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 .
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{\ displaystyle F = \ mathbb {Q} ({\ sqrt {a}})}
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Let be the wholeness ring of , i.e. with if congruent 2 or 3 mod 4 and if congruent 1 mod 4.
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{\ displaystyle {\ mathcal {O}} _ {F} = \ mathbb {Z} \ left [x_ {a} \ right]}
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{\ displaystyle x_ {a} = {\ sqrt {a}}}
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{\ displaystyle x_ {a} = {\ frac {1 + {\ sqrt {a}}} {2}}}
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Be the embeddings of , so
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{\ displaystyle \ left \ {\ sigma _ {+}, \ sigma _ {-}: {\ mathcal {O}} _ {F} \ rightarrow \ mathbb {R} \ right \}}
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{\ displaystyle {\ mathcal {O}} _ {F}}
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{\ displaystyle \ sigma _ {\ pm} (m + nx_ {a}) = m \ pm nx_ {a}}
for everyone .
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{\ displaystyle m, n \ in \ mathbb {Z}}
The images define embeddings .
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{\ 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}}}
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{\ displaystyle \ sigma _ {\ pm}: SL (2, {\ mathcal {O}} _ {F}) \ rightarrow SL (2, \ mathbb {R})}
The Hilbert module group is the image from under the embedding
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{\ displaystyle SL (2, {\ mathcal {O}} _ {F})}
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{\ 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.
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{\ displaystyle SL (2, \ mathbb {R}) \ times SL (2, \ mathbb {R})}
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{\ displaystyle SL (2, {\ mathcal {O}} _ {F})}
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{\ 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 .
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{\ displaystyle \ Gamma \ subset SL (2, {\ mathcal {O}} _ {F})}
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{\ displaystyle \ Gamma \ backslash (\ mathbb {H} ^ {2} \ times \ mathbb {H} ^ {2})}
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{\ 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 .
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{\ displaystyle \ Gamma \ subset SL (2, {\ mathcal {O}} _ {F})}
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{\ 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 .
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{\ displaystyle 2 \ zeta _ {F} (- 1)}
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↑ 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)
↑ 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
↑ 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
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