Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg最新文献

A comparison among different constructions in (mathbb {H}^2 cong {mathbb {R}}^8) of the quaternionic 4-form (Phi _{text {Sp}(2)text {Sp}(1)}) and of the Cayley calibration (Phi _{text {Spin}(7)}) shows that one can start for them from the same collections of “Kähler 2-forms”, entering both in quaternion Kähler and in (text {Spin}(7)) geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension 16, similar constructions allow to write explicit formulas in (mathbb {R}^{16}) for the canonical 4-forms (Phi _{text {Spin}(8)}) and (Phi _{text {Spin}(7)text {U}(1)}), associated with Clifford systems related with the subgroups (text {Spin}(8)) and (text {Spin}(7)text {U}(1)) of (text {SO}(16)). We characterize the calibrated 4-planes of the 4-forms (Phi _{text {Spin}(8)}) and (Phi _{text {Spin}(7)text {U}(1)}), extending in two different ways the notion of Cayley 4-plane to dimension 16.

The set of unrestricted homotopy classes ([M,S^n]) where M is a closed and connected spin ((n+1))-manifold is called the n-th cohomotopy group (pi ^n(M)) of M. Using homotopy theory it is known that (pi ^n(M) = H^n(M;{mathbb {Z}}) oplus {mathbb {Z}}_2). We will provide a geometrical description of the ({mathbb {Z}}_2) part in (pi ^n(M)) analogous to Pontryagin’s computation of the stable homotopy group (pi _{n+1}(S^n)). This ({mathbb {Z}}_2) number can be computed by counting embedded circles in M with a certain framing of their normal bundle. This is a similar result to the mod 2 degree theorem for maps (M rightarrow S^{n+1}). Finally we will observe that the zero locus of a section in an oriented rank n vector bundle (E rightarrow M) defines an element in (pi ^n(M)) and it turns out that the ({mathbb {Z}}_2) part is an invariant of the isomorphism class of E. At the end we show that if the Euler class of E vanishes this ({mathbb {Z}}_2) invariant is the final obstruction to the existence of a nowhere vanishing section.

We take another look at the so-called quasi-derivation relations in the theory of multiple zeta values, by giving a certain formula for the quasi-derivation operator. In doing so, we are not only able to prove the quasi-derivation relations in a simpler manner but also give an analog of the quasi-derivation relations for finite multiple zeta values.

We present explicit formulas for Hecke eigenforms as linear combinations of q-analogues of modified double zeta values. As an application, we obtain period polynomial relations and sum formulas for these modified double zeta values. These relations have similar shapes as the period polynomial relations of Gangl, Kaneko, and Zagier and the usual sum formulas for classical double zeta values.

We study analogues of Sturm’s bounds for vector valued Siegel modular forms of type (k, 2), which was already studied by Sturm in the case of an elliptic modular form and by Choi–Choie–Kikuta, Poor–Yuen and Raum–Richter in the case of scalar valued Siegel modular forms.

We show that a dihedral congruence prime for a normalised Hecke eigenform f in (S_k(Gamma _0(D),chi _D)), where (chi _D) is a real quadratic character, appears in the denominator of the Bloch–Kato conjectural formula for the value at 1 of the twisted adjoint L-function of f. We then use a formula of Zagier to prove that it appears in the denominator of a suitably normalised (L(1,{mathrm {ad}}^0(g)otimes chi _D)) for some (gin S_k(Gamma _0(D),chi _D)).

We show that a hypersurface of the regularized, spatial circular restricted three-body problem is of contact type whenever the energy level is below the first critical value (the energy level of the first Lagrange point) or if the energy level is slightly above it. A dynamical consequence is that there is no blue sky catastrophe in this energy range.

We study finite dimensional almost- and quasi-effective prolongations of nilpotent ({mathbb {Z}})-graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize effectiveness and algebraicity and are appropriate to obtain Levi–Malčev and Levi–Chevalley decompositions and precisions on the heigth and other properties of the prolongations in a very natural way. In a last section we consider the semisimple case and discuss some examples in which the structural algebras are central extensions of orthogonal Lie algebras and their degree ((-,1)) components arise from spin representations.

We derive the existence of p-adic Hurwitz–Lerch L-function by means of a method provided by Washington. This function is a generalization of the one-variable p-adic L-function of Kubota and Leopoldt, and two-variable p-adic L-function of Fox. We also deduce divisibility properties of generalized Apostol–Bernoulli polynomials, in particular establish Kummer-type congruences for them.

Inspired by the work of Lu and Tian (Duke Math J 125(2):351–387, 2004), in this paper we address the problem of studying those Kähler manifolds satisfying the (Delta)-property, i.e. such that on a neighborhood of each of its points the kth power of the Kähler Laplacian is a polynomial function of the complex Euclidean Laplacian, for all positive integer k (see below for its definition). We prove two results: (1) if a Kähler manifold satisfies the (Delta)-property then its curvature tensor is parallel; (2) if an Hermitian symmetric space of classical type satisfies the (Delta)-property then it is a complex space form (namely it has constant holomorphic sectional curvature). In view of these results we believe that if a Kähler manifold satisfies the (Delta)-property then it is a complex space form.