Modular Curves on Modular Surfaces

Abstract

In this chapter we introduce the modular curves on Hilbert modular surfaces. It is the presence of these curves that makes the geometry and arithmetic of these surfaces so rich. If one considers the surface (PSL(2, ℤ)\ℌ)² as a degenerate Hilbert modular surface, then on this surface the modular curves are just the Hecke correspondences T _N . If not by this analogy, one arrives at these modular curves by viewing Hilbert modular surfaces as moduli spaces of abelian surfaces with real multiplication: the points of the modular curves correspond to abelian surfaces whose endomorphism ring contains an order of a quaternion algebra, see Ch. IX. In spite of these heuristic indications and of the fact that the equations for these curves already occur in Hecke’s thesis, nobody looked at them seriously before the seventies. Hirzebruch then studied them in connection with the classification of Hilbert modular surfaces. The development was then triggered by a letter of Serre to Hirzebruch (8 dec. 1971), where Serre noticed a remarkable coincidence of the dimension of a space of modular forms in one variable and the dimension of a certain part of the cohomology group ${{\rm H}^2}({\Upsilon _{{\Gamma _{\rm K}}}},\mathbb{C})$ containing the classes of these curves. The outcome of this will be the topic of the next chapter.