334 research outputs found
A Carlitz module analogue of the Grunwald--Wang theorem
The classical Grunwald--Wang theorem is an example of a local--global (or
Hasse) principle stating that except in some special cases which are precisely
determined, an element in a number field is an -th power in
if and only if it is an -th power in the completion
for all but finitely many primes of . In this
paper, we prove a Carlitz module analogue of the Grunwald--Wang theorem
Carlitz module analogues of Mersenne primes, Wieferich primes, and certain prime elements in cyclotomic function fields
In this paper, we introduce a Carlitz module analogue of Mersenne primes, and
prove Carlitz module analogues of several classical results concerning Mersenne
primes. In contrast to the classical case, we can show that there are
infinitely many composite Mersenne numbers. We also study the acquaintances of
Mersenne primes including Wieferich and non-Wieferich primes in the Carlitz
module context that were first introduced by Dinesh Thakur
Certain sets over function fields are polynomial families
In 1938, Skolem conjectured that is not a
polynomial family for any . Carter and Keller disproved Skolem's
conjecture for all by proving that is
boundedly generated by the elementary matrices, and hence a polynomial family
for any . Only recently, Vaserstein refuted Skolem's conjecture
completely by showing that is a polynomial family.
An immediate consequence of Vaserstein's theorem also implies that
is a polynomial family for any . In this
paper, we prove a function field analogue of Vaserstein's theorem: that is, if
is the ring of polynomials over a finite field of odd
characteristic, then is a polynomial family in 52
variables. A consequence of our main result also implies that
is a polynomial family for any .Comment: Final versio
Generalized Mordell curves, generalized Fermat curves, and the Hasse principle
A generalized Mordell curve of degree over \bQ is the smooth
projective model of the affine curve of the form , where are nonzero integers. A generalized Fermat curve of signature
with over \bQ is the smooth projective curve of the form for some nonzero integers . In this paper, we show
that for each prime with and ,
there exists a threefold \cX_p \subseteq \bP^6 such that certain rational
points on \cX_p produce infinite families of non-isomorphic generalized
Mordell curves of degree and infinite families of generalized Fermat
curves of signature for each that are
counterexamples to the Hasse principle explained by the Brauer-Manin
obstruction. We also show that the set of special rational points on \cX_p
producing generalized Mordell curves and generalized Fermat curves that are
counterexamples to the Hasse principle is infinite, and can be constructed
explicitly.Comment: 45 page
Algebraic families of hyperelliptic curves violating the Hasse principle
In , Colliot-Th\'el\`ene and Poonen showed how to construct algebraic
families of genus one curves violating the Hasse principle. Poonen explicitly
constructed an algebraic family of genus one cubic curves violating the Hasse
principle using the general method developed by Colliot-Th\'el\`ene and
himself. The main result in this paper generalizes the result of
Colliot-Th\'el\`ene and Poonen to arbitrarily high genus hyperelliptic curves.
More precisely, for and , we show that there
is an algebraic family of hyperelliptic curves of genus that is
counterexamples to the Hasse principle explained by the Brauer-Manin
obstruction
Representation of units in cyclotomic function fields
Hilbert's Satz 90 tells us that for a given cyclic extension , a unit of
norm in can be written as a quotient of conjugate elements in . For
the extensions with prime , Newman
proved a refinement of Hilbert's Satz 90 that gives a sufficient and necessary
condition for which a unit of norm in can be written
as a quotient of conjugate units. In order to obtain this result, Newman proved
a stronger result that gives a unique representation of units of norm as a
product of a power of with a quotient of
conjugate units, where is a given primitive root modulo . In this paper,
we obtain a function field analogue of Newman's result for the -th
cyclotomic function field extensions , where
is a monic prime in . As a consequence, we proved a
refinement of Hilbert's Satz 90 for the extensions
that gives a sufficient and necessary
condition for which a unit of norm in can be written as
a quotient of conjugate units
The classical umbral calculus, and the flow of a Drinfeld module
David Goss developed a very general Fourier transform in additive harmonic
analysis in the function field setting. In order to introduce the Fourier
transform for continuous characteristic valued functions on ,
Goss introduced and studied an analogue of flows in finite characteristic. In
this paper, we use another approach to study flows in finite characteristic. We
recast the notion of a flow in the language of the classical umbral calculus,
which allows to generalize the formula for flows first proved by Goss to a more
general setting. We study duality between flows using the classical umbral
calculus, and show that the duality notion introduced by Goss seems a natural
one. We also formulate a question of Goss about the exact relationship between
two flows of a Drinfeld module in the language of the classical umbral
calculus, and give a partial answer to it.Comment: Final versio
Function field analogues of Bang-Zsigmondy's theorem and Feit's theorem
In the number field context, Bang-Zsigmondy's theorem states that for any
integers , there exists a prime divisor of such that
does not divide for every integer except in some
exceptional cases that can be explicitly determined. A prime satisfying the
conditions in Bang-Zsigmondy's theorem is called a Zsigmondy prime for .
In 1988, Feit introduced the notion of large Zsigmondy primes as follows: A
Zsigmondy prime for is called a large Zsigmondy prime if either or divides . In the same year, Feit proved a refinement
of Bang-Zsigmondy's theorem which states that for any integers ,
there exists a large Zsigmondy prime for except in some exceptional
cases that can be explicitly determined.
In this paper, we introduce notions of Zsigmondy primes and large Zsigmondy
primes in the Carlitz module context, and prove function field analogues of
Bang-Zsigmondy's theorem and Feit's theorem.Comment: Final versio
Waring's problem for unipotent algebraic groups
In this paper, we formulate an analogue of Waring's problem for an algebraic
group . At the field level we consider a morphism of varieties and ask whether every element of is the product of a
bounded number of elements . We give an affirmative
answer when is unipotent and is a characteristic zero field which is
not formally real.
The idea is the same at the integral level, except one must work with
schemes, and the question is whether every element in a finite index subgroup
of can be written as a product of a bounded number of elements
of . We prove this is the case when is unipotent and
is the ring of integers of a totally imaginary number field
Towards one-shot learning for rare-word translation with external experts
Neural machine translation (NMT) has significantly improved the quality of
automatic translation models. One of the main challenges in current systems is
the translation of rare words. We present a generic approach to address this
weakness by having external models annotate the training data as Experts, and
control the model-expert interaction with a pointer network and reinforcement
learning. Our experiments using phrase-based models to simulate Experts to
complement neural machine translation models show that the model can be trained
to copy the annotations into the output consistently. We demonstrate the
benefit of our proposed framework in outof-domain translation scenarios with
only lexical resources, improving more than 1.0 BLEU point in both translation
directions English to Spanish and German to EnglishComment: 2nd Workshop on Neural Machine Translation and Generation, ACL 201
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