76,774 research outputs found
Computing in Jacobians of projective curves over finite fields
We give algorithms for computing with divisors on projective curves over
finite fields, and with their Jacobians, using the algorithmic representation
of projective curves developed by Khuri-Makdisi. We show that many desirable
operations can be done efficiently in this setting: decomposing divisors into
prime divisors; computing pull-backs and push-forwards of divisors under finite
morphisms, and hence Picard and Albanese maps on Jacobians; generating
uniformly random divisors and points on Jacobians; computing Frobenius maps and
Kummer maps; and finding a basis for the -torsion of the Picard group, where
is a prime number different from the characteristic of the base field.Comment: 42 page
Computing Super Matrix Invariants
In [Trace identities and -graded invariants, {\it Trans.
Amer. Math. Soc. \bf309} (1988), 581--589] we generalized the first and second
fundamental theorems of invariant theory from the general linear group to the
general linear Lie superalgebra. In the current paper we generalize the
computations of the the numerical invariants (multiplicities and Poincar\'e
series) to the superalgebra case. The results involve either inner products of
symmetric functions in two sets of variables, or complex integrals. we
generalized the first and second fundamental theorems of invariant theory from
the general linear group to the general linear Lie superalgebra. In the current
paper we generalize the computations of the the numerical invariants
(multiplicities and Poincar\'e series) to the superalgebra case. The results
involve either inner products of symmetric functions in two sets of variables,
or complex integrals
Modular Invariance and Uniqueness of Conformal Characters
We show that the conformal characters of various rational models of
W-algebras can be already uniquely determined if one merely knows the central
charge and the conformal dimensions. As a side result we develop several tools
for studying representations of SL(2,Z) on spaces of modular functions. These
methods, applied here only to certain rational conformal field theories, may be
useful for the analysis of many others.Comment: 21 pages (AMS TeX), BONN-TH-94-16, MPI-94-6
Diophantine definability of infinite discrete non-archimedean sets and Diophantine models over large subrings of number fields
We prove that infinite p-adically discrete sets have Diophantine definitions
in large subrings of some number fields. First, if K is a totally real number
field or a totally complex degree-2 extension of a totally real number field,
then there exists a prime p of K and a set of K-primes S of density arbitrarily
close to 1 such that there is an infinite p-adically discrete set that is
Diophantine over the ring O_{K,S} of S-integers in K. Second, if K is a number
field over which there exists an elliptic curve of rank 1, then there exists a
set of K-primes S of density 1 and an infinite Diophantine subset of O_{K,S}
that is v-adically discrete for every place v of K. Third, if K is a number
field over which there exists an elliptic curve of rank 1, then there exists a
set of K-primes S of density 1 such that there exists a Diophantine model of Z
over O_{K,S}. This line of research is motivated by a question of Mazur
concerning the distribution of rational points on varieties in a
non-archimedean topology and questions concerning extensions of Hilbert's Tenth
Problem to subrings of number fields.Comment: 17 page
Point counting on curves using a gonality preserving lift
We study the problem of lifting curves from finite fields to number fields in
a genus and gonality preserving way. More precisely, we sketch how this can be
done efficiently for curves of gonality at most four, with an in-depth
treatment of curves of genus at most five over finite fields of odd
characteristic, including an implementation in Magma. We then use such a lift
as input to an algorithm due to the second author for computing zeta functions
of curves over finite fields using -adic cohomology
Fusion Algebras and Characters of Rational Conformal Field Theories
We introduce the notion of (nondegenerate) strongly-modular fusion algebras.
Here strongly-modular means that the fusion algebra is induced via Verlinde's
formula by a representation of the modular group whose kernel contains a
congruence subgroup. Furthermore, nondegenerate means that the conformal
dimensions of possibly underlying rational conformal field theories do not
differ by integers. Our first main result is the classification of all
strongly-modular fusion algebras of dimension two, three and four and the
classification of all nondegenerate strongly-modular fusion algebras of
dimension less than 24. Secondly, we show that the conformal characters of
various rational models of W-algebras can be determined from the mere knowledge
of the central charge and the set of conformal dimensions. We also describe how
to actually construct conformal characters by using theta series associated to
certain lattices. On our way we develop several tools for studying
representations of the modular group on spaces of modular functions. These
methods, applied here only to certain rational conformal field theories, are in
general useful for the analysis rational models.Comment: 87 pages, AMS TeX, one postscript figure, one exceptional case added
to Main theorem 2, some typos correcte
Quantum cohomology of flag manifolds and Toda lattices
We discuss relations of Vafa's quantum cohomology with Floer's homology
theory, introduce equivariant quantum cohomology, formulate some conjectures
about its general properties and, on the basis of these conjectures, compute
quantum cohomology algebras of the flag manifolds. The answer turns out to
coincide with the algebra of regular functions on an invariant lagrangian
variety of a Toda lattice.Comment: 35 page
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