Bolza Quaternion Order and Asymptotics of Systoles Along Congruence Subgroups

Abstract

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 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.

Keywords:

• arithmetic Fuchsian group
• Bolza curve
• hyperbolic surface
• invariant trace field
• order
• principal congruence subgroup
• quaternion algebra
• systole
• totally real field
• triangle group

2000 AMS Subject Classification:

• 11R52
• 53C23
• 16K20

Acknowledgments

We are grateful to M. Belolipetsky and J. Voight for their helpful comments.

Funding

M. Katz is partially supported by ISF grant 1517/12. M. M. Schein is partially supported by GIF grant 1246/2014. U. Vishne is partially supported by a BSF grant 206940 and an ISF grant 1207/12.

Notes

1 Recall that in the Hurwitz case there are two Archimedean ramified places and no non-Archimedean ones (see [Katz et al. 07]).