Introduction

For some time now Hirzebruch and others have studied certain fields of Hilbert modular functions from a geometric point of view (see [2], [3], [5]). This leads to the introduction of a non-singular algebraic surface Y0(p) for all square-free positive integers p. In [5] the question is settled how the surfaces Y0(p) fit into the rough classification of algebraic surfaces, at least for those values of p which are prime and congruent 1 mod 4. It turns out that for p=5, 13 and 17 the surface Y0(p) is rational, that for p = 29, 37 and 41 this surface is an elliptic K3-surface, that for p = 53, 61 and 73 it is a minimal honestly elliptic surface, and that for p ≥ 89 the surface Y0(p) is of general type. As already follows from this description, the surfaces Y0(p) are minimal (i.e. without exceptional curves of the first kind) for 29 ≤ p ≤ 73. Now it is stated as a conjecture in [5] (p. 21) that this remains true for all p ≥ 89. Of course, it would be very interesting if this conjecture could be proved. In fact, if you know that a certain (simply connected) surface is of general type, and even if you know in addition its arithmetical genus and its Euler characteristic, but you don't know whether it is minimal, your knowledge does not amount to very much.