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Real-dihedral harmonic Maass forms and CM-values of Hilbert modular functions

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  02 February 2016

Yingkun Li
Yingkun Li*
Affiliation:
Fachbereich Mathematik, TU Darmstadt, Schloßgartenstr. 7, 64289 Darmstadt, Germany email li@mathematik.tu-darmstadt.de
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Abstract

In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math. 163 (2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

Keywords

harmonic Maass formautomorphic Green’s functionGalois representation

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F80: Galois representations
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 6 , June 2016 , pp. 1159 - 1197
Copyright
© The Author 2016 

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