Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg最新文献

Let (s,t,u in {{mathbb {C}}}) and T(s, t, u) be the Tornheim double zeta function. In this paper, we investigate some properties of symmetric Tornheim double zeta functions which can be regarded as a desingularization of the Tornheim double zeta function. As a corollary, we give explicit evaluation formulas or rapidly convergent series representations for T(s, t, u) in terms of series of the gamma function and the Riemann zeta function.

The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of (E_infty )-ring spectra in various ways. In this work we first establish, in the context of (infty )-categories and using Goodwillie’s calculus of functors, that various definitions of the cotangent complex of a map of (E_infty )-ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an (E_infty )-ring spectrum and (mathrm {Pic}(R)) denote its Picard (E_infty )-group. Let Mf denote the Thom (E_infty )-R-algebra of a map of (E_infty )-groups (f:Grightarrow mathrm {Pic}(R)); examples of Mf are given by various flavors of cobordism spectra. We prove that the cotangent complex of (Rrightarrow Mf) is equivalent to the smash product of Mf and the connective spectrum associated to G.

Let (pod_3(n)) denote the number of 3-regular partitions of n with distinct odd parts (and even parts are unrestricted). In this article, we prove an infinite family of congruences for (pod_3(n)) using the theory of Hecke eigenforms. We also study the divisibility properties of (pod_3(n)) using arithmetic properties of modular forms.

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.