Automorphic Green functions on Hilbert modular surfaces
Abstract
In this paper, we generalize results of Bruinier on automorphic Green functions on Hilbert modular surfaces to arbitrary ideals. For instance, we compute the Fourier expansion of the unregularized Green functions, use it to regularize them, obtain the Fourier expansion of the regularized Green functions and evaluate integrals of unregularized and regularized Green functions. Furthermore, we investigate their growth behavior at the cusps in the Hirzebruch compactification by computing the precise vanishing orders along the exceptional divisors. This makes the arithmetic HirzebruchZagier theorem from Bruinier, Burgos Gil and Kühn more explicit. To this end, we generalize the theory of local Borcherds products. Lastly, we investigate a new decomposition of the Green functions into smooth functions and compute and estimate the Fourier coefficients of those smooth functions. Finally, this is employed to prove the welldefinedness and almost everywhere convergence of the generating series of the Green functions and the modularity of its integral.
 Publication:

arXiv eprints
 Pub Date:
 April 2023
 DOI:
 10.48550/arXiv.2304.13370
 arXiv:
 arXiv:2304.13370
 Bibcode:
 2023arXiv230413370B
 Keywords:

 Mathematics  Number Theory;
 11F41;
 11G18 (Primary) 14G35;
 11F30 (Secondary)
 EPrint:
 36 pages