Hilbert module area

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

construction

Let be a totally real square number field , i.e. for a square-free natural number . ${\displaystyle F}$ ${\displaystyle F=\mathbb {Q} ({\sqrt {a}})}$ ${\displaystyle a}$

Let be the wholeness ring of , i.e. with if congruent 2 or 3 mod 4 and if congruent 1 mod 4. ${\displaystyle {\mathcal {O}}_{F}\subset F}$ ${\displaystyle F}$ ${\displaystyle {\mathcal {O}}_{F}=\mathbb {Z} \left[x_{a}\right]}$ ${\displaystyle x_{a}={\sqrt {a}}}$ ${\displaystyle a}$ ${\displaystyle x_{a}={\frac {1+{\sqrt {a}}}{2}}}$ ${\displaystyle a}$

Be the embeddings of , so ${\displaystyle \left\{\sigma _{+},\sigma _{-}:{\mathcal {O}}_{F}\rightarrow \mathbb {R} \right\}}$ ${\displaystyle {\mathcal {O}}_{F}}$

${\displaystyle \sigma _{\pm }(m+nx_{a})=m\pm nx_{a}}$for everyone . ${\displaystyle m,n\in \mathbb {Z} }$

The images define embeddings . ${\displaystyle {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}\rightarrow {\begin{pmatrix}\sigma _{\pm }(a_{11})&\sigma _{\pm }(a_{12})\\\sigma _{\pm }(a_{21})&\sigma _{\pm }(a_{22})\end{pmatrix}}}$ ${\displaystyle \sigma _{\pm }:SL(2,{\mathcal {O}}_{F})\rightarrow SL(2,\mathbb {R} )}$

The Hilbert module group is the image from under the embedding ${\displaystyle SL(2,{\mathcal {O}}_{F})}$

${\displaystyle (\sigma _{+},\sigma _{-}):SL(2,{\mathcal {O}}_{F})\rightarrow SL(2,\mathbb {R} )\times SL(2,\mathbb {R} )}$.

The group SL (2, R) acts on the hyperbolic level through fractional-linear transformations . By means of the embedding after acts on , the product of two hyperbolic planes. ${\displaystyle SL(2,\mathbb {R} )\times SL(2,\mathbb {R} )}$ ${\displaystyle SL(2,{\mathcal {O}}_{F})}$ ${\displaystyle \mathbb {H} ^{2}\times \mathbb {H} ^{2}}$

If a subgroup is of finite index , then the quotient space is called Hilbert's module area and Hilbert's module group . Hilbert module groups are examples of arithmetic groups . ${\displaystyle \Gamma \subset SL(2,{\mathcal {O}}_{F})}$ ${\displaystyle \Gamma \backslash (\mathbb {H} ^{2}\times \mathbb {H} ^{2})}$ ${\displaystyle \Gamma }$

If a Hilbert module group is torsion-free , then the Hilbert module area is a locally symmetrical space , otherwise the Hilbert module area has singularities . ${\displaystyle \Gamma \subset SL(2,{\mathcal {O}}_{F})}$ ${\displaystyle \Gamma \backslash (\mathbb {H} ^{2}\times \mathbb {H} ^{2})}$

Algebraic surfaces

Friedrich Hirzebruch and Don Zagier give a classification of Hilbert module surfaces from the point of view of algebraic geometry .

Number theory

The geometry of the Hilbert module surface encodes properties of the body . For example, the number of ends of Hilbert's module area is equal to the class number of . The volume of Hilbert's module area is , where is the Dedekind zeta function of the body . ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle 2\zeta _{F}(-1)}$ ${\displaystyle \zeta _{F}}$ ${\displaystyle F}$

swell

1. ↑ Hirzebruch, Zagier: Classification of Hilbert modular surfaces , in: WL Baily, T. Shioda (Ed.): Complex analysis and algebraic geometry, Cambridge University Press, 1977, pp. 43-77, online . (PDF; 1.4 MB)
2. ↑ Chapter III.2.7. in: Armand Borel , Lizhen Ji : Compactifications of symmetric and locally symmetric spaces. Mathematics: Theory & Applications. Birkhauser Boston, Inc., Boston, MA, 2006. ISBN 978-0-8176-3247-2
3. ↑ Gerard van der Geer : Hilbert modular surfaces. Results of mathematics and their border areas (3), 16. Springer-Verlag, Berlin, 1988. ISBN 3-540-17601-2