Hilbert Modular Surface - Definitions

Definitions

If R is the ring of integers of a real quadratic field, then the Hilbert modular group SL2(R) acts on the product H×H of two copies of the upper half plane H. There are several birationally equivalent surfaces related to this action, any of which may be called Hilbert modular surfaces:

  • The surface X is the quotient of H×H by SL2(R); it is not compact and usually has quotient singularities coming from points with non-trivial isotropy groups.
  • The surface X* is obtained from X by adding a finite number of points corresponding to the cusps of the action. It is compact, and has not only the quotient singularities of X, but also singularities at its cusps.
  • The surface Y is obtained from .X* by resolving the singularities in a minimal way. It is a compact smooth algebraic surface, but is not in general minimal.
  • The surface Y0 is obtained from Y by blowing down certain exceptional −1-curves. It is smooth and compact, and is often (but not always) minimal.

There are several variations of this construction:

  • The Hilbert modular group may be replaced by some subgroup of finite index, such as a congruence subgroup.
  • One can extend the Hilbert modular group by a group of order 2, acting on the Hilbert modular group via the Galois action, and exchanging the two copies of the upper half plane.

Read more about this topic:  Hilbert Modular Surface

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