In [l] Bagchi and Bagchi conjectured that one of the hypotheses of their theorem 2(d) is superfluous. This turns out to be true: PROPOSITION. Let 4 = 9 mod 16 be a prime power such that 2 is a fourth power in Fq. Then 1 f fi are nonsquares in Fp. Pruc$ Let p be a primitive eighth root of unity in Fq (note that /3 is a nonsquare). Because of (02 + 1)2 = p4 + 202 + 1 = 202 we obtain that (0 * + 1...

Single mass scale diagrams: construction of a basis for the ε-expansion. Abstract Exploring the idea of Broadhurst on the " sixth root of unity " we present an ansatz for construction of a basis of transcendental numbers for the ε-expansion of single mass scale diagrams with two particle massive cut. As example, several new two-and three-loop master integrals are calculated.

As usual, we write Z,Q,Fp,C for the ring of integers, the field of rational numbers, the finite field with p elements and the field of complex numbers respectively. If Z is a smooth algebraic variety over an algebraically closed field then we write Ω(Z) for the space of differentials of the first kind on Z. If Z is an abelian variety then we write End(Z) for its ring of (absolute) endomorphisms...

We prove a Morita reduction theorem for the cyclotomic Hecke algebras Hr,p,n(q,Q) of type G(r, p, n). As a consequence, we show that computing the decomposition numbers of Hr,p,n(Q) reduces to computing the psplittable decomposition numbers (see Definition 1.1) of the cyclotomic Hecke algebras Hr′,p′,n′ (Q ), where 1 ≤ r ≤ r, 1 ≤ n ≤ n, p | p and where the parameters Q are contained in a single...

d The action of the derivation e = q ~ on the q-expansions of modular forms in characteristic p is one of the fundamental tools in the Serre/Swinnerton-Dyer theory of mod p modular forms. In this note, we extend the basic results about this action, already known for P > 5 and level one, to arbitrary p and arbitrary prime-to-p level. !. Review of modular forms in characteristic p We fix an algeb...

In [1], we proposed a model for quantized discrete general relativity with a Euclidean signature. The model was constructed by combining the structure of a certain tensor category and of a full subcategory of it with the combinatorics of a triangulated 4-manifold. The category was the representations of Uqso(4), for q a 4n-th root of unity and the subcategory the representations which are calle...

Let Q(ζ) be the cyclotomic field obtained from Q by adjoining a primitive seventh root of unity ζ. Normalized primary elements of this field are characterized and related to Jacobi sums and to solutions of a system of quadratic Diophantine equations of Dickson type involving a rational prime p ≡ 1 (mod 7). These objects and their connection are then used to give another formulation of the compl...

Let p ≡ 1 (mod 4) be a prime number and let ζ = e be a primitive root of unity. Then there exists a unique biquadratic extension eld Q(y)/Q that is a sub eld of Q(ζ). The aim of this work is to construct an algorithm for nding such y explicitly. Finally we state a general conjecture about the y we found.

In this paper, we show that the problem of computing the complex roots of unity is not as simple as it seems at rst. In particular, the formulas given in a standard programmer's reference book Knuth, Seminumerical Algorithms, 1981] are shown to be numerically unstable , giving unacceptably large error for moderate sized sequences. We give alternative formulas, which we show to be superior both ...