We show that the A-polynomial of the J(k,l) knots is a single resultant. A consequence is that for the twist knots and the J(3,2n) knots, the A-polynomial can be written in terms of Fibonacci polynomials and satisfies a simple recursion (this recovers work of Hoste and Shanahan). Moreover, we write the A-polynomial of the J(4,2n) and J(5,2n) knots explicitly in terms of symmetric functions.
The classical theorem of Bombieri and Vinogradov is generalized to a non-abelian, non-Galois setting. This leads to a prime number theorem of ``mixed-type" for arithmetic progressions ``twisted" by splitting conditions in number fields. One can view this as an extension of earlier work of M. R. Murty and V. K. Murty on a variant of the Bombieri-Vinogradov theorem. We develop this theory with a view to applications in the study of the Euclidean algorithm in number fields and arithmetic orbifolds.
Let K be a number field of unit rank greater than 3 with a subfield M of K such that K/M is Galois with group G of order greater than 3. For such K we prove a number field version of Artin's primitive root conjecture. Specifically, we show that there are >> x/(log x)^2 prime ideals P in O_K such that with f_P denoting the reduction modulo P map, f_P(O_K^x) surjects f_P(O_K)^x. We conclude that for such K, O_K is Euclidean if and only if it is a principal ideal domain. This was previously know under the assumption of the GRH.
We investigate a connection between two important classes of Euclidean lattices: well-rounded lattices and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal non-trivial vectors spans the whole space. We consider lattices coming from full rings of integers in number fields, proving that only cyclotomic fields give rise to well-rounded lattices. We further study the well-rounded lattices coming from ideals in quadratic rings of integers, showing that there exist infinitely many quadratic number fields containing ideals which give rise to well-rounded lattices in the plane.
We find explicit models for the PSL_2(C)- and SL_2(C)-character varieties of the fundamental groups of complements in S^3 of an infinite family of two-bridge knots that contains the twist knots. We compute the genus of the components of these character varieties, and deduce upper bounds on the degree of the associated trace fields. We also show that these knot complements are fibered if and only if they are commensurable to a fibered knot complement in a Z/2Z-homology sphere, resolving a conjecture of Hoste and Shanahan.
Given a fixed quadratic extension K of Q, we consider the distribution of elements in K of norm 1 (denoted N ). When K is an imaginary quadratic extension, N is naturally embedded in the unit circle in C and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in N can be written as a/a* (where a* is the complex conjugate) for some a in O_K, which yields another ordering of N given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that N is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that N is equidistributed with respect to norm, under the map b -> log |b| mod log |e^2| where e is a fundamental unit of O_K .
Let K be a number field with r real places and s complex places, and class number h_K. The quotient [H^2]^r x [H^3]^s/ PSL_2(O_K) has h_K cusps. For a prime ideal P of O_K of norm q, let f_P denote the reduction modulo P map. We provide an un-conditional proof that if K is Galois with unit rank greater than three, then there are infinitely many such P with the property that f_P(O_K^x)=(O_K/P)^x. Our result establishes an un-conditional proof that for such K, PSL_2(O_K) has infinitely many maximal subgroups, G, such that the quotient [H^2]^r x [H^3]^s/G has exactly h_K cusps.
Let K be a number field with r real places and s complex places, and let O_K be the ring of integers of K. The quotient [H^2]^r x [H^3]^s / PSL_2(O_K) has h_K cusps, where h_K is the class number of K.We show that under the assumption of the generalized Riemann hypothesis that if K is not Q or an imaginary quadratic field and if i is not in K then PSL_2(O_K) has infinitely many maximal subgroups with h_K cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.
We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree N is naturally associated to a subset of R^(N-1). We calculate the volume of this set, prove the set is homeomorphic to the N-1 ball and that its isometry group is isomorphic to the dihedral group of order 2N.
We show that there are only finitely many maximal congruence subgroups of the Bianchi groups such that the quotient by hyperbolic upper half space has only one cusp.
This note explains the construction of a class of arithmetic groups which act on products of the hyperbolic half plane and hyperbolic half space. The constuction centers around orders in quaternion algebras. These arithemtic groups include the arithmetic Fuchsian groups, the arithmetic Kleinian groups, and the arithmetic groups PSL(2,O_K) where O_K is the rung of integers of the number field K.
Like the modular group, the Bianchi groups fail to have the congruence subgroup property. We realize this topologically for those Bianchi groups whose quotients have only one cusp. We show that whereas there are many subgroups whose quotients have only one cusp, few of these are congruence subgroups. We ask the analogous question for other groups of the form PSL_2(O_K). These groups have the congruence subgroup property. We show that assuming the GRH, there are infinitely many (necessarily congruence) subgroups with a minimal number of cusps.