Kathleen L. Petersen

On character varieties of two-bridge knot groups

We find explicit models for the PSL2(C)and SL2(C)-character varieties of the fundamental groups of complements in S 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.

A Bombieri-Vinogradov theorem for all number fields

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. Dirichlet’s density theorem gives an asymptotic estimate for the density of primes in arithmetic progressions. Let π(x) denote the number of primes p ≤ x, and for positive integers a ≤ q such that (a, q) = 1, denote by π(x, q, a) the number of primes p ≤ x which satisfy the congruence p ≡ a (mod q). Dirichlet’s theorem indicates that as x→∞ π(x, q, a) ∼ π(x) φ(q) where φ is Euler’s totient function, and f ∼ g means that f/g → 1. The Riemann hypothesis for all Dirichlet L-functions implies that the error term satisfies the estimate ∣∣∣∣π(x, q, a)− π(x) φ(q) ∣∣∣∣ x 1 2 log qx, where f g (equivalently f = O(g)) means that |f/g| is bounded, and will be referred to by saying that f is of order g. The celebrated theorem of Bombieri [3] and Vinogradov [24] shows that this estimate holds on the average. Theorem 0.1 (Bombieri, Vinogradov). Let A > 0 be given. Then there is a B = B(A) > 0 so that for Q = x 1 2 (log x)−B ∑ q≤Q max (a,q)=1 max y≤x ∣∣∣π(y, q, a)− π(y) φ(q) ∣∣∣ x (log x)A . In [17], a variant of Theorem 0.1 is proven for K/Q a Galois extension of number fields. The goal of this paper is to prove an analogous theorem without the Galois assumption. Specifically, let K be a number field and M a subfield of K (possibly M = K) such that K/M is Galois. Let G = Gal(K/M) and let C be a conjugacy class in G. Let p be a prime ideal of M unramified in K and let σp denote the conjugacy class of Frobenius automorphisms corresponding to prime ideals q of K lying over p. Define π(x,C) to be the number of prime ideals p of M unramified in K with σp = C and Np ≤ x where N = NM/Q. With a, q as above let π(x,C, q, a) denote the number of primes ideals p of M unramified in K with σp = C, Np ≤ x and Np ≡ a (mod q). By the Chebotarev density theorem [23] π(x,C, q, a) ∼ d(C, q, a)π(x) for a density d(C, q, a) ≥ 0. If K ∩Q(ζq) = Q where ζq is a primitive q-th root of unity, then d(C, q, a) = |C| |G| 1 φ(q) . We prove the following theorem, which is equivalent to Theorem 4.1 proven in §4.

Conjugate Reciprocal Polynomials with all Roots on the Unit Circle

AbstractWe study the geometry, topology and Lebesgue measure of the set of monic conju-gate reciprocal polynomials of fixed degree with all roots on the unit circle. The set ofsuch polynomials of degree N is naturally associated to a subset of R N−1 . We calculatethe volume of this set, prove the set is homeomorphic to the N − 1 ball and that itsisometry group is isomorphic to the dihedral group of order 2N. 1 Introduction Let N be a positive integer and suppose f(x) is a polynomial in C[x] of degree N. If fsatisfies the identity,(1.1) f(x) = x N f(1/x),then fis said to be conjugate reciprocal, or simply CR. Furthermore, if fis given byf(x) = x N +X Nn=1 c n x N−n .then (1.1) implies that c N = 1,c N−n = c n for 1 ≤ n≤ N− 1 and, if αis a zero of fthen so too is 1/α. The purpose of this manuscript is to study the set of CR polynomialswith all roots on the unit circle. The interplay between the symmetry condition on thecoefficients and the symmetry of the roots allows for a number of interesting theoremsabout the geometry, topology and Lebesgue measure of this set.CR polynomials have various names in the literature including reciprocal, self-reciprocaland self-inversive (though we reserve the term reciprocal for polynomials which satisfy anidentity akin to (1.1) except without both instances of complex conjugation).Thecondition onthe coefficientsofa conjugatereciprocalpolynomialallowsusto identifythe set of CR polynomials with R

Character varieties of double twist links

We compute both natural and smooth models for the

The Euclidean algorithm for number fields and primitive roots

SL_2(\mathbb C)

Lower Bounds for Heights in Relative Galois Extensions

character varieties of the two component double twist links, an infinite family of two-bridge links indexed as

EQUIDISTRIBUTION OF ALGEBRAIC NUMBERS OF NORM ONE IN QUADRATIC NUMBER FIELDS

J(k,l)

Vibrational and quantum chemical studies of 1,2-difluoroethylenes: Spectra of 1,2-13C2H2F2 species, scaled force fields, and dipole derivatives

. For each

Non-profit alternatives to commercial academic journals: Success stories from mathematics

J(k,l)

COUNTING CUSPS OF SUBGROUPS OF PSL2(OK)

, the component(s) of the character variety containing characters of irreducible representations are birational to a surface of the form