This paper is a continuation of our papers EK1, EK2]. In EK2] we showed that for the root system A n1 one can obtain Macdonald's polynomials { a new interesting class of symmetric functions recently deened by I. Macdonald M1] { as weighted traces of intertwining operators between certain nite-dimensional representations of U q sl n. The main goal of the present paper is to use this construction...

We present and analyze two algorithms for computing the Hilbert class polynomial HD. The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CM-curves over finite fields. Our run time analysis gives tighter bounds for the complexity of all known algorithms for comput...

It is known that six Painlevé equations are in general irreducible, namely, their solutions cannot be expressed by “classical functions” in the sense of Umemura [18]. However, it is also known that they admit classical solutions for special values of parameters except for PI. Much effort have been made for the investigation of classical solutions. As a result, it has been recognized that there ...

The exact solution of the Dorokhov-Mello-Pereyra-Kumar-equation for quasi one-dimensional disordered conductors in the unitary symmetry class is employed to calculate all m-point correlation functions by a generalization of the method of orthogonal polynomials. We obtain closed expressions for the first two conductance moments which are valid for the whole range of length scales from the metall...

Superconformal Ward identities are derived for the the four point functions of chiral primary BPS operators for N = 2, 4 superconformal symmetry in four dimensions. Manipulations of arbitrary tensorial fields are simplified by introducing a null vector so that the four point functions depend on two internal R-symmetry invariants as well as two conformal invariants. The solutions of these identi...

— In [22], the authors proved an explicit formula for the arithmetic intersection number (CM(K).G1) on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field K. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. One of the...

We give an algorithm that computes the genus two class polynomials of a primitive quartic CM field K, and we give a runtime bound and a proof of correctness of this algorithm. This is the first proof of correctness and the first runtime bound of any algorithm that computes these polynomials. Our algorithm uses complex analysis and runs in time e O(∆), where ∆ is the discriminant of K.