Arithmetic Groups and Lehmer’s Conjecture

Arithmetic groups are a rich class of groups where connections between topology and number theory are showcased in a particularly striking way. One construction of these groups is motivated by the modular group, PSL2(Z). The group of orientation preserving isometries of the hyperbolic upper half plane, H, is isomorphic to PSL2(R). Since Z is a discrete subgroup of R it follows that PSL2(Z) is discrete in PSL2(R). The modular group acts on H by linear fractional transformations, and the quotient H/PSL2(Z) is a finite volume hyperbolic orbifold. The modular group has deep connections to many branches of mathematics and to number theory in particular. The modular group encodes the moduli space of elliptic curves. Modular forms, which are analytic functions on H satisfying a functional equation with respect to the modular group, have far-reaching connections between geometry, number theory, and analysis. In particular, Wiles’ proof of the Taniyama Shimura conjecture (the modularity theorem) established a proof of Fermat’s Last Theorem, one of the most famous conjectures of our time. The geometry of the action of the modular group on H can also be used to provide a proof of Roth’s theorem (the Thue-Siegel-Roth theorem). This theorem essentially says that an algebraic integer (which is not in Z) does not have many ‘good’ rational approximations. Precisely, Roth’s theorem says that if α is an irrational algebraic integer, then for any > 0 ∣∣∣∣α− pq ∣∣∣∣ < 1