n-Torsion of Brauer groups as relative Brauer groups of abelian extensions

It is now known [H. Kisilevsky, J. Sonn, Abelian extensions of global fields with constant local degrees, Math. Res. Lett. 13 (4) (2006) 599-607; C.D. Popescu, Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields, J. Number Theory 115 (2005) 27-44] that if F is a global field, then the n-torsion subgroup of its Brauer group Br(F) equals the relative Brauer group Br(Ln/F) of an abelian extension Ln/F, for all n∈Z?1. We conjecture that this property characterizes the global fields within the class of infinite fields which are finitely generated over their prime fields. In the first part of this paper, we make a first step towards proving this conjecture. Namely, we show that if F is a non-global infinite field, which is finitely generated over its prime field and ?≠char(F) is a prime number such that μ?²⊆F^×, then there does not exist an abelian extension L/F such that . The second and third parts of this paper are concerned with a close analysis of the link between the hypothesis μ?²⊆F^× and the existence of an abelian extension L/F such that , in the case where F is a Henselian valued field.     

Higher rank subgroups in the class groups of imaginary function fields

Let F be a finite field and T a transcendental element over F. In this paper, we construct, for integers m and n relatively prime to the characteristic of F(T), infinitely many imaginary function fields K of degree m over F(T) whose class groups contain subgroups isomorphic to (Z/nZ)^m. This increases the previous rank of m−1 found by the authors in [Y. Lee, A. Pacelli, Class groups of imaginary function fields: The inert case, Proc. Amer. Math. Soc. 133 (2005) 2883-2889].     

Wild ramification in number field extensions of prime degree

We show that if L/ K is a degree p extension of number fields which is wildly ramified at a prime ${\frak p}$ of K of residue characteristic p, then the ramification groups of ${\frak p}$ (in the splitting field of L over K) are uniquely determined by the ${\frak p}$-adic valuation of the discriminant of L /K.Received: 3 July 2002     

The distribution of class groups of function fields

For any sufficiently general family of curves over a finite field F_q and any elementary abelian ?-group H with ? relatively prime to q, we give an explicit formula for the proportion of curves C for which Jac(C)[?](F_q)≅H. In doing so, we prove a conjecture of Friedman and Washington.     

Reflection theorem for divisor class groups of relative quadratic function fields

We present the reflection theorem for divisor class groups of relative quadratic function fields. Let K be a global function field with constant field Fq. Let L₁ be a quadratic geometric extension of K and let L₂ be its twist by the quadratic constant field extension of K. We show that for every odd integer m that divides q+1 the divisor class groups of L₁ and L₂ have the same m-rank.     

Calculating formula and divisibility for relative class numbers of abelian function fields

In this paper abelian function fields are restricted to the subfields of cyclotomic function fields. For any abelian function field K/k with conductor an irreducible polynomial over a finite field of odd characteristic, we give a calculating formula of the relative divisor class number of K. And using the given calculating formula we obtain a criterion for checking whether or not the relative divisor class number is divisible by the characteristic of k.     

Analogies between group actions on 3-manifolds and number fields

Let a cyclic group $G$ act either on a number field $\mathbb L$ or on a $3$-manifold $M$. Let $s_{\mathbb L}$ be the number of ramified primes in the extension $\mathbb L^G\subset \mathbb L$ and $s_M$ be the number of components of the branching set of the branched covering $M\to M/G$. In this paper, we prove several formulas relating $s_{\mathbb L}$ and $s_M$ to the induced $G$-action on $Cl(\mathbb L)$ and $H_1(M),$ respectively. We observe that the formulas for $3$-manifolds and number fields are almost identical, and therefore, they provide new evidence for the correspondence between $3$-manifolds and number fields postulated in arithmetic topology.     

Class number indivisibility for quadratic function fields

Let M?5. For any odd prime power q and any prime ??q, we show that there are at least pairwise coprime D∈Fq[T] which are square-free and of odd degree ?M, such that ? does not divide the class number of the complex quadratic functions fields .     

On some questions related to the Gauss conjecture for function fields

We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.     

On class numbers of quadratic extensions¶over function fields

We consider class numbers of quadratic extensions over a fixed function field. We will show that there exist infinitely many quadratic extensions which have class numbers not being divisible by 3 and satisfy prescribed ramification conditions. Received: 24 October 1997 / Revised version: 26 February 1998     

Arithmetic actions on cyclotomic function fields

We derive the group structure for cyclotomic function fields obtained by applying the Carlitz action for extensions of an initial constant field. The tame and wild structures are isolated to describe the Galois action on differentials. We show that the associated invariant rings are not polynomial.     

The ideal class groups of dihedral extensions over imaginary quadratic fields and the special values of the Artin L-function

We study the relation between the minus part of the p-class subgroup of a dihedral extension over an imaginary quadratic field and the special value of the Artin L-function at 0.     

Tower of the maximal abelian extensions of local fields and its application

In this paper, we study the algebraic structure of principal units in the tower of the maximal abelian extensions of local fields of characteristic zero and the corresponding Galois groups at each level. As an application, we show the finiteness result for the number of coverings with a given degree of the maximal abelian extension of a local field in characteristic zero. The number of p-coverings for is computed explicitly.     

The 3-Sylow subgroup of the tame kernel of real number fields

Let F be a cubic cyclic field with exactly one ramified prime p,p>7, or , a real quadratic field with . In this paper, we study the 3-primary part of K₂O_F. If 3 does not divide the class number of F, we get some results about the 9-rank of K₂O_F. In particular, in the case of a cubic cyclic field F with only one ramified prime p>7, we prove that four conclusions concerning the 3-primary part of K₂O_F, obtained by J. Browkin by numerical computations for primes p, 7≤p≤5000, are true in general.     

The Rubin-Stark conjecture for a special class of function field extensions

We prove a strong form of the Brumer-Stark Conjecture and, as a consequence, a strong form of Rubin's integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum k_p∞?k_p·k_∞ of the maximal pro-p abelian extension k_p/k and the maximal constant field extension k_∞/k of k, which happens to sit inside the maximal abelian extension k^ab of k with a quasi-finite index. This way, we extend the results obtained by the present author in (Comp. Math. 116 (1999) 321-367).     

Sharp conditions for global stability of Lotka-Volterra systems with distributed delays

We give a criterion for the global attractivity of a positive equilibrium of n-dimensional non-autonomous Lotka-Volterra systems with distributed delays. For a class of autonomous Lotka-Volterra systems, we show that such a criterion is sharp, in the sense that it provides necessary and sufficient conditions for the global asymptotic stability independently of the choice of the delay functions. The global attractivity of positive equilibria is established by imposing a diagonal dominance of the instantaneous negative feedback terms, and relies on auxiliary results showing the boundedness of all positive solutions. The paper improves and generalizes known results in the literature, namely by considering systems with distributed delays rather than discrete delays.     

Contributions to coincidence degree theory of asymptotically homogeneous operators

In this paper we give contributions to the coincidence degree theory of asymptotically homogeneous operators. Applications are given to the periodic problem for second-order functional differential equations.     

Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks

This paper addresses the local and global stability of n-dimensional Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks. Necessary and sufficient conditions for local stability independent of the choice of the delay functions are given, by imposing a weak nondelayed diagonal dominance which cancels the delayed competition effect. The global asymptotic stability of positive equilibria is established under conditions slightly stronger than the ones required for the linear stability. For the case of monotone interactions, however, sharper conditions are presented. This paper generalizes known results for discrete delays to systems with distributed delays. Several applications illustrate the results.     

Rational places in extensions and sequences of function fields of Kummer type

In this paper we prove results on the number of rational places in extensions of Kummer type over finite fields and give sufficient conditions for non-trivial lower bounds on the number of rational places at each step of sequences of function fields over a finite field, that we call (a, b)-sequences. In the case of a prime field, we apply these results to the study of rational places in certain sequences of function fields of Kummer type.