Note on linear relations in Galois cohomology and étale K-theory of curves

In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [G. Banaszak and P. Krasoń, On a local to global principle in étale K-groups of curves, J. K-Theory Appl. Algebra Geom. Topol. 12 (2013) 183–201], G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for étale [Formula: see text]-theory of a curve. This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases, this result is the best possible i.e. if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for étale [Formula: see text]-theory of a curve. The dynamical local to global principle for the groups of Mordell–Weil type has recently been considered by S. Barańczuk in [S. Barańczuk, On a dynamical local-global principle in Mordell-Weil type groups, Expo. Math. 35(2) (2017) 206–211]. We show that all our results remain valid for Quillen [Formula: see text]-theory of [Formula: see text] if the Bass and Quillen–Lichtenbaum conjectures hold true for [Formula: see text]

UNEXPECTED AVERAGE VALUES OF GENERALIZED VON MANGOLDT FUNCTIONS IN RESIDUE CLASSES

In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more important cases, $k=2$ , was considered by Selberg [‘An elementary proof of the prime-number theorem’, Ann. of Math. (2)50 (1949), 305–313]. For $k\geq 2$ , it was studied by Bombieri [‘The asymptotic sieve’, Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec [‘On Bombieri’s asymptotic sieve’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve. Let $\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$ , where $\unicode[STIX]{x1D707}_{j}$ denotes the Liouville function for $(j+1)$ -free integers, and $0$ otherwise. In this paper we evaluate the average value of $\unicode[STIX]{x1D6EC}_{j,k}$ in a residue class $n\equiv a\text{ mod }q$ , $(a,q)=1$ , uniformly on $q$ . When $j\geq 2$ , we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for $\unicode[STIX]{x1D6EC}_{k}(n)$ involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported ${\mathcal{C}}^{2}$ function.

Compact Tori Associated to Hyperkähler Manifolds of Kummer Type

Dedicato alla piccola Mia. For $X$ a hyperkähler manifold of Kummer type, let $J^3(X)$ be the intermediate Jacobian associated to $H^3(X)$. We prove that $H^2(X)$ can be embedded into $H^2(J^3(X))$. We show that there exists a natural smooth quadric $Q(X)$ in the projectivization of $H^3(X)$, such that Gauss–Manin parallel transport identifies the set of projectivizations of $H^{2,1}(Y)$, for $Y$ a deformation of $X$, with an open subset of a linear section of $Q^{+}(X)$, one component of the variety of maximal linear subspaces of $Q(X)$. We give a new proof of a result of Mongardi restricting the action of monodromy on $H^2(X)$. Lastly, we show that if $X$ is projective, then $J^3(X)$ is an abelian fourfold of Weil type.

INTERTEMPORAL PORTFOLIO ALLOCATION AND HEDGING DEMAND: AN APPLICATION TO SOUTH AFRICA

This paper analyses the intertemporal hedging demand for stocks and bonds in South Africa, the United Kingdom and the United States. The analysis is done using an approximate solution method for the optimal consumption and wealth portfolio problem of an infinitely long-lived investor. Investors are assumed to have Epstein-Zin-Weil-type preferences and face asset returns described by a first-order vector autoregression in returns and state variables. The results show that the mean intertemporal hedging demands for stocks are considerably smaller in SA than in the UK or the US, whilst the mean intertemporal hedging demand for bonds are not significantly different from zero in any of the countries considered. Furthermore, it is found that stocks in the US and the UK do not present a useful hedging opportunity for an investor in SA, nor do SA stocks present a useful hedging opportunity for investors from the UK or the US.

PROJECTIVE SPECTRUM IN BANACH ALGEBRAS

For a tuple A = (A1, A2, …, An) of elements in a unital algebra [Formula: see text] over ℂ, its projective spectrumP(A) or p(A) is the collection of z ∈ ℂn, or respectively z ∈ ℙn-1 such that A(z) = z1A1 + z2A2 + ⋯ + znAn is not invertible in [Formula: see text]. In finite dimensional case, projective spectrum is a projective hypersurface. When A is commuting, P(A) looks like a bundle over the Taylor spectrum of A. In the case [Formula: see text] is reflexive or is a C*-algebra, the projective resolvent setPc(A) := ℂn \ P(A) is shown to be a disjoint union of domains of holomorphy. [Formula: see text]-valued 1-form A-1(z)dA(z) reveals the topology of Pc(A), and a Chern–Weil type homomorphism from invariant multilinear functionals to the de Rham cohomology [Formula: see text] is established.

Algebraicity of some Weil Hodge Classes

We show that the Prym map for 4-th cyclic étale covers of curves of genus 4 is a dominant morphism to a Shimura variety for a family of Abelian 6-folds of Weil type. According to the result of Schoen, this implies algebraicity of Weil classes for this family.