Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields

We show that, for all characteristic p global fields k and natural numbers n coprime to the order of the non-p-part of the Picard group Pic⁰(k) of k, there exists an abelian extension L/k whose local degree at every prime of k is equal to n. This answers in the affirmative in this context a question recently posed by Kisilevsky and Sonn. As a consequence, we show that, for all n and k as above, the n-torsion subgroup Br_n(k) of the Brauer group Br(k) of k is algebraic, answering a question of Aldjaeff and Sonn in this context.     

Computation of the Iwasawa invariants of certain real abelian fields

Let p be a prime number and k a finite extension of . It is conjectured that the Iwasawa invariants λ_p(k) and μ_p(k) vanish for all p and totally real number fields k. Some methods to verify the conjecture for each real abelian field k are known, in which cyclotomic units and a set of auxiliary prime numbers are used. We give an effective method, based on the previous one, to compute the exact value of the other Iwasawa invariant ν_p(k) by using Gauss sums and another set of auxiliary prime numbers. As numerical examples, we compute the Iwasawa invariants associated to in the range 1<f<200 and 5?p<10000.     

The hyperring of adèle classes

Text

We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK=AK/K^× of a global field K. After promoting F₁ to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G⊂R^× is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G∪{0} is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension HK of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field K of positive characteristic, the groupoid of the prime elements of the hyperring HK is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field K.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3LSKD_PfJyc.     

On the independence of Heegner points in the function field case

Let ∞ be a fixed place of a global function field k. Let E be an elliptic curve defined over k which has split multiplicative reduction at ∞ and fix a modular parametrization ΦE:X₀(N)→E. Let be Heegner points associated to the rings of integers of distinct quadratic “imaginary” fields K₁,…,Kr over (k,∞). We prove that if the “prime-to-2p” part of the ideal class numbers of ring of integers of K₁,…,Kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in . Moreover, when k is rational, we show that there are infinitely many imaginary quadratic fields for which the prime-to-2p part of the class numbers are larger than C.     

On the torsion of group extensions and group actions

In this note, we study the torsion of extensions of finitely generated abelian by elementary abelian groups. When the action is trivial , we make a specific choice of a 1-cochain for a vanishing multiple of the cohomology class defining the extension and use it to completely describe the torsion of central extensions. As an application, one gets that, under the assumption of trivial action on homology, Z_p^r may act freely on (S¹)^k if and only if r?k, providing an alternative proof of the main theorem in [Trans. Amer. Math. Soc. 352 (6) (2000) 2689-2700] for central extensions.     

Galois module structure for dihedral extensions of degree 8: Realizable classes over the group ring

Let k be a number field with ring of integers O_k, and let Γ be the dihedral group of order 8. For each tame Galois extension N/k with group isomorphic to Γ, the ring of integers O_N of N determines a class in the locally free class group Cl(O_k[Γ]). We show that the set of classes in Cl(O_k[Γ]) realized in this way is the kernel of the augmentation homomorphism from Cl(O_k[Γ]) to the ideal class group Cl(O_k), provided that the ray class group of O_k for the modulus 4O_k has odd order. This refines a result of the second-named author (J. Algebra 223 (2000) 367-378) on Galois module structure over a maximal order in k[Γ].     

Conjectures de Greenberg et extensions¶pro-p-libres d'un corps de nombres

For an algebraic number field k and a prime number p (if p=2, we assume that μ₄⊂k), we study the maximal rank ρ _p of a free pro-p-extension of k. This problem is related to deep conjectures of Greenberg in Iwasawa theory. We give different equivalent formulations of these conjectures and we apply them to show that, essentially, ρ _k =r ₂(k)+1 if and only if k is a so-called p-rational field. Received: 29 April 1999 / Revised version: 31 January 2000     

How often can a finite group be realized as a Galois group over a field?

Let G be any finite group and any class of fields. By we denote the minimal number of realizations of G as a Galois group over some field from the class . For G abelian and the class of algebraic extensions of ℚ we give an explicit formula for . Similarly we treat the case of an abelian p-group G and the class which is conjectured to be the class of all fields of characteristic ≠p for which the Galois group of the maximal p-extension is finitely generated. For non-abelian groups G we offer a variety of sporadic results. Received: 27 October 1998 / Revised version: 3 February 1999     

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.     

On a Normal Integral Basis Problem over Cyclotomic Zp-extensions, II

Let p be an odd prime number, K an imaginary abelian field with ζ_p∈K^×, and K_∞/K the cyclotomic Z_p-extension with its nth layer K_n. In the previous paper, we showed that for any n and any unramified cyclic extension L/K_n of degree p, LK_n+1/K_n+1 does have a normal integral basis (NIB) even if L/K_n has no NIB, under the assumption that p does not divide the class number of the maximal real subfield K⁺ (and some additional assumptions on K). In this paper, we show that similar but more delicate phenomena occur for a certain class of tamely ramified extensions of degree p.     

On the Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field

We prove that for every field k and every positive integer n there exists an absolutely simple n-dimensional abelian variety over k. We also prove an asymptotic result for finite fields: For every finite field k and positive integer n, we let S(k, n) denote the fraction of the isogeny classes of n-dimensional abelian varieties over k that consist of absolutely simple ordinary abelian varieties. Then for every n we have S(F_q, n)→1 as q→∞ over the prime powers.     

Deformations of locally abelian Galois representations and unramified extensions

We study the deformation theory of Galois representations whose restriction to every decomposition subgroup is abelian. As an application, we construct unramified non-solvable extensions over the field obtained by adjoining all p-power roots of unity to the field of rational numbers.     

Square classes of totally positive units

In this article, we investigate some conditions for a real cyclic extension K over Q to satisfy the property that every totally positive unit of K is a square. As an application, we give a partial answer to Taussky's conjecture. We then extend our result to real abelian extensions of certain type.     

Generalized radix representations and dynamical systems. IV

For r = (r₁,…, r_d) ∈ ?^d the mapping τ_r:?^d →?^d given byτ_r(a₁,…,a_d) = (a₂, …, a_d,−⌊r₁a₁+…+ r_da_d⌋)where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ?^d there exists an integer k > 0 with τ_r^k(a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).     

A description of continued fraction expansions of quadratic surds represented by polynomials

The principal thrust of this investigation is to provide families of quadratic polynomials , where e_k²−f_k²C=n (for any given nonzero integer n) satisfying the property that for any , the period length of the simple continued fraction expansion of is constant for fixed k and lim_k→∞?_k=∞. This generalizes, and completes, numerous results in the literature, where the primary focus was upon |n|=1, including the work of this author, and coauthors, in Mollin (Far East J. Math. Sci. Special Vol. 1998, Part III, 257-293; Serdica Math. J. 27 (2001) 317) Mollin and Cheng (Math. Rep. Acad. Sci. Canada 24 (2002) 102; Internat Math J 2 (2002) 951) and Mollin et al. (JP J. Algebra Number Theory Appl. 2 (2002) 47).     

The class number of cyclotomic function fields

Let k be a rational function field over a finite field. Carlitz and Hayes have described a family of extensions of k which are analogous to the collection of cyclotomic extensions {Q(ζm)| m ≥ 2} of the rational field Q. We investigate arithmetic properties of these “cyclotomic function fields.” We introduce the notion of the maximal real subfield of the cyclotomic function field and develop class number formulas for both the cyclotomic function field and its maximal real subfield. Our principal result is the analogue of a classical theorem of Kummer which for a prime p and positive integer n relates the class number of Q(ζpn + ζpn?1), the maximal real subfield of Q(ζpn), to the index of the group of cyclotomic units in the full unit group of Z[ζpn].     

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.     

Noyaux de Tate et capitulation

Following Kahn, and Assim and Movahhedi, we look for bounds for the order of the capitulation kernels of higher K-groups of S-integers into abelian S-ramified p-extensions. The basic strategy is to change twists inside some Galois-cohomology groups, which is done via the comparison of Tate Kernels of higher order. We investigate two ways: a global one, valid for twists close to 0 (in a certain sense), and a local one, valid for twists close to 1 in cyclic extensions. The global method produces lower bounds for abelian p-extensions which are S-ramified, but not Zp-embeddable. The local method is close to that of [J. Assim, A. Movahhedi, Bounds for étale capitulation kernels, K-Theory 33 (2004) 199-213], but is improved to take into consideration what happens when S consists of only the p-places. In contrast to the first one, one can expect this second method to produce nontrivial lower bounds in certain Zp-extensions. For example, we construct Zp-extensions in which the capitulation kernel is as big as we want (when letting the twist vary). We also include a complete solution to the problem of comparing Tate Kernels.