-адические ряды Гекке мнимых квадратичных полей
Parameterized families of quadratic number fields with 3-rank at least 2
Parametric form of an eight class field
Partitions and indefinite quadratic forms.
Periodic expansions and units in quadratic and cubic number fields.
Points of Order p on Elliptic Curves Over Q(...p).
Points on Shimura curves rational over number fields.
Polynomials of Pellian Type and Continued Fractions
We investigate infinite families of integral quadratic polynomials {fk (X)} k∈N and show that, for a fixed k ∈ N and arbitrary X ∈ N, the period length of the simple continued fraction expansion of √fk (X) is constant. Furthermore, we show that the period lengths of √fk (X) go to infinity with k. For each member of the families involved, we show how to determine, in an easy fashion, the fundamental unit of the underlying quadratic field. We also demonstrate how the simple continued fraction ex-...
Prime divisors of Lucas sequences
Prime factors of values of polynomials
We prove that for every quadratic binomial f(x) = rx² + s ∈ ℤ[x] there are pairs ⟨a,b⟩ ∈ ℕ² such that a ≠ b, f(a) and f(b) have the same prime factors and min{a,b} is arbitrarily large. We prove the same result for every monic quadratic trinomial over ℤ.
Prime valued polynomials and class numbers of quadratic fields.
Principalization algorithm via class group structure
For an algebraic number field with -class group of type , the structure of the -class groups of the four unramified cyclic cubic extension fields , , of is calculated with the aid of presentations for the metabelian Galois group of the second Hilbert -class field of . In the case of a quadratic base field it is shown that the structure of the -class groups of the four -fields frequently determines the type of principalization of the -class group of in . This provides...
Prolongement analytique et valeurs aux entiers négatifs de certaines sacute;ries arithmétiques relatives à des formes quadratiques.
Propagation de la 2-birationalité
Let L/K be a 2-birational CM-extension of a totally real 2-rational number field. We characterize in terms of tame ramification totally real 2-extensions K’/K such that the compositum L’=LK’ is still 2-birational. In case the 2-extension K’/K is linearly disjoint from the cyclotomic ℤ₂-extension , we prove that K’/K is at most quadratic. Furthermore, we construct infinite towers of such 2-extensions.
Pure quartic number fields whose class numbers are even.
p-адическая дзета-функция мнимого квадратичного поля и регулятор Леопольдта