Products of Vector Valued Eisenstein Series

We prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin-Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagier in the context of their previously mentioned result.Comment: accepted for publication in Forum Mathematicu

Sturm Bounds for Siegel Modular Forms

We establish Sturm bounds for degree g Siegel modular forms modulo a prime p, which are vital for explicit computations. Our inductive proof exploits Fourier-Jacobi expansions of Siegel modular forms and properties of specializations of Jacobi forms to torsion points. In particular, our approach is completely different from the proofs of the previously known cases g=1,2, which do not extend to the case of general g

Harmonic Maa{\ss}-Jacobi forms of degree 1 with higher rank indices

We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of order 4. In ranks exceeding 1, the notions of H-harmonicity and semi-holomorphicity are the same.Comment: 28 page

Almost holomorphic Poincar\ue9 series corresponding to products of harmonic Siegel–Maass forms

\ua9 2016, The Author(s). We investigate Poincar\ue9 series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincar\ue9 series are almost holomorphic as well. In general, this is not the case. The main point of this paper is the study of Siegel–Poincar\ue9 series of degree\ua02 attached to products of terms of Fourier series of harmonic Siegel–Maass forms and holomorphic Siegel modular forms. We establish conditions on the convergence and nonvanishing of such Siegel–Poincar\ue9 series. We surprisingly discover that these Poincar\ue9 series are almost holomorphic Siegel modular forms, although the product of terms of Fourier series of harmonic Siegel–Maass forms and holomorphic Siegel modular forms (in contrast to the elliptic case) is not almost holomorphic. Our proof employs tools from representation theory. In particular, we determine some constituents of the tensor product of Harish-Chandra modules with walls

Hyper-Algebras of Vector-Valued Modular Forms

We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of Q, acting on these hyper-algebras. These definitions bridge the classical and representation theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein series. As a byproduct, we obtain inclusions of cuspidal automorphic representations into the tensor product of global principal series

Formal Fourier Jacobi expansions and special cycles of codimension two

We prove that formal Fourier Jacobi expansions of degree two are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension two, which were defined by Kudla. A second application is the proof of termination of an algorithm to compute Fourier expansions of arbitrary Siegel modular forms of degree two. Combining both results enables us to determine relations of special cycles in the second Chow group

Modular completions of indefinite theta series on tetrahedral cones

Holomorphic indefinite theta series are approximately the sum over the intersection of a lattice and a closed cone in the associated real quadratic space. It is necessary for convergence that this cone is non-negative. Polyhedral cones are cones which correspond to hyperbolic polyhedra in the projectivisation of the real quadratic space. They can be used to approximate all other cones, and on the other hand can be built up from tetrahedral cones. Zwegers's thesis contains the case of a 1-tetrahedron in the projectivisation of a quadratic space of signature $(n,1)$. In summer, the case of positive, rectangular cones that are positive in signature $(n,2)$ was treated.Non UBCUnreviewedAuthor affiliation: Chalmers University of TechnologyFacult