Eisenstein series for the orthogonal group $O(2,n)$

  • Eisensteinreihen zur orthogonalen Gruppe $O(2,n)$

Schaps, Felix; Krieg, Aloys (Thesis advisor); Heim, Bernhard (Thesis advisor); Alfes-Neumann, Claudia (Thesis advisor)

Aachen : RWTH Aachen University (2022)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022

Abstract

In this thesis we study various scalar-valued Eisenstein series for the orthogonal group O(2,n) where the underlying lattice contains two hyperbolic planes. We consider the Fourier coefficients of the Eisenstein series defined in analogy to elliptic Eisenstein series, Jacobi Eisenstein series, and Siegel Eisenstein series, respectively. The main focus is on the Fourier coefficients of Eisenstein series of Siegel type for the standard cusp. For maximal lattices, we prove the rationality of the Fourier coefficients, as well as that they belong to the Maaß space. Moreover, we prove that the results hold for all localizations of the lattice which are maximal ones. Thus, this also holds true for non-maximal lattices except for finitely many local places. The Fourier-Jacobi coefficient of index 1 turns out to be a Jacobi-Eisenstein series. We give explicit formulas for the Fourier expansion in some cases and links to other known Eisenstein series. We also consider Eisenstein series of Klingen type, prove their absolute convergence, observe their behavior under the Petersson inner product and prove that they generate the space of non-cusp forms, at least whenever the lattice is Euclidean. Lastly, we deal with Hecke theory for the O(2,n+2) and its applications to Eisenstein series which are Hecke eigenforms. We show that the Eisenstein series are the only non-cusp forms which are eigenforms of some Hecke operators. Finally, using the methods of Heim and Krieg (2020), the Maaß relations of the Eisenstein series hold for non-maximal lattices, too.

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