2,205 research outputs found
Functional Decomposition using Principal Subfields
Let be a univariate rational function. It is well known that any
non-trivial decomposition , with , corresponds to a
non-trivial subfield and vice-versa. In
this paper we use the idea of principal subfields and fast
subfield-intersection techniques to compute the subfield lattice of
. This yields a Las Vegas type algorithm with improved complexity
and better run times for finding all non-equivalent complete decompositions of
.Comment: 8 pages, accepted for ISSAC'1
Regulator constants and the parity conjecture
The p-parity conjecture for twists of elliptic curves relates multiplicities
of Artin representations in p-infinity Selmer groups to root numbers. In this
paper we prove this conjecture for a class of such twists. For example, if E/Q
is semistable at 2 and 3, K/Q is abelian and K^\infty is its maximal pro-p
extension, then the p-parity conjecture holds for twists of E by all orthogonal
Artin representations of Gal(K^\infty/Q). We also give analogous results when
K/Q is non-abelian, the base field is not Q and E is replaced by an abelian
variety. The heart of the paper is a study of relations between permutation
representations of finite groups, their "regulator constants", and
compatibility between local root numbers and local Tamagawa numbers of abelian
varieties in such relations.Comment: 50 pages; minor corrections; final version, to appear in Invent. Mat
Infinite dimensional non-positively curved symmetric spaces of finite rank
This paper concerns a study of three families of non-compact type symmetric
spaces of infinite dimension. Although they have infinite dimension they have
finite rank. More precisely, we show they have finite telescopic dimension. We
also show the existence of Furstenberg maps for some group actions on these
spaces. Such maps appear as a first step toward superrigidity results.Comment: Some references have been adde
On a BSD-type formula for L-values of Artin twists of elliptic curves
This is an investigation into the possible existence and consequences of a
Birch-Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted
by Artin representations. We translate expected properties of L-functions into
purely arithmetic predictions for elliptic curves, and show that these force
some peculiar properties of the Tate-Shafarevich group, which do not appear to
be tractable by traditional Selmer group techniques. In particular we exhibit
settings where the different p-primary components of the Tate-Shafarevich group
do not behave independently of one another. We also give examples of
"arithmetically identical" settings for elliptic curves twisted by Artin
representations, where the associated L-values can nonetheless differ, in
contrast to the classical Birch-Swinnerton-Dyer conjecture.Comment: 27 pages, new versio
Anticyclotomic Iwasawa theory of CM elliptic curves
We study the Iwasawa theory of a CM elliptic curve in the anticyclotomic
-extension of the CM field, where is a prime of good,
ordinary reduction for . When the complex -function of vanishes to
even order, the two variable main conjecture of Rubin implies that the
Pontryagin dual of the -power Selmer group over the anticyclotomic extension
is a torsion Iwasawa module. When the order of vanishing is odd, work of
Greenberg shows that it is not a torsion module. In this paper we show that in
the case of odd order of vanishing the dual of the Selmer group has rank
exactly one, and we prove a form of the Iwasawa main conjecture for the torsion
submodule.Comment: Final version. To appear in the Annales de L'Institut Fourie
Faces of weight polytopes and a generalization of a theorem of Vinberg
The paper is motivated by the study of graded representations of Takiff
algebras, cominuscule parabolics, and their generalizations. We study certain
special subsets of the set of weights (and of their convex hull) of the
generalized Verma modules (or GVM's) of a semisimple Lie algebra \lie g. In
particular, we extend a result of Vinberg and classify the faces of the convex
hull of the weights of a GVM. When the GVM is finite-dimensional, we ask a
natural question that arises out of Vinberg's result: when are two faces the
same? We also extend the notion of interiors and faces to an arbitrary subfield
\F of the real numbers, and introduce the idea of a weak \F-face of any
subset of Euclidean space. We classify the weak \F-faces of all lattice
polytopes, as well as of the set of lattice points in them. We show that a weak
\F-face of the weights of a finite-dimensional \lie g-module is precisely
the set of weights lying on a face of the convex hull.Comment: Statement changed in Section 4. Typos fixed and some proofs updated.
Submitted to "Algebra and Representation Theory." 18 page
Exact and efficient top-K inference for multi-target prediction by querying separable linear relational models
Many complex multi-target prediction problems that concern large target
spaces are characterised by a need for efficient prediction strategies that
avoid the computation of predictions for all targets explicitly. Examples of
such problems emerge in several subfields of machine learning, such as
collaborative filtering, multi-label classification, dyadic prediction and
biological network inference. In this article we analyse efficient and exact
algorithms for computing the top- predictions in the above problem settings,
using a general class of models that we refer to as separable linear relational
models. We show how to use those inference algorithms, which are modifications
of well-known information retrieval methods, in a variety of machine learning
settings. Furthermore, we study the possibility of scoring items incompletely,
while still retaining an exact top-K retrieval. Experimental results in several
application domains reveal that the so-called threshold algorithm is very
scalable, performing often many orders of magnitude more efficiently than the
naive approach
Toroidal automorphic forms, Waldspurger periods and double Dirichlet series
The space of toroidal automorphic forms was introduced by Zagier in the
1970s: a GL_2-automorphic form is toroidal if it has vanishing constant Fourier
coefficients along all embedded non-split tori. The interest in this space
stems (amongst others) from the fact that an Eisenstein series of weight s is
toroidal for a given torus precisely if s is a non-trivial zero of the zeta
function of the quadratic field corresponding to the torus.
In this paper, we study the structure of the space of toroidal automorphic
forms for an arbitrary number field F. We prove that it decomposes into a space
spanned by all derivatives up to order n-1 of an Eisenstein series of weight s
and class group character omega precisely if s is a zero of order n of the
L-series corresponding to omega at s, and a space consisting of exactly those
cusp forms the central value of whose L-series is zero.
The proofs are based on an identity of Hecke for toroidal integrals of
Eisenstein series and a result of Waldspurger about toroidal integrals of cusp
forms combined with non-vanishing results for twists of L-series proven by the
method of double Dirichlet series.Comment: 14 page
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