Hilbert modular surface

In mathematics, Hilbert module surfaces are certain complex algebraic surfaces, obtained as the quotient of the product of two hyperbolic planes.

Construction

Be a totally real quadratic number fields, ie for a square-free natural number.

Be the wholeness of the ring, so if congruent with 2 or 3 mod 4 and if congruent 1 mod 4

Be the embeddings of so

The images define embeddings.

The Hilbert modular group is the image of under the embedding

The group SL (2, R) acts on the hyperbolic plane by fractional- linear transformations. By means of embedding by then acts on the product of two hyperbolic planes.

If a subgroup of finite index, then the quotient space is called Hilbert module surface and Hilbert modular group. Hilbert module groups are examples of arithmetic groups.

If a Hilbert modular group is torsion-free, then the Hlbertsche module surface is a locally symmetric space, otherwise, the Hilbert module surface singularities.

Algebraic Surfaces

A classification Hilbert module surfaces from the viewpoint of algebraic geometry give Hirzebruch - Zagier.

Number Theory

The geometry of the Hilbert module area encodes properties of the body. For example, the number of ends of the Hilbert module surface is equal to the number of class. The volume of the Hilbert module surface, wherein the Dedekindian zeta function of the body respectively.

Swell

• Algebraic Geometry
• Geometric topology