Let E be an elliptic curve, and let Ln be the Kummer extension generated by a primitive pnth root of unity and a pn-th root of a for a fixed a ∈ Q − {±1}. A detailed case study by Coates, Fukaya, Kato and Sujatha and V. Dokchitser has led these authors to predict unbounded and strikingly regular growth for the rank of E over Ln in certain cases. The aim of this note is to explain how some of th...

We consider the use of generalized Ding-Helleseth cyclotomy to design sequences over the finite field of order four. Using generalized cyclotomic classes of order four we obtain the family of balanced sequences of odd period with high linear complexity. Also we present a method of constructing sequences with high linear complexity and arbitrary even period over the finite field of order four. T...

Let K be a cyclic Galois extension of degree p over a field F containing a primitive pth root of unity. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F ) such that corresponding normal subgroups are indecomposable Fp[Gal(K/F )]modules. For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on au...

Let K be a cyclic Galois extension of degree p over a field F containing a primitive pth root of unity. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F ) such that corresponding normal subgroups are indecomposable Fp[Gal(K/F )]modules. For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on au...

In 1977 Kervaire and Murthy presented three conjectures regarding K0ZCpn , where Cpn is the cyclic group of order p n and p is a semi-regular prime that is p does not divide h (regular p does not divide the class number h = hh). The Mayer-Vietoris exact sequence provides the following short exact sequence 0 → Vn → PicZCpn → ClQ(ζn−1)× PicZCpn−1 → 0 Here ζn−1 is a primitive p -th root of unity. ...

The algebra su 2 is derived from two commuting quon algebras for which the parameter q is a root of unity. This leads to a polar decomposition of the shift operators J + and J − of the group SU 2 (with J + = J † − = HUr where H is Hermitean and Ur unitary). The Wigner-Racah algebra of SU 2 is developed in a new basis arising from the simultanenous diagonalization of the commuting operators J 2 ...

In his work on Wiener-Hopf determinants, A. Böttcher came across what he termed a “mysterious” identity that evaluates a certain sum of a rational function of a primitive root of unity in terms of the Barnes Gfunction, which was later generalized by Basor and Forrester. We give a direct proof of Böttcher’s identity and its many generalizations by using elements of the theory of symmetric functi...

One trivial zero phenomenon for p-adic analytic function is considered. We then prove that the first derivative of this function is essentially the Kummer class associated with p. 1. Introduction. In this paper we always fix an odd prime p > 2. For n ≥ 1, fix a p n th primitive root of unity ζ p n such that ζ