Class numbers of imaginary quadratic fields

Starting from the analytic class number formula involving its L-function, we first give an expression for the class number of an imaginary quadratic field which, in the case of large discriminants, provides us with a much more powerful numerical technique than that of counting the number of reduced definite positive binary quadratic forms, as has been used by Buell in order to compute his class number tables. Then, using class field theory, we will construct a periodic character χ \chi , defined on the ring of integers of a field K that is a quadratic extension of a principal imaginary quadratic field k, such that the zeta function of K is the product of the zeta function of k and of the L-function L ( s , χ ) L(s,\chi ) . We will then determine an integral representation of this L-function that enables us to calculate the class number of K numerically, as soon as its regulator is known. It will also provide us with an upper bound for these class numbers, showing that Hua’s bound for the class numbers of imaginary and real quadratic fields is not the best that one could expect. We give statistical results concerning the class numbers of the first 50000 quadratic extensions of Q ( i ) {\mathbf {Q}}(i) with prime relative discriminant (and with K/Q a non-Galois quartic extension). Our analytic calculation improves the algebraic calculation used by Lakein in the same way as the analytic calculation of the class numbers of real quadratic fields made by Williams and Broere improved the algebraic calculation consisting in counting the number of cycles of reduced ideals. Finally, we give upper bounds for class numbers of K that is a quadratic extension of an imaginary quadratic field k which is no longer assumed to be of class number one.

1. Introduction. It is known (cf. [1], [3], [4], [5], [7]) that there exist infinitely many imaginary quadratic fields each with class number divisible by a given integer. In this paper we show another simple proof of the above theorem. By a similar method we also construct infinitely many imaginary cyclic quartic fields whose relative class numbers are divisible by l, where l is a given prime congruent to 1 modulo 4. Further we characterize the imaginary quadratic fields whose class numbers are multiples of a given prime, by describing them explicitly. In the following, for an integer :r in an algebraic number field K we denote by (~) the principal ideal in K generated by ~. We write p ]] n for a prime p and an

We prove that the j -invariant of an elliptic curve defined over an imaginary quadratic number field having good reduction everywhere satisfies certain Diophantine equations under some hypothesis on the arithmetic of the quadratic field. By solving the Diophantine equations explicitly in the rings of quadratic integers, we show the non-existence of such elliptic curve for certain imaginary quadratic fields. This extends the results due to Setzer and Stroeker.

Class numbers of quadratic fields have been the object of attention for many years, and there exist a large number of interesting results. This is a survey aimed at reviewing results concerning the divisibility of class numbers of both real and imaginary quadratic fields. More precisely, to review the question ‘do there exist infinitely many real (respectively imaginary) quadratic fields whose class numbers are divisible by a given integer?’ This survey also contains the current status of a quantitative version of this question.

Let $n$ be an integer. Then, it is well known that there are infinitely many\nimaginary quadratic fields with an ideal class group having a subgroup isomorphic to\n$\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$. Less is known for real\nquadratic fields, other than the cases that $n=3,5,$ or $7$, due to Craig [3] and\nMestre [4, 5]. In this article, we will prove that there exist infinitely many real\nquadratic number fields with the ideal class group having a subgroup isomorphic to\n$\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$ In addition, we will prove\nthat there exist infinitely many imaginary quadratic number fields with the ideal\nclass group having a subgroup isomorphic to $\\mathbb{Z}/n\\mathbb{Z} \\times\n\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$.

Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one. The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, ℒ -function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number h ( d ) formula in real quadratic fields claims that we have h ( d ) . log ε d = Δ . ℒ ( 1 , χ d ) h\left(d \right).log {\varepsilon _d} = \sqrt {\Delta} {\scr L} \left({1,\;{\chi _d}}\right) for positive d > 0 and the fundamental unit ɛ d of ℚ ( d ) {\rm{\mathbb Q}}\left({\sqrt d} \right) . It is seen that discriminant, ℒ -function and fundamental unit ɛ d are significant and necessary tools for determining the structure of real quadratic fields. The focus of this paper is to determine structure of some special real quadratic fields for d > 0 and d ≡ 2,3 ( mod 4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element w d , fundamental unit ɛ d , and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.

We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of Goldston–Graham–Pintz–Yildirim (Proc Lond Math Soc 98:741–774, 2009), and Maynard (Ann Math 181:383–413, 2015). An important consequence of our main theorem is existence of infinitely many pairs alpha _1, alpha _2 which are product of two primes in the imaginary quadratic field K such that |sigma (alpha _1-alpha _2)|le 2 for all embeddings sigma of K if the class number of K is one and |sigma (alpha _1-alpha _2)|le 8 for all embeddings sigma of K if the class number of K is two.

Let k be an imaginary quadratic number field (with class number 1). We describe a new, essentially linear-time algorithm, to list all isomorphism classes of cubic extensions L/k up to a bound X on the norm of the relative discriminant ideal. The main tools are Taniguchi's generalization of Davenport-Heilbronn parametrisation of cubic extensions, and reduction theory for binary cubic forms over imaginary quadratic fields. Finally, we give numerical data for k=Q(i), and we compare our results with ray class field algorithm ones, and with asymptotic heuristics, based on a generalization of Roberts' conjecture.

In this chapter, we consider the simplest of all number fields that are different from ℚ, i.e. quadratic fields. Since n = 2 = r 1 + 2r 2, the signature (r 1, r 2) of a quadratic field K is either (2, 0), in which case we will speak of real quadratic fields, or (0, 1), in which case we will speak of imaginary (or complex) quadratic fields. By Proposition 4.8.11 we know that imaginary quadratic fields are those of negative discriminant, and that real quadratic fields are those with positive discriminant.

We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary quadratic number fields with class number one, J. Théor. Nombres Bordx. 32 (2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for ℚ (i) by extending his method which gave this exponent for ℚ [On the distribution of αp modulo one. II, Proc. London Math. Soc. 72, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Größencharacters.

Hodge structures of type (n, 0, . . . , 0, n) Burt Totaro Completing earlier work by Albert, Shimura found all the possible endomor- phism algebras (tensored with the rationals) for complex abelian varieties of a given dimension [12, Theorem 5]. In five exceptional cases, every abelian variety on which a certain algebra acts has “extra endomorphisms”, so that the full endomorphism algebra is bigger than expected. Complex abelian varieties X up to isogeny are equivalent to polarizable Q-Hodge structures of weight 1, with Hodge numbers (n, n) (where n is the dimension of X). In this paper, we generalize Shimura’s classification to determine all the possible endomorphism algebras for polarizable Q-Hodge structures with Hodge numbers (n, 0, . . . , 0, n). For Hodge structures of odd weight, the answer is the same as for abelian varieties. For Hodge structures of even weight, the answer is similar but not identical. The proof combines ideas from Shimura with Green-Griffiths-Kerr’s approach to computing Mumford-Tate groups [4, Proposition VI.A.5]. As with abelian varieties, the most interesting feature of the classification is that in certain cases, every Hodge structure on which a given algebra acts must have extra endomorphisms. (Throughout this discussion, we only consider polarizable Hodge structures.) One known case (pointed out to me by Beauville) is that every Q- Hodge structure with Hodge numbers (1, 0, 1) has endomorphisms by an imaginary quadratic field and hence is of complex multiplication (CM) type, meaning that its Mumford-Tate group is commutative. More generally, every Q-Hodge structure with Hodge numbers (n, 0, n) that has endomorphisms by a totally real field F of degree n has endomorphisms by a totally imaginary quadratic extension field of F , and hence is of CM type. Another case, which seems to be new, is that a Q-Hodge structure V with Hodge numbers (2, 0, 2) that has endomorphisms by an imaginary quadratic field F 0 must have endomorphisms by a quaternion algebra over Q. In this case, V need not be of CM type; there is a period space isomorphic to CP 1 of Hodge structures of this type, whereas there are only countably many Hodge structures of CM type. To motivate the results of this paper on endomorphism algebras, consider the geometric origin of Hodge structures. A Hodge structure comes from geometry if it is a summand of the cohomology of a smooth complex projective variety defined by an algebraic correspondence. Griffiths found (“Griffiths transversality”) that a family of Hodge structures coming from geometry can vary only in certain directions, expressed by the notion of a variation of Hodge structures [15, Theorem 10.2]. In particular, any variation of Hodge structures of weight at least 2 with Hodge numbers (n, 0, . . . , 0, n) (so there is at least one 0) is locally constant; more generally, this holds whenever there are no two adjacent nonzero Hodge numbers. This has the remarkable consequence that only countably many Hodge structures of weight at least 2 with Hodge numbers (n, 0, . . . , 0, n) come from geometry. Very little is

On the Ono invariants of imaginary quadratic number fields

Let Q ( − d ) \mathbf {Q}(\sqrt {-d}) and Q ( 3 d ) \mathbf {Q}(\sqrt {3d}) be quadratic fields with d ≡ d \equiv 2 (mod 3) a positive integer. Let λ − , λ + \lambda ^-, \lambda ^+ be the respective Iwasawa λ \lambda -invariants of the cyclotomic Z 3 \mathbf {Z}_3 -extension of these fields. We show that if λ − = 1 \lambda ^- =1 , then 3 does not divide the class number of Q ( 3 d ) \mathbf {Q}(\sqrt {3d}) and λ + = 0 \lambda ^+ = 0 .

For imaginary quadratic number fields F = <TEX>$\mathbb{Q}(\sqrt{{\varepsilon}p_1{\ldots}p_{t-1}})$</TEX>, where <TEX>${\varepsilon}{\in}$</TEX>{-1,-2} and distinct primes <TEX>$p_i{\equiv}1$</TEX> mod 4, we give condition of 8-ranks of class groups C(F) of F equal to 1 or 2 provided that 4-ranks of C(F) are at most equal to 2. Especially for F = <TEX>$\mathbb{Q}(\sqrt{{\varepsilon}p_1p_2)$</TEX>, we compute densities of 8-ranks of C(F) equal to 1 or 2 in all such imaginary quadratic fields F. The results are stated in terms of congruence relation of <TEX>$p_i$</TEX> modulo <TEX>$2^n$</TEX>, the quartic residue symbol <TEX>$(\frac{p_1}{p_2})4$</TEX> and binary quadratic forms such as <TEX>$p_2^{h+(2_{p_1})/4}=x^2-2p_1y^2$</TEX>, where <TEX>$h+(2p_1)$</TEX> is the narrow class number of <TEX>$\mathbb{Q}(\sqrt{2p_1})$</TEX>. The results are also very useful for numerical computations.

The distribution of $$\alpha p$$ modulo one, where p runs over the rational primes and $$\alpha $$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $$\nu >0$$ one can establish the infinitude of primes p satisfying $$||\alpha p||\le p^{-\nu }$$ . The latest record in this regard is Kaisa Matomäki’s landmark result $$\nu =1/3-\varepsilon $$ which presents the limit of currently known technology. Recently, Glyn Harman, and, jointly, Marc Technau and the first-named author, investigated the same problem in the context of imaginary quadratic fields. Glyn Harman obtained an analog for $$\mathbb {Q}(i)$$ of his result in the context of $$\mathbb {Q}$$ , which yields an exponent of $$\nu =7/22$$ . Marc Technau and the first-named author produced an analogue of Bob Vaughan’s result $$\nu =1/4-\varepsilon $$ for all imaginary quadratic number fields of class number 1. In the present article, we establish an analog of the last-mentioned result for real quadratic fields of class number 1 under a certain Diophantine restriction. This setting involves the additional complication of an infinite group of units in the ring of integers. Moreover, although the basic sieve approach remains the same (we use an ideal version of Harman’s sieve), the problem takes a different flavor since it becomes truly 2-dimensional. We reduce it eventually to a counting problem which is, interestingly, related to roots of quadratic congruences. To approximate them, we use an approach by Christopher Hooley based on the theory of binary quadratic forms.