28,274 research outputs found
Computation of 2-groups of positive classes of exceptional number fields
We present an algorithm for computing the 2-group of the positive divisor
classes of a number field F in case F has exceptional dyadic places. As an
application, we compute the 2-rank of the wild kernel WK2(F) in K2(F) for such
number fields
The complexity of class polynomial computation via floating point approximations
We analyse the complexity of computing class polynomials, that are an
important ingredient for CM constructions of elliptic curves, via complex
floating point approximations of their roots. The heart of the algorithm is the
evaluation of modular functions in several arguments. The fastest one of the
presented approaches uses a technique devised by Dupont to evaluate modular
functions by Newton iterations on an expression involving the
arithmetic-geometric mean. It runs in time for any , where
is the CM discriminant and is the degree of the class polynomial.
Another fast algorithm uses multipoint evaluation techniques known from
symbolic computation; its asymptotic complexity is worse by a factor of . Up to logarithmic factors, this running time matches the size of the
constructed polynomials. The estimate also relies on a new result concerning
the complexity of enumerating the class group of an imaginary-quadratic order
and on a rigorously proven upper bound for the height of class polynomials
The Computation of the Logarithmic Cohomology for Plane Curves
We give algorithms of computing bases of logarithmic cohomology groups for
square-free polynomials in two variables. (Fixed typos of v1)Comment: 19 page
Computing Hilbert Class Polynomials
We present and analyze two algorithms for computing the Hilbert class
polynomial . 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 computing , and we show that all methods have comparable
run times
Logarithmic Conformal Field Theory: Beyond an Introduction
This article aims to review a selection of central topics and examples in
logarithmic conformal field theory. It begins with a pure Virasoro example,
critical percolation, then continues with a detailed exposition of symplectic
fermions, the fractional level WZW model on SL(2;R) at level -1/2 and the WZW
model on the Lie supergroup GL(1|1). It concludes with a general discussion of
the so-called staggered modules that give these theories their logarithmic
structure, before outlining a proposed strategy to understand more general
logarithmic conformal field theories. Throughout, the emphasis is on continuum
methods and their generalisation from the familiar rational case. In
particular, the modular properties of the characters of the spectrum play a
central role and Verlinde formulae are evaluated with the results compared to
the known fusion rules. Moreover, bulk modular invariants are constructed, the
structures of the corresponding bulk state spaces are elucidated, and a
formalism for computing correlation functions is discussed.Comment: Invited review by J Phys A for a special issue on LCFT; v2 updated
references; v3 fixed a few minor typo
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