Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg最新文献

A scale Hilbert space is a natural generalization of a Hilbert space which considers not only a single Hilbert space but a nested sequence of subspaces. Scale structures were introduced by H. Hofer, K. Wysocki, and E. Zehnder as a new concept of smooth structures in infinite dimensions. In this paper, we prove that scale structures on mapping spaces are completely determined by the dimension of domain manifolds. We also give a complete description of the local invariant introduced by U. Frauenfelder for these spaces. Product mapping spaces and relative mapping spaces are also studied. Our approach is based on the spectral resolution of Laplace type operators together with the eigenvalue growth estimate.

We construct isomorphisms between spaces of vector-valued modular forms for the dual Weil representation and certain spaces of scalar-valued modular forms in the case that the underlying finite quadratic module A has order p or 2p, where p is an odd prime. The isomorphisms are given by twisted sums of the components of vector-valued modular forms. Our results generalize work of Bruinier and Bundschuh to the case that the components (F_{gamma }) of the vector-valued modular form are antisymmetric in the sense that (F_{gamma } = -F_{-gamma }) for all (gamma in A). As an application, we compute restrictions of Doi–Naganuma lifts of odd weight to components of Hirzebruch–Zagier curves.

We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we use ideas of Wuthrich and Bianchi), a parity result of orders of vanishing, and invariance of Iwasaswa invariants under conjugate twists of the p-adic L-functions.

In this paper, we study the non-vanishing of the Miyawaki type lift in various situations. In the case of GSpin(2, 10) constructed in Kim and Yamauchi (Math Z 288(1–2):415–437, 2018), we use the fact that the Fourier coefficient at the identity is closely related to the Rankin–Selberg L-function of two elliptic cusp forms. In the case of the original Miyawaki lifts of Siegel cusp forms, we use the fact that certain Fourier coefficients are the Petersson inner product which is non-trivial. This provides infinitely many examples of non-zero Miyawaki lifts. We give explicit examples of degree 24 and weight 24. We also prove a similar result for Miyawaki lifts for unitary groups. Especially, we obtain an unconditional result on non-vanishing of Miyawaki lifts for (U(n+1,n+1)) for each (nequiv 3) mod 4. In the last section, we prove the non-vanishing of the Miyawaki lifts for infinitely many half-integral weight Siegel cusp forms. We give explicit examples of degree 16 and weight (frac{29}{2}).

We determine the structure over (mathbb {Z}) of a ring of symmetric Hermitian modular forms of degree 2 with integral Fourier coefficients whose weights are multiples of 4 when the base field is the Gaussian number field (mathbb {Q}(sqrt{-1})). Namely, we give a set of generators consisting of 24 modular forms. As an application of our structure theorem, we give the Sturm bounds for such Hermitian modular forms of weight k with (4mid k), for (p=2), 3. We remark that the bounds for (pge 5) are already known.

We prove the meromorphic continuation and the functional equation of a twisted real-analytic Hermitain Eisenstein series of Klingen type, and as a consequence, deduce similar properties for the twisted Dirichlet series associated to a pair of Hermitian modular forms involving their Fourier–Jacobi coefficients. As an application of our result, we prove that infinitely many of the Fourier–Jacobi coefficients of a non-zero Hermitian cusp form do not vanish in any non-trivial arithmetic progression.

The paper builds a DPW approach of Willmore surfaces via conformal Gauss maps. As applications, we provide descriptions of minimal surfaces in ({mathbb {R}}^{n+2}), isotropic surfaces in (S^4) and homogeneous Willmore tori via the loop group method. A new example of a Willmore two-sphere in (S^6) without dual surfaces is also presented.

We prove that at least one of the six numbers (beta (2i)) for (i=1,ldots ,6) is irrational. Here (beta (s)=sum _{k=0}^{infty }(-1)^k(2k+1)^{-s}) denotes Dirichlet’s beta function, so that (beta (2)) is Catalan’s constant.

We consider the Siegel upper half space (H_{2m}) of degree 2m and a subset (H_mtimes H_m) of (H_{2m}) consisting of two (mtimes m) diagonal block matrices. We consider two actions of (Sp(m,{mathbb R})times Sp(m,{mathbb R}) subset Sp(2m,{mathbb R})), one is the action on holomorphic functions on (H_{2m}) defined by the automorphy factor of weight k on (H_{2m}) and the other is the action on vector valued holomorphic functions on (H_mtimes H_m) defined on each component by automorphy factors obtained by (det^k otimes rho ), where (rho ) is a polynomial representation of (GL(n,{mathbb C})). We consider vector valued linear holomorphic differential operators with constant coefficients on holomorphic functions on (H_{2m}) which give an equivariant map with respect to the above two actions under the restriction to (H_mtimes H_m). In a previous paper, we have already shown that all such operators can be obtained either by a projection of the universal automorphic differential operator or alternatively by a vector of monomial basis corresponding to the partition (2m=m+m). Here in this paper, based on a completely different idea, we give much simpler looking one-line formula for such operators. This is obtained independently from our previous results. The proofs also provide more algorithmic approach to our operators.