Imaginary Number Field Research Articles

Class numbers, Ono invariants and some interesting primes

Our aim is to find all the prime numbers p such that p+x2 has at most two different prime factors, for all the odd integers x such that x2≤p. We solve entirely the cases p≡1,3,5(mod8), using the knowledge of the quadratic imaginary number fields with class numbers 4, 1 and 2 respectively. The case p≡7(mod8) is not completely solved. Taking into account a result of Stéphane Louboutin, we prove that there is at most one value p≡7(mod8) besides our list. Assuming a Restricted Riemann Hypothesis, the list is complete. In the last section of the paper we give a short sketch for the general problem: find all odd integers n>1 such that n+x2 has at most two different prime factors, for all the odd integers x such that x2≤n.

Composition of Bhargava’s cubes over number fields

In this paper, the composition of Bhargava’s cubes is generalized to the ring of integers of a number field of narrow class number one, excluding the case of totally imaginary number fields. The exclusion of the latter case arises from the nonexistence of a bijection between (classes of) binary quadratic forms and an ideal class group. This problem, together with a related mistake in another paper of the author, is addressed in the appendix.

Weighted badly approximable complex vectors and bounded orbits of certain diagonalizable flows

We show an analog of a theorem of [J. An, A. Ghosh, L. Guan and T. Ly, Bounded orbits of diagonalizable flows on finite volume quotients of products of [Formula: see text], Adv. Math. 354 (2019) 106743] on weighted badly approximable vectors for totally imaginary number fields. We show that for [Formula: see text] and [Formula: see text] a lattice subgroup, the points of [Formula: see text] with bounded orbits under a one-parameter Ad-semisimple subgroup of [Formula: see text] form a hyperplane-absolute-winning set. As an application, we also provide a generalization of a result of [R. Esdahl-Schou and S. Kristensen, On badly approximable complex numbers, Glasgow Math. J. 52 (2010) 349–355] about the set of badly approximable complex numbers.

Arithmetic Chern–Simons theory with real places

The goal of this paper is two-fold: we generalize the arithmetic Chern–Simons theory over totally imaginary number fields studied in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3; M. Kim, Arithmetic Chern–Simons theory I, preprint (2015) arXiv:1510.05818v4] to arbitrary number fields (with real places) and provide new examples of non-trivial arithmetic Chern–Simons invariant with coefficient [Formula: see text] [Formula: see text] associated to a non-abelian gauge group. The main idea for the generalization is to use cohomology with compact support (see [17]) to deal with real places. Before the results of this paper, non-trivial examples were limited to some non-abelian gauge group with coefficient [Formula: see text] in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3] and the abelian cyclic gauge group with coefficient [Formula: see text] in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3]. Our non-trivial examples (with non-abelian gauge group and general coefficient [Formula: see text]) will be given by a simple twisting argument based on examples of [F. Bleher, T. Chinburg, R. Greenberg, M. Kakde, G. Pappas and M. Taylor, Cup products in the étale cohomology of number fields, New York J. Math. 24 (2018) 514–542].

On the rank of the 2-class group of some imaginary biquadratic number fields

On the rank of the 2-class group of some imaginary biquadratic number fields

Chromatic Selmer groups and arithmetic invariants of elliptic curves

Chromatic Selmer groups are modified Selmer groups with local information for supersingular primes p. We sketch their role in establishing the p-primary part of the Birch–Swinnerton-Dyer formula in Sections 2–5, and then study the growth of the Mordell–Weil rank along the ℤ p 2 -extension of a quadratic imaginary number field in which p splits in Section 6.

On the $2$-class group of some number fields of $2$-power degree

Let $K$ be an imaginary cyclic quartic number field whose $2$-class group is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$, and let $K^*$ denote the genus field of $K$. In this paper, we compute the rank of the $2$-class group of $K^*_n$ the $n$-th layer of the cyclotomic $Z_2$-extension of $K^*$.

2-Class groups of cyclotomic towers of imaginary biquadratic fields and applications

Let [Formula: see text] be an odd positive square-free integer. In this paper, we shall investigate the structure of the [Formula: see text]-class group of the cyclotomic [Formula: see text]-extension of the imaginary biquadratic number field [Formula: see text] if [Formula: see text] is of specifiic form. Furthermore, we deduce the structure of the [Formula: see text]-class group of the cyclotomic [Formula: see text]-extension of [Formula: see text].

Imaginary multiquadratic number fields with class group of exponent $3$ and $5$

Imaginary multiquadratic number fields with class group of exponent $3$ and $5$

On the rank of the 2-class group of some imaginary triquadratic number fields

Let d be an odd square-free integer and \(\zeta _8\) a primitive 8-th root of unity. The purpose of this paper is to investigate the rank of the 2-class group of the fields \(L_d={\mathbb {Q}}(\zeta _8,\sqrt{d})\).

English

Let $$\mathbb{k} = \mathbb{Q} \left( {\sqrt 2 ,\; \sqrt d } \right)$$ be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and $$\mathbb{k}_2^{\left( 2 \right)}$$ its second Hilbert 2-class field. Denote by $$G = {\rm{Gal}}\left( {\mathbb{k}_2^{\left( 2 \right)}/ \mathbb{k}} \right)$$ the Galois group of $${\mathbb{k}_2^{\left( 2 \right)}/ \mathbb{k}}$$ . The purpose of this note is to investigate the Hilbert 2-class field tower of $$\mathbb{k}$$ and then deduce the structure of G.

Distribution of Modular Symbols in ℍ3

AbstractWe introduce a new technique for the study of the distribution of modular symbols, which we apply to the congruence subgroups of Bianchi groups. We prove that if $K$ is a quadratic imaginary number field of class number one and $\mathcal{O}_K$ is its ring of integers, then for certain congruence subgroups of $\textrm{PSL}_2(\mathcal{O}_K)$, the periods of a cusp form of weight two obey asymptotically a normal distribution. These results are specialisations from the more general setting of quotient surfaces of cofinite Kleinian groups where our methods apply. We avoid the method of moments. Our new insight is to use the behaviour of the smallest eigenvalue of the Laplacian for spaces twisted by modular symbols. Our approach also recovers the first and second moments of the distribution.

Computations in the Pre-Bloch group

For compute the five term relations in the pre-Bloch group for specify an infinite-order- element in , m ∈ ℕ square- free. For the quadratic imaginary number fields \U0001d53d of discriminant (−1,−2,−3,−7,−17,−19). We use the GAP Programming software to implement our method.

Third Galois cohomology group of function fields of curves over number fields

Let F be the function field of a curve over totally imaginary number field. Let p be a prime. If F contains a primitive p th root of unity, then every element in the third Galois cohomology group of F with values in the group of p th roots of unity, is a symbol.

Third Galois cohomology group of function fields of curves over number fields

Let F be the function field of a curve over totally imaginary number field. Let p be a prime. If F contains a primitive p th root of unity, then every element in the third Galois cohomology group of F with values in the group of p th roots of unity, is a symbol.

On the cyclicity of the 2-class group of the fields $\mathbb{Q}(i,\sqrt{p_1},\dots,\sqrt{p_n})$

Let $n\geq 2$ be an integer and $p_1,\ldots ,p_n$ be distinct odd prime integers. We give the list of all imaginary multiquadratic number fields $ K_n=\mathbb {Q}(i,\sqrt {p_1},\ldots ,\sqrt {p_n})$ that have a cyclic 2-class group.

Two remarks on sums of squares with rational coefficients

There exist homogeneous polynomials $f$ with $\mathbb Q$-coefficients that are sums of squares over $\mathbb R$ but not over $\mathbb Q$. The only systematic construction of such polynomials that is known so far uses as its key ingredient totally imaginary number fields $K/\mathbb Q$ with specific Galois-theoretic properties. We first show that one may relax these properties considerably without losing the conclusion, and that this relaxation is sharp at least in a weak sense. In the second part we discuss the open question whether any $f$ as above necessarily has a (non-trivial) real zero. In the minimal open cases $(3,6)$ and $(4,4)$, we prove that all examples without a real zero are contained in a thin subset of the boundary of the sum of squares cone.

On the 2-class field towers of some imaginary quartic cyclic number fields

On the 2-class field towers of some imaginary quartic cyclic number fields

Waring’s problem for unipotent algebraic groups

In this paper, we formulate an analogue of Waring's problem for an algebraic group $G$. At the field level we consider a morphism of varieties $f\colon \mathbb{A}^1\to G$ and ask whether every element of $G(K)$ is the product of a bounded number of elements $f(\mathbb{A}^1(K)) = f(K)$. We give an affirmative answer when $G$ is unipotent and $K$ is a characteristic zero field which is not formally real. The idea is the same at the integral level, except one must work with schemes, and the question is whether every element in a finite index subgroup of $G(\mathcal{O})$ can be written as a product of a bounded number of elements of $f(\mathcal{O})$. We prove this is the case when $G$ is unipotent and $\mathcal{O}$ is the ring of integers of a totally imaginary number field.

On the rank of the 2-class group of an extension of degree 8 over $$\mathbb {Q}$$ Q

Let K be an imaginary cyclic quartic number field whose 2-class group is nontrivial, it is known that there exists at least one unramified quadratic extension F of K. In this paper, we compute the rank of the 2-class group of the field F.