Green functions and arithmetic generating series on Hilbert modular surfaces
Johannes J. Buck
In this thesis we define and investigate two types of Green functions on Hilbert modular surfaces associated to real quadratic number fields. Both types possess logarithmic singularities along Hirzebruch-Zagier divisors. On the one hand, we consider the automorphic Green functions, originally introduced by Bruinier, and on the other hand Kudla's Green functions, which go back to Kudla. We calculate associated Fourier expansions, investigate their growth at the boundary, obtain integrability statements and determine associated integrals.
Especially for the automorphic Green functions we find a valuable decomposition into smooth functions with many applications.
When examining Kudla's Green functions, we find that they do not fit into the arithmetic intersection theory generalized by Burgos Gil, Kramer and Kühn, which is due to their strong growth at the cusps. We then present a modification that subtracts the undesired growth at the boundary using a partition of unity. This is done in such an elegant way that the resulting functions are not only actual Green functions in the sense of Burgos Gil, Kramer, and Kühn, but the generating series of the subtracted error terms is modular. We use this to prove our main result, the modularity of the generating series of the arithmetic Hirzebruch-Zagier divisors equipped with the modified Green functions. In the proof, we trace its modularity back to the modularity of the generating series of the arithmetic Hirzebruch-Zagier divisors equipped with the automorphic Green functions whose modularity was already shown by Bruinier, Burgos Gil and Kühn.