Examples
Let F be a totally real number field and D a quaternion division algebra over F. The multiplicative group D× gives rise to a canonical Shimura variety. Its dimension d is the number of infinite places over which D splits. In particular, if d = 1 (for example, if F = Q and D ⊗ R ≅ M2(R)), fixing a sufficiently small arithmetic subgroup of D×, one gets a Shimura curve, and curves arising from this construction are already compact (i.e. projective).
Some examples of Shimura curves with explicitly known equations are given by the Hurwitz curves of low genus:
- Klein quartic (genus 3)
- Macbeath surface (genus 7)
- First Hurwitz triplet (genus 14)
and by the Fermat curve of degree 7.
Other examples of Shimura varieties include Picard modular surfaces and Hilbert–Blumenthal varieties.
Read more about this topic: Shimura Variety
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