with a simpler and faster method for computing class numbers of quadratic number fields. We also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields.

     

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.     

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.     

Explicit construction of a class of infinitely many imaginary quadratic fields whose class number is divisible by 3

The results of the article are summarized by the title.     

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 .     

Class numbers of imaginary abelian number fields

Let be an imaginary abelian number field. We know that , the relative class number of , goes to infinity as , the conductor of , approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CM-fields with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of . Second, we have proved in this paper that there are exactly 48 such fields.

     

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...     

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 .

     

Theta series of imaginary quadratic function fields

LetL be an imaginary quadratic extension of the rational function field . We prove transformation rules for the theta series corresponding to partial zeta functions of the extension .