Narrow Results

Classification of binary indefinite primitive Hermitian forms modulo the action of the extended Bianchi group (or Hilbert modular group) B_d is given. When the discriminant of the quadratic field (and d) is negative, the results obtained can be applied to classify the maximal non-elementary Fuchsian subgroups of B_d, and to find the Hermitian points in the Markov spectrum of B_d. If ν is a Hermitian point in the spectrum, then there is a set of extremal geodesics in H³, the upper half-space model of the three-dimensional hyperbolic space, with diameter 1/ν, which depends on one continuous parameter.

This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur as such Galois groups.

Let be a real quadratic number field with a prime P lying above p in ℚ. If p splits in K/ℚ and p does not divide the big class number of K, then any pro-p extension of K ramified only at P is finite cyclic. If p is inert in K/ℚ, then there exist infinite extensions of K ramified only at P. Furthermore, for big enough integer k, the ray class field (mod P^k+1) is obtained from the ray class field (mod P^k) by adjoining ζ_p^k+1.

In the case of a regular cyclotomic number field K = ℚ(ζ_p), the explicit structure of ray class groups (mod P^k) is given for any positive integer k, where P is the unique prime in K above p.

We consider a variation of the unit sum number problem for quadratic fields and prove various results.

For a prime p ≥ 2 and a number field K with p-class group of type (p, p) it is shown that the class, coclass, and further invariants of the metabelian Galois group of the second Hilbert p-class field of K are determined by the p-class numbers of the unramified cyclic extensions N_i|K, 1 ≤ i ≤ p + 1, of relative degree p. In the case of a quadratic field and an odd prime p ≥ 3, the invariants of G are derived from the p-class numbers of the non-Galois subfields L_i|ℚ of absolute degree p of the dihedral fields N_i. As an application, the structure of the automorphism group of the second Hilbert 3-class field is analyzed for all quadratic fields K with discriminant -10⁶ < D < 10⁷ and 3-class group of type (3, 3) by computing their principalization types. The distribution of these metabelian 3-groups G on the coclass graphs , 1 ≤ r ≤ 6, in the sense of Eick and Leedham-Green is investigated.

It is well known that for any prime p ≡ 3 (mod 4), the class numbers of the quadratic fields and and h(-p), respectively, are odd. It is natural to ask whether there is a formula for h(p)/h(-p) modulo powers of 2. We show the formula h(p) ≡ h(-p)m(p) (mod 16), where m(p) is an integer defined using the "negative" continued fraction expansion of . Our result solves a conjecture of Richard Guy.

Let $p>3$ be a prime. Gauss first introduced the polynomial $S_p(x)={\prod }_c(x-{\zeta }_p^c),$ where $0<c<p$ and $c$ varies over all quadratic residues modulo p and ${\zeta }_p=e^{2\pi i/p}$. Later Dirichlet investigated this polynomial and used this to solve problems involving the Pell equations. Recently, Sun studied some trigonometric identities involving this polynomial. In this paper, we generalize their results. As applications of our result, we extend Chowla’s result on the congruence concerning the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and give an equivalent form of the extended Ankeny et al. conjecture.

We construct an infinite family of imaginary bicyclic biquadratic number fields 𝕜 with the 2-ranks of their 2-class groups are ≥ 3, whose strongly ambiguous classes of 𝕜/ℚ(i) capitulate in the absolute genus field 𝕜^(*), which is strictly included in the relative genus field (𝕜/ℚ(i))* and we study the capitulation of the 2-ideal classes of 𝕜 in its quadratic extensions included in 𝕜^(*).

Let $k$ be a number field and $k_2^{(1)}$ (respectively, $k_2^{(2)}$) its first (respectively, second) Hilbert $2$-class field. Let $G=\mathrm{\text{Gal}}(k_2^{(2)}/k)$ be the Galois group of $k_2^{(2)}/k$. The purpose of this note is to determine the structure of $G$ for some special Dirichlet fields $k=\mathbb{Q}(\sqrt{d},\sqrt{-1})$.