Bolza Quaternion Order and Asymptotics of Systoles Along Congruence Subgroups. Issue 4 (1st October 2016)

• Record Type: Journal Article
• Authors: Katz, Karin; Katz, Mikhail; Schein, Michael M.; Vishne, Uzi
• Is Part Of: Experimental mathematics. Volume 25:Issue 4(2016)
• Page Start: 399
• Page End: 415
• Publication Date: 2016-10-01
• Subjects: arithmetic Fuchsian group -- Bolza curve -- hyperbolic surface -- invariant trace field -- order -- principal congruence subgroup -- quaternion algebra -- systole -- totally real field -- triangle group; 11R52 -- 53C23 -- 16K20; Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2015.1073642
• Languages: English
• ISSNs: 1058-6458

Abstract

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" (usingmagma ). 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 inmagma to generate these surfaces.