Bolza quaternion order and asymptotics of systoles along congruence subgroups

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We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound sys(X) > 4/3 log g(X) on the systole, where g is the genus. We also exhibit the Bolza group as a congruence subgroup, and calculate out a few examples of “Bolza twins” (using magma). Like the Hurwitz triplets, these correspond to the factoring of certain rational primes in the ring of integers of the invariant trace field of the surface.We exploit random sampling combined with the Reidemeister–Schreier algorithm as implemented in magma to generate these surfaces.

Original languageEnglish
Pages (from-to)399-415
Number of pages17
JournalExperimental Mathematics
Issue number4
StatePublished - 2016

Bibliographical note

Publisher Copyright:
© 2016 Taylor & Francis.


  • Arithmetic Fuchsian group
  • Bolza curve
  • Hyperbolic surface
  • Invariant trace field
  • Order
  • Principal congruence subgroup
  • Quaternion algebra
  • Systole
  • Totally real field
  • Triangle group


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