Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg最新文献

We look at genera of even unimodular lattices of rank 12 over the ring of integers of ({{mathbb {Q}}}(sqrt{5})) and of rank 8 over the ring of integers of ({{mathbb {Q}}}(sqrt{3})), using Kneser neighbours to diagonalise spaces of scalar-valued algebraic modular forms. We conjecture most of the global Arthur parameters, and prove several of them using theta series, in the manner of Ikeda and Yamana. We find instances of congruences for non-parallel weight Hilbert modular forms. Turning to the genus of Hermitian lattices of rank 12 over the Eisenstein integers, even and unimodular over ({{mathbb {Z}}}), we prove a conjecture of Hentschel, Krieg and Nebe, identifying a certain linear combination of theta series as an Hermitian Ikeda lift, and we prove that another is an Hermitian Miyawaki lift.

Inspired by the work of Lu and Tian (Duke Math J 125:351--387, 2004), Loi et al. address in (Abh Math Semin Univ Hambg 90: 99-109, 2020) the problem of studying those Kähler manifolds satisfying the (Delta )-property, i.e. such that on a neighborhood of each of its points the k-th power of the Kähler Laplacian is a polynomial function of the complex Euclidean Laplacian, for all positive integer k. In particular they conjectured that if a Kähler manifold satisfies the (Delta )-property then it is a complex space form. This paper is dedicated to the proof of the validity of this conjecture.

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.