Bolza quaternion order and asymptotics of systoles along congruence subgroups

TL;DR

This paper analyzes the arithmetic properties of the Bolza surface, describing its quaternion order, and establishes a lower bound on systoles of congruence covers, with computational examples illustrating the theory.

Contribution

It provides a detailed description of the Bolza surface's quaternion order and proves a systole lower bound for its congruence covers, including explicit examples of Bolza twins.

Findings

Principal congruence covers satisfy sys(X) > 4/3 log g(X).

Bolza group is identified as a congruence subgroup.

Examples of Bolza twins are computed using magma.

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 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.