The 4-class ranks of quadratic fields related to the Weyl group algebra.
The class number one problem for some non-abelian normal CM-fields of degree
We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to , the alternating group of degree and order . There are two such fields with Galois group (see Theorem 14) and at most one with Galois group SL (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number .
The class number one problem for the dihedral and dicyclic CM-fields
We recall the determination of all the dihedral CM-fields with relative class number one, and prove that dicyclic CM-fields have relative class numbers greater than one.
The class number one problem for the non-abelian normal CM-fields of degree 24 and 40
The class number one problem for the non-normal sextic CM-fields. Part 2
The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields
1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension of k for p (cf. [10]). Then Greenberg’s conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of . We know by the Ferrero-Washington...
The minimal resultant locus
Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map factors through a function on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in , or on a segment, and the minimal resultant locus is contained in the tree in spanned by the fixed points and poles...
The narrow class groups of some ℤₚ-extensions over the rationals
The non-abelian normal CM-fields of degree 36 with class number one
The non-normal quartic CM-fields and the dihedral octic CM-fields with ideal class groups of exponent
The R₂ measure for totally positive algebraic integers
Let α be a totally positive algebraic integer of degree d, i.e., all of its conjugates are positive real numbers. We study the set ₂ of the quantities . We first show that √2 is the smallest point of ₂. Then, we prove that there exists a number l such that ₂ is dense in (l,∞). Finally, using the method of auxiliary functions, we find the six smallest points of ₂ in (√2,l). The polynomials involved in the auxiliary function are found by a recursive algorithm.
The extensions of degree 6 with minimum discriminant.
The size of the fundamental solutions of consecutive Pell equations.
The splitting of primes in division fields of elliptic curves.
The totally real extension of degree 6 with minimum discriminant.
The totally real extension of degree 6 with minimum discriminant.
Topics in computational algebraic number theory
We describe practical algorithms from computational algebraic number theory, with applications to class field theory. These include basic arithmetic, approximation and uniformizers, discrete logarithms and computation of class fields. All algorithms have been implemented in the Pari/Gp system.
Totally indefinite Euclidean quaternion fields
We study the Euclidean property for totally indefinite quaternion fields. In particular, we establish a complete list of norm-Euclidean such fields over imaginary quadratic number fields. This enables us to exhibit an example which gives a negative answer to a question asked by Eichler. The proofs are both theoretical and algorithmic.