185 research outputs found
Computing the Characteristic Polynomial of a Finite Rank Two Drinfeld Module
Motivated by finding analogues of elliptic curve point counting techniques,
we introduce one deterministic and two new Monte Carlo randomized algorithms to
compute the characteristic polynomial of a finite rank-two Drinfeld module. We
compare their asymptotic complexity to that of previous algorithms given by
Gekeler, Narayanan and Garai-Papikian and discuss their practical behavior. In
particular, we find that all three approaches represent either an improvement
in complexity or an expansion of the parameter space over which the algorithm
may be applied. Some experimental results are also presented
A Riemann Hypothesis for characteristic p L-functions
We propose analogs of the classical Generalized Riemann Hypothesis and the
Generalized Simplicity Conjecture for the characteristic p L-series associated
to function fields over a finite field. These analogs are based on the use of
absolute values. Further we use absolute values to give similar reformulations
of the classical conjectures (with, perhaps, finitely many exceptional zeroes).
We show how both sets of conjectures behave in remarkably similar ways.Comment: This is the final version (with new title) as it will appear in the
Journal of Number Theor
Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields
The fastest known algorithm for factoring univariate polynomials over finite
fields is the Kedlaya-Umans (fast modular composition) implementation of the
Kaltofen-Shoup algorithm. It is randomized and takes time to factor polynomials of degree over the finite field
with elements. A significant open problem is if the
exponent can be improved. We study a collection of algebraic problems and
establish a web of reductions between them. A consequence is that an algorithm
for any one of these problems with exponent better than would yield an
algorithm for polynomial factorization with exponent better than
Algorithms for computing norms and characteristic polynomials on general Drinfeld modules
We provide two families of algorithms to compute characteristic polynomials
of endomorphisms and norms of isogenies of Drinfeld modules. Our algorithms
work for Drinfeld modules of any rank, defined over any base curve. When the
base curve is , we do a thorough study of the
complexity, demonstrating that our algorithms are, in many cases, the most
asymptotically performant. The first family of algorithms relies on the
correspondence between Drinfeld modules and Anderson motives, reducing the
computation to linear algebra over a polynomial ring. The second family,
available only for the Frobenius endomorphism, is based on a new formula
expressing the characteristic polynomial of the Frobenius as a reduced norm in
a central simple algebra
Zero modes' fusion ring and braid group representations for the extended chiral su(2) WZNW model
The zero modes' Fock space for the extended chiral WZNW model gives
room to a realization of the Grothendieck fusion ring of representations of the
restricted quantum universal enveloping algebra (QUEA) at an even
(-th) root of unity, and of its extension by the Lusztig operators. It is
shown that expressing the Drinfeld images of canonical characters in terms of
Chebyshev polynomials of the Casimir invariant allows a streamlined
derivation of the characteristic equation of from the defining relations of
the restricted QUEA. The properties of the fusion ring of the Lusztig's
extension of the QUEA in the zero modes' Fock space are related to the braiding
properties of correlation functions of primary fields of the extended
current algebra model.Comment: 36 pages, 1 figure; version 3 - improvements in Sec. 2 and 3:
definitions of the double, as well as R- (and M-)matrix changed to fit the
zero modes' one
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