Theorem 1.1. There exists a holomorphic map σ of C of the form ξ → λξ + O(2), η → λη + O(2), with λ not a root of unity and |λ| = 1, such that σ is reversible by an antiholomorphic involution and by a formal holomorphic involution, and is however not reversible by any C-smooth involution of which the linear part is holomorphic. In particular, the σ is not reversible by any holomorphic involution.

In [1], Vazirani and I gave a new interpretation of what we called ℓ-partitions, also known as (ℓ, 0)-Carter partitions. The primary interpretation of such a partition λ is that it corresponds to a Specht module S λ which remains irreducible over the finite Hecke algebra Hn(q) when we specialize q to a primitive ℓ th root of unity. In [1], we relied heavily on the description of such a partitio...

Abstract: We provide, with proofs, a complete description of the authors’ construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of Re...

Let p be an odd prime number and F a field containing a primitive pth root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group GF of F . Namely, the third subgroup G (3) F in the descending p-central sequence of GF is the intersection of all open normal subgroups N such that GF /N is 1, Z/p , or the extra-special group Mp3 of order p and exponent p.

It is shown that a linear combination of roots of unity with rational coefficients generates a large subfield of the field generated by the set of roots of unity involved, except when certain partial sums vanish. Some related results about polygons with all sides and angles rational are also proved.

Let p be the characteristic of Fq and let q be a primitive root modulo a prime r = 2n + 1. Let β ∈ Fq2n be a primitive rth root of unity. We prove that the multiplicative order of the Gauss period β + β−1 is at least (log p)c logn for some c > 0. This improves the bound obtained by Ahmadi, Shparlinski and Voloch when p is very large compared with n. We also obtain bounds for ”most” p.

This paper gives examples of hyperbolic 3-manifolds whose SL(2,C) character varieties have ideal points for which the associated roots of unity are not ±1. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to whether roots of unity other than ±1 occur.

A series all of whose coeecients have unit modulus is called an Hadamard square root of unity. We investigate and partially characterise the algebraic Hadamard square roots of unity. The investigation makes use of a result about the asymptotic behavior of the coeecients of algebraic series and the Weyl-von Neumann theorem.

We study the structure of the Kauffman algebra of a surface with parameter equal to √ −1. We obtain an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat SU(2)-connections over the surface. We analyse the asymptotics of traces of curve-operators in TQFT in non standard regimes where the root of unity par...

For fixed complex q with |q| > 1, the q-logarithm Lq is the meromorphic continuation of the series ∑ n>0 z / q −1 , |z| < |q|, into the whole complex plane. IfK is an algebraic number field, one may ask if 1, Lq 1 , Lq c are linearly independent over K for q, c ∈ K× satisfying |q| > 1, c / q, q2, q3, . . .. In 2004, Tachiya showed that this is true in the Subcase K Q, q ∈ Z, c −1, and the prese...