An explicit example of a noncrossed product division algebra

The paper presents an explicit example of a noncrossed product division algebra of index and exponent 8 over the field ?(s)(t). It is an iterated twisted function field in two variables D(x, σ)(y, τ ) over a quaternion division algebra D which is defined over the number field ?(√3,√?7). The automorphisms σ and τ are computed by solving relative norm equations in extensions of number fields. The example is explicit in the sense that its structure constants are known. Moreover, it is pointed out that the same arguments also yield another example, this time over the field ?((s))((t)), given by an iterated twisted Laurent series ring D((x, σ))((y, τ )) over the same quaternion division algebra D. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Free pro-p groups as galois groups over ℚ(p)(t)

Letp be a prime and let ℚ(p) denote the maximalp-extension of ℚ. We prove that for every primep, the free pro-p group on countably many generators is realizable as a regular extension of ℚ(p)(t). As a consequence, if ℚ _nil denotes the maximal nilpotent extension of ℚ, then every finite nilpotent group is realizable as a regular extension of ℚ _nil (t).     

Fields for which the projective schur subgroup is the whole Brauer group

LetBr(K) denote the Brauer group of a fieldK andPS(K) the projective Schur subgroup. 1. LetK be a finitely generated infinite field. ThenPS(K)=Br(K) if and only ifK is a global field. 2.|LetK be a finitely generated infinite field, and letK((t)) denote the field of formal power series int overK. ThenPS(K((t)))=Br(K((t))) if and only ifK=ℚ.     

Hilbert’s irreducibility theorem for prime degree and general polynomials

Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt ₀εℤ such thatf (X, t ₀) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t ₀) is irreducible for all but finitely manyt ₀εℤ unless the curve given byf (X, t)=0 has infinitely many points (x ₀,t ₀) withx ₀εℚ,t ₀εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others. Supported by the DFG.     

Brauer groups,embedding problems,and nilpotent groups as Galois groups

Let ℚ _ab denote the maximal abelian extension of the rationals ℚ, and let ℚ_abnil denote the maximal nilpotent extension of ℚ _ab . We prove that for every primep, the free pro-p group on countably many generators is realizable as the Galois group of a regular extension of ℚ_abnil(t). We also prove that ℚ_abnil is not PAC (pseudo-algebraically closed). This paper was inspired by the author's participation in a special year on the arithmetic of fields at the Institute for Advanced Studies at the Hebrew University of Jerusalem. I would like to express my appreciation to the Institute for its hospitality, and to the organizers, especially Moshe Jarden. Partially supported by the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund-Japan Technion Society Research Fund.     

Path integrals for quadratic lagrangians on <Emphasis Type="Italic">p</Emphasis>-adic and adelic spaces

Feynman’s path integrals in ordinary, p-adic and adelic quantum mechanics are considered. The corresponding probability amplitudes K(x″, t″; x′, t′) for two-dimensional systems with quadratic Lagrangians are evaluated analytically and obtained expressions are generalized to any finite-dimensional spaces. These general formulas are presented in the form which is invariant under interchange of the number fields ℝ ↔ ℚ _p and ℚ ↔ ℚ _p , p ≠ p′. According to this invariance we have that adelic path integral is a fundamental object in mathematical physics of quantum phenomena.     

Laurent rings

This is a study of ring-theoretic properties of a Laurent ring over a ring A, which is defined to be any ring formed from the additive group of Laurent series in a variable x over A, such that left multiplication by elements of A and right multiplication by powers of x obey the usual rules, and such that the lowest degree of the product of two nonzero series is not less than the sum of the lowest degrees of the factors. The main examples are skew-Laurent series rings A((x; ϕ)) and formal pseudo-differential operator rings A((t ⁻¹; δ)), with multiplication twisted by either an automorphism ϕ or a derivation δ of the coefficient ring A (in the latter case, take x = t ⁻¹). Generalized Laurent rings are also studied. The ring of fractional n-adic numbers (the localization of the ring of n-adic integers with respect to the multiplicative set generated by n) is an example of a generalized Laurent ring. Necessary and/or sufficient conditions are derived for Laurent rings to be rings of various standard types. The paper also includes some results on Laurent series rings in several variables. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 151–224, 2006.     

Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves

We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves over perfect fields. For example, if k is finitely generated over ℚ and X → C is a quadric fibration of odd relative dimension at least 11, then CH ⁱ (X) is finitely generated for i ≤ 4.     

Algebraic independence over ℚ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> of values of analytic functions at points from ℂ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

General theorems on the algebraic independence over ℚ _p of the values of analytic functions at points from ℂ _p and their applications to particular examples are presented in the paper.     

Multivalued nonlinear equations on the half line: A fixed point approach

New fixed point theorems of the authors are used to establish the existence of one (or more) C[0, ∞) solutions to the nonlinear integral inclusion y(t)∈ ∫₀^∞ K(t, s) F(s, y(s))ds fort ∈ [0,∞).     

Regulator constants and the parity conjecture

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p ^∞-Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/ℚ is semistable at 2 and 3, K/ℚ is abelian and K ^∞ is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of . We also give analogous results when K/ℚ is non-abelian, the base field is not ℚ and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations. T. Dokchitser is supported by a Royal Society University Research Fellowship.     

Analytic functions over a field of power series

 We extend the notion of absolute convergence for real series in several variables to a notion of convergence for series in a power series field ℝ((t ^Γ)) with coefficients in ℝ. Subsequently, we define a natural notion of analytic function at a point of ℝ((t ^Γ))^m. Then, given a real function f analytic on a open box I of ℝ ^m , we extend f to a function f ^★ which is analytic on a subset of ℝ((t ^Γ)) ^m containing I. We prove that the functions f ^★ share with real analytic functions certain basic properties: they are , they have usual Taylor development, they satisfy the inverse function theorem and the implicit function theorem. Received: 5 October 2000 / Revised version: 19 June 2001 / Published online: 12 July 2002     

Nonsmooth generalized guiding functions for periodic differential inclusions

We consider the periodic problem for differential inclusions in $$ \user2{\mathbb{R}}^{\rm N} $$ with a nonconvex-valued orientor field F(t, ζ), which is lower semicontinuous in $$ \zeta \in \user2{\mathbb{R}}^{\rm N} $$ Using the notion of a nonsmooth, locally Lipschitz generalized guiding function, we prove that the inclusion has periodic solutions. We have two such existence theorems. We also study the “convex” periodic problem and prove an existence result under upper semicontinuity hypothesis on F(t, ·) and using a nonsmooth guiding function. Our work was motivated by the recent paper of Mawhin-Ward [23] and extends the single-valued results of Mawhin [19] and the multivalued results of De Blasi-Górniewicz-Pianigiani [4], where either the guiding function is C¹ or the conditions on F are more restrictive and more difficult to verify.     

The group of endo-permutation modules

The group D(P) of all endo-permutation modules for a finite p-group P is a finitely generated abelian group. We prove that its torsion-free rank is equal to the number of conjugacy classes of non-cyclic subgroups of P. We also obtain partial results on its torsion subgroup. We determine next the structure of ℚ⊗D(-) viewed as a functor, which turns out to be a simple functor S _{E, ℚ}, indexed by the elementary group E of order p ² and the trivial Out(E)-module ℚ. Finally we describe a rather strange exact sequence relating ℚ⊗D(P), ℚ⊗B(P), and ℚ⊗R(P), where B(P) is the Burnside ring and R(P) is the Grothendieck ring of ℚP-modules. Oblatum 6-VII-1998 & 27-V-1999 / Published online: 22 September 1999     

Linear independence measures for infinite products

Let f be an entire transcendental function with rational coefficients in its power series about the origin. Further, let f satisfy a functional equation f(qz)= (z−c)f(z)+Q(z) with and some particular c∈ℚ. Then the linear independence of 1,f(α), f(−α) over ℚ for non-zero α∈ℚ is proved, and a linear independence measure for these numbers is given. Clearly, for Q= 0 the function f can be written as an infinite product. Received: 19 September 2000 / Revised version: 14 March 2001     

Discrete circle actions: A note using non-standard analysis

In this note we prove that the existence of effective uniformly Lipschitz ℚ/ℤ actions on manifolds (and other spaces) follows from the existence of suchZ _nactions. The method of approach is non-standard analysis with all non-trivial transformation group theoretical information concentrated in Newman's theorem; this results in a completely elementary argument. We give examples showing that, in contrast, there are spaces with noS ¹. effective actions despiteZ _nand hence ℚ/ℤ effective actions.     

3–global attractivity of the zero solution of the “food limited” type functional differential equation

We obtain 3/2-condition for global attractivity to occur in the “food-limited” type functional differential equationx′ (t) + [1 +x(t)][1 −cx(t)]F(t, x(·)) = 0. These results contain and improve all corresponding theorems in literature.     

The number of elliptic curves over ℚ with conductorNwith conductorN

We prove that the number of elliptic curves E/ℚ with conductorN isO(N ^1/2+ε). More generally, we prove that the number of elliptic curves E/ℚ with good reduction outsideS isO(M ^1/2+ε), whereM is the product of the primes inS. Assuming various standard conjectures, we show that this bound can be improved toO(M ^c/loglogM ). Research partially supported by NSF DMS-9424642.     

Convergence theorems for semigroup-type families of non-self mappings

Motivated by a paper Chidume and Zegeye [Strong convergence theorems for common fixed points of uniformly L-Lipschitzian pseudocontractive semi-groups, Applicable Analysis, 86 (2007), 353–366], we prove several strong convergence theorems for a family (not necessarily a semigroup) ℱ = {T(t): t ∈ G} of nonexpansive or pseudocontractive non-self mappings in a reflexive strictly convex Banach space with a uniformly Gateaux differentiable norm, where G is an unbounded subset of ℝ⁺. Our results extend and improve the corresponding ones byMatsushita and Takahashi [Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions,Nonlinear Analysis, 68 (2008), 412–419],Morales and Jung [Convergence of paths for pseudo-contractive mappings in Banach spaces, Proceedings of American Mathematical Society, 128 (2000), 3411–3419], Song [Iterative approximation to common fixed points of a countable family of nonexpansive mappings, Applicable Analysis, 86 (2007), 1329–1337], Song and Xu [Strong convergence theorems for nonexpansive semigroup in Banach spaces, Journal of Mathematical Analysis and Applications, 338 (2008), 152–161], Wong, Sahu, and Yao [Solving variational inequalities involving nonexpansive type mappings, Nonlinear Analysis, (2007) doi:10.1016/j.na. 2007.11.025] in the context of a non-semigroup family of non-self mappings.