Quaternion models for Hurwitz
The existence of a quaternion algebra construction of Hurwitz surfaces
is due to G. Shimura ('67). An explicit presentation was described by
N. Elkies in a pair of articles ('98 and '99).
R. Vogeler applied group-theoretic methods to study Hurwitz surfaces
in '03, and obtained differential-geometric, and systolic,
consequences. In particular, he calculated the number of systolic
loops in each genus below a million.
In '07, Kobi Gurkan developed magma models for the Hurwitz quaternion
order and for Hurwitz surfaces of genus 3 and 14.
Hurwitz surfaces of genus 14: see wikipedia page "First Hurwitz triplet"
Hurwitz surface of genus 3: see wikipedia page "Klein quartic"
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