Upon quotienting by units, the elements of norm 1 in a number field K form a countable subset of a torus of dimension r1 +r2 +1 where r1 and r2 are the numbers of real and pairs of complex embeddings. When K is Galois with cyclic Galois group we demonstrate that this countable set is equidistributed in this torus with respect to a natural partial ordering.
The goal of this paper is to obtain lower bounds on the height of an algebraic number in a relative setting, extending previous work of Amoroso and Masser. Specifically, in our first theorem we obtain an effective bound for the height of an algebraic number a when the base field K is a number field and K(a)/K is Galois. Our second result establishes an explicit height bound for any non-zero element a which is not a root of unity in a Galois extension F/K, depending on the degree of K/Q and the number of conjugates of a which are multiplicatively independent over K. As a consequence, we obtain a height bound for such a that is independent of the multiplicative independence condition.
The number F(h) of imaginary quadratic fields with a given class number h is of classical interest: Gauss' class number problem asks for a determination of those fields counted by F(h). The unconditional computation of F(h) for h up to 100 was completed by M. Watkins, using ideas of Goldfeld and Gross-Zagier; Soundararajan has more recently made conjectures about the order of magnitude of F(h) as h increases without bound, and determined its average order. In the present paper, we refine Soundararajan's conjecture to a conjectural asymptotic formula and also consider the subtler problem of determining the number F(G) of imaginary quadratic fields with class group isomorphic to a given finite abelian group G. Using Watkins' tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance the elementary abelian group of order 27 does not). This observation is explained in part by the Cohen-Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine heuristics of Cohen-Lenstra together with our refinement of Soundararajan's conjecture to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of "missing" class groups, for the case of p-groups as p tends to infinity. Furthermore, conditionally on the Generalized Riemann Hypothesis, we extend Watkins' data, tabulating F(h) for odd h up to 10^6 and F(G) for G a p-group of odd order with |G| up to 10^6. The numerical evidence matches quite well with our conjectures.
We explore the behavior of character varieties under Dehn filling. Specifically, if M is a two-cusped hyperbolic 3-manifold, we consider the families of Dehn filling of one cusp of M. We write M(-,p/q) for these one cusped manifolds. First, we show that the gonality of the canonical component X_0(M(-,p/q)) of the SL(2,C) character variety is bounded by a constant independent of p/q. (The gonality of an algebraic curve C is the minimal degree of a dense map from C to C.) We show that the genus of X_0(M(-,p/q)) is bounded above by a constant times max{p^2,q^2} and that the degree of the image of the canonical component in the A-polynomial curve has degree bounded above by a constant times max{|p|, |q|}.
We compute the full SL(2,C) character varieties of the two-component double twist links. The canonical components are surfaces birational to the product of a curve and C. We compute these curves and their genera, and the degree of irrationality of the surface. For the J (2m + 1, 3) links, we go into more depth. Namely, we show that the character variety of the double twist link J(2m + 1, 3) is the conic bundle over the projective line P1 which is isomorphic to the surface obtained from P1 P1 by repeating a one-point blowup 9m timesif m>0,and (6+9m) times if m<-1. For m>0,J(2m+1,3)is the link obtained by 1/m Dehn surgery on the Magic manifold, confirming a conjecture of E. Landes.
We determine the A-polynomial of the double twist knots (denoted J(k,l)) as 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 recusrively, and determine a recursive formula for the canonical component of the A-polynomial for the J(2n,2n) knots.
Arithmetic Groups and Lehmers Conjecture Institut Mittag-Leffler preprint series, Workshop on Mahler Measure and Growth in Geometry and Topology (2013).
In this non-refereed expository article, we discuss the construction of arithmetic Fuch- sian and arithmetic Kleinian groups. We extend this to a generalized family of arith- metic groups, using a construction from quaternion algebras. A connection between the geodesic lengths and Lehmers conjecture is then discussed.
(with K. Baker) Character Varieties of Once-Punctured Torus Bundles with Tunnel Number One Internat. J. Math., 24 (2013), No. 06. 1350048, 57pp.
We determine the PSL(2,C) and SL(2,C) character varieties of the once-punctured torus bundles with tunnel number one, i.e. the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link exterior. In particular, we determine `natural' models for these algebraic sets, identify them up to birational equivalence with smooth models, and compute the genera of the canonical components. This enables us to compare dilatations of the monodromies of these bundles with these genera. We also determine the minimal polynomials for the trace fields of these manifolds. Additionally we study the action of the symmetries of these manifolds upon their character varieties, identify the characters of their lens space fillings, and compute the twisted Alexander polynomials for their representations to SL(2,C).
(with M. R. Murty) A Bombieri-Vinogradov Theorem for All Number Fields Trans. Amer. Math. Soc. 365 (2013), no. 9, 4987-5032.
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.
(with M. R. Murty) The Euclidean Algorithm for Number Fields and Primitive Roots Proc. Amer. Math. Soc. 141 (2013), 181-190.
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.
(with L. Fukshansky) On Well-Rounded Ideal Lattices Int. J. Number Theory Vol 8 (2012) Issue 1:189-206.
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.
(with M. Macasieb and R. van Luijk) Character Varieties of a Family of Two-Bridge Knots Proc. London Math. Soc. (2011) 103(3): 473-507
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.
(with C. D. Sinclair) Equidistribution of Algebraic Numbers of Norm One in Quadratic Number Fields Int. J. Number Theory Volume: 7, Issue: 7(2011) pp. 1841-1861.
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 .
On Character Varieties of Certain Families of 3-Manifolds. Oberwolfach reports Volume 7, Issue 3, 2010
(with M. R. Murty) The Generalized Artin Conjecture and Arithmetic Orbifolds. Groups and symmetries, 259--265, CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009.
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.
Counting Cusps of Subgroups of PSL(2,OK) Proc. Amer. Math. Soc. 136 (2008), 2387-2393
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.
(with C. D. Sinclair) Conjugate Reciprocal Polynomials with all Roots on the Unit Circle Canad. J. Math. (2008) 60, no. 5, 1149-1167.
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.
One-Cusped Congruence Groups of Bianchi Groups Mathematische Annalen (2007) 338, no 2, 249 - 282.
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.
Non-Profit Alternatives to Commercial Academic Journals: Success Stories from Mathematics. Political Geography Volume 31, Issue 5, June 2012, Pages 263-265
This is a solicited editorial response concerning the current publishing situation in mathematics.
(with N.C. Craig and D. C. McKean) Vibrational and Quantum Chemical Studies of 1,2-Difluoroethylenes: Spectra of 1,2-13 C2H2F2 Species, Scaled Force Fields, and Dipole Derivatives The Journal of Physical Chemistry A, (2002) Vol. 106, No. 26, pp. 6358-6369
These molecules are interesting, as contrary to most, the cis- structure is more common than the trans- structure. Using IR and Raman spectroscopy, we compute the exact structure of these molecules.
One-cusped Congruence Subgroups of PSL(2,OK). PhD thesis, University of Texas, 2005
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.