Arithmetic Fuchsian Groups of Genus Zero

Finiteness of arithmetic hyperbolic reflection groups

“…Our results are independent of their work in the sense that we obtain a finiteness result in each dimension and then appeal to the result of Prokhorov that reflection groups do not exist in high enough dimensions [30]. The argument of this paper generalizes an argument of LongMaclachlan-Reid [21], which implies that there are only finitely many arithmetic minimal (or congruence) hyperbolic 2-orbifolds with bounded genus. Their argument is in fact a generalization of an argument of Zograf [43], who reproved that there are only finitely many congruence groups commensurable with PSL.2; Z/ such that H 2 = has genus 0 (This was proven originally by Dennin [9], [10], and was known as Rademacher's conjecture).…”

On the Hasse principle for Shimura curves

“…Acknowledgement: I am grateful to D. Lorenzini for a careful reading of this final section (written more than a year after the rest of the paper) and to J. Voight for making me aware of [12], which was not published until after this paper was first submitted.…”

Commensurability classes of discrete arithmetic hyperbolic groups

“…Necessary and sufficient conditions for such Coxeter groups to be arithmetic are due to Vinberg and these groups are always of the simplest type [28]. Furthermore, it is now known that there are only finitely many commensurability classes of hyperbolic arithmetic Coxeter groups in all dimensions [21], [22], [16], [1], [2], [23]. We show here that Vinberg's methods determine the quadratic spaces and hence the commensurability class parameters, thus partitioning the arithmetic Coxeter groups into commensurability classes.…”