Kiefer, Paul (2022)
Orthogonal Eisenstein Series of Singular Weight.
Technische Universität Darmstadt
doi: 10.26083/tuprints-00020368
Dissertation, Erstveröffentlichung, Verlagsversion

Kurzbeschreibung (Abstract)

In this thesis we investigate (non-)holomorphic orthogonal Eisenstein series by using Borcherds' additive theta lift.

Therefore we start by looking at the boundary components of the orthogonal upper half-plane and its quotients by congruence subgroups. In particular we investigate the case of prime level and square-free level.

Afterwards we consider the additive theta lift of non-holomorphic vector-valued Eisenstein series with respect to the Weil representation of a lattice of signature (b⁺, b⁻). We will derive the meromorphic continuation and functional equation of the theta lifts. Moreover, we will calculate their Fourier expansion.

In the last part we will specialise to signature (2, l) and show, that additive theta lifts of non-holomorphic vector-valued Eisenstein series are non-holomorphic orthogonal Eisenstein series. This yields a new proof of their meromorphic continuation and functional equation. Moreover, we will investigate if the theta lift is injective or surjective. Afterwards we consider the holomorphic Eisenstein series by evaluating the non-holomorphic Eisenstein series at special values. Again, we investigate, if the theta lift is injective or surjective and show, that if the lattice splits two hyperbolic planes, then all holomorphic modular forms of singular weight κ = l/2 - 1, that are linear combinations of Eisenstein series on the boundary, can be written as theta lifts.

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