Expand all Collapse all | Results 201 - 220 of 220 |
201. CJM 1999 (vol 51 pp. 176)
Values of the Dedekind Eta Function at Quadratic Irrationalities Let $d$ be the discriminant of an imaginary quadratic field. Let
$a$, $b$, $c$ be integers such that
$$
b^2 - 4ac = d, \quad a > 0, \quad \gcd (a,b,c) = 1.
$$
The value of $\bigl|\eta \bigl( (b + \sqrt{d})/2a \bigr) \bigr|$ is
determined explicitly, where $\eta(z)$ is Dedekind's eta function
$$
\eta (z) = e^{\pi iz/12} \prod^\ty_{m=1} (1 - e^{2\pi imz})
\qquad \bigl( \im(z) > 0 \bigr). %\eqno({\rm im}(z)>0).
$$
Keywords:Dedekind eta function, quadratic irrationalities, binary quadratic forms, form class group Categories:11F20, 11E45 |
202. CJM 1998 (vol 50 pp. 1253)
Integral representation of $p$-class groups in ${\Bbb Z}_p$-extensions and the Jacobian variety For an arbitrary finite Galois $p$-extension $L/K$ of
$\zp$-cyclotomic number fields of $\CM$-type with Galois group $G =
\Gal(L/K)$ such that the Iwasawa invariants $\mu_K^-$, $ \mu_L^-$
are zero, we obtain unconditionally and explicitly the Galois
module structure of $\clases$, the minus part of the $p$-subgroup
of the class group of $L$. For an arbitrary finite Galois
$p$-extension $L/K$ of algebraic function fields of one variable
over an algebraically closed field $k$ of characteristic $p$ as its
exact field of constants with Galois group $G = \Gal(L/K)$ we
obtain unconditionally and explicitly the Galois module structure
of the $p$-torsion part of the Jacobian variety $J_L(p)$ associated
to $L/k$.
Keywords:${\Bbb Z}_p$-extensions, Iwasawa's theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structure Categories:11R33, 11R23, 11R58, 14H40 |
203. CJM 1998 (vol 50 pp. 1323)
L'invariant de Hasse-Witt de la forme de Killing Nous montrons que l'invariant de Hasse-Witt de la forme de Killing
d'une alg{\`e}bre de Lie semi-simple $L$ s'exprime {\`a} l'aide de
l'invariant de Tits de la repr{\'e}sentation irr{\'e}ductible de
$L$ de poids dominant $\rho=\frac{1}{2}$ (somme des racines
positives), et des invariants associ{\'e}s au groupe des
sym{\'e}tries du diagramme de Dynkin de $L$.
Categories:11E04, 11E72, 17B10, 17B20, 11E88, 15A66 |
204. CJM 1998 (vol 50 pp. 1007)
Galois module structure of ambiguous ideals in biquadratic extensions Let $N/K$ be a biquadratic extension of algebraic number fields, and
$G=\Gal (N/K)$. Under a weak restriction on the ramification filtration
associated with each prime of $K$ above $2$, we explicitly describe the
$\bZ[G]$-module structure of each ambiguous ideal of $N$. We find under
this restriction that in the representation of each ambiguous ideal as a
$\bZ[G]$-module, the exponent (or multiplicity) of each indecomposable
module is determined by the invariants of ramification, alone.
For a given group, $G$, define ${\cal S}_G$ to be the set of
indecomposable $\bZ[G]$-modules, ${\cal M}$, such that there
is an extension, $N/K$, for which $G\cong\Gal (N/K)$, and ${\cal M}$
is a $\bZ[G]$-module summand of an ambiguous ideal of $N$. Can
${\cal S}_G$ ever be infinite? In this paper we answer this
question of Chinburg in the affirmative.
Keywords:Galois module structure, wild ramification Categories:11R33, 11S15, 20C32 |
205. CJM 1998 (vol 50 pp. 1105)
Tempered representations and the theta correspondence Let $V$ be an even dimensional nondegenerate symmetric bilinear
space over a nonarchimedean local field $F$ of characteristic zero,
and let $n$ be a nonnegative integer. Suppose that $\sigma \in
\Irr \bigl(\OO (V)\bigr)$ and $\pi \in \Irr \bigl(\Sp (n,F)\bigr)$
correspond under the theta correspondence. Assuming that $\sigma$
is tempered, we investigate the problem of determining the
Langlands quotient data for $\pi$.
Categories:11F27, 22E50 |
206. CJM 1998 (vol 50 pp. 794)
Upper bounds on $|L(1,\chi)|$ and applications We give upper bounds on the modulus of the values at $s=1$ of
Artin $L$-functions of abelian extensions unramified at all
the infinite places. We also explain how we can compute better
upper bounds and explain how useful such computed bounds are
when dealing with class number problems for $\CM$-fields. For
example, we will reduce the determination of all the
non-abelian normal $\CM$-fields of degree $24$ with Galois
group $\SL_2(F_3)$ (the special linear group over the finite
field with three elements) which have class number one to the
computation of the class numbers of $23$ such $\CM$-fields.
Keywords:Dedekind zeta function, Dirichlet series, $\CM$-field, relative class number Categories:11M20, 11R42, 11Y35, 11R29 |
207. CJM 1998 (vol 50 pp. 465)
Six primes and an almost prime in four linear equations There are infinitely many triplets of primes $p,q,r$ such that the
arithmetic means of any two of them, ${p+q\over2}$, ${p+r\over2}$,
${q+r\over2}$ are also primes. We give an asymptotic formula for
the number of such triplets up to a limit. The more involved
problem of asking that in addition to the above the arithmetic mean
of all three of them, ${p+q+r\over3}$ is also prime seems to be out
of reach. We show by combining the Hardy-Littlewood method with the
sieve method that there are quite a few triplets for which six of
the seven entries are primes and the last is almost prime.}
Categories:11P32, 11N36 |
208. CJM 1998 (vol 50 pp. 563)
Primes in short segments of arithmetic progressions Consider the variance for the number of primes that are both in the
interval $[y,y+h]$ for $y \in [x,2x]$ and in an arithmetic
progression of modulus $q$. We study the total variance
obtained by adding these variances over all the reduced residue
classes modulo $q$. Assuming a strong form of the twin prime
conjecture and the Riemann Hypothesis one can obtain an asymptotic
formula for the total variance in the range when $1 \leq h/q \leq
x^{1/2-\epsilon}$, for any $\epsilon >0$. We show that one can still
obtain some weaker asymptotic results assuming the Generalized Riemann
Hypothesis (GRH) in place of the twin prime conjecture. In their
simplest form, our results are that on GRH the same asymptotic formula
obtained with the twin prime conjecture is true for ``almost all'' $q$
in the range $1 \leq h/q \leq h^{1/4-\epsilon}$, that on averaging
over $q$ one obtains an asymptotic formula in the extended range $1
\leq h/q \leq h^{1/2-\epsilon}$, and that there are lower bounds with
the correct order of magnitude for all $q$ in the range $1 \leq h/q
\leq x^{1/3-\epsilon}$.
Category:11M26 |
209. CJM 1998 (vol 50 pp. 412)
Asymptotic transformations of $q$-series For the $q$-series $\sum_{n=0}^\infty a^nq^{bn^2+cn}/(q)_n$
we construct a companion $q$-series such that the asymptotic
expansions of their logarithms as $q\to 1^{\scriptscriptstyle -}$
differ only in the dominant few terms. The asymptotic expansion
of their quotient then has a simple closed form; this gives rise
to a new $q$-hypergeometric identity. We give an asymptotic
expansion of a general class of $q$-series containing some of
Ramanujan's mock theta functions and Selberg's identities.
Categories:11B65, 33D10, 34E05, 41A60 |
210. CJM 1998 (vol 50 pp. 74)
Elementary proof of the fundamental lemma for a unitary group The fundamental lemma in the theory of automorphic forms is proven
for the (quasi-split) unitary group $U(3)$ in three variables
associated with a quadratic extension of $p$-adic fields, and its
endoscopic group $U(2)$, by means of a new, elementary technique.
This lemma is a prerequisite for an application of the trace
formula to classify the automorphic and admissible representations
of $U(3)$ in terms of those of $U(2)$ and base change to $\GL(3)$.
It compares the (unstable) orbital integral of the characteristic
function of the standard maximal compact subgroup $K$ of $U(3)$ at
a regular element (whose centralizer $T$ is a torus), with an
analogous (stable) orbital integral on the endoscopic group $U(2)$.
The technique is based on computing the sum over the double coset
space $T\bs G/K$ which describes the integral, by means of an
intermediate double coset space $H\bs G/K$ for a subgroup $H$ of
$G=U(3)$ containing $T$. Such an argument originates from
Weissauer's work on the symplectic group. The lemma is proven for
both ramified and unramified regular elements, for which endoscopy
occurs (the stable conjugacy class is not a single orbit).
Categories:22E35, 11F70, 11F85, 11S37 |
211. CJM 1997 (vol 49 pp. 1139)
Majorations effectives pour l'Ã©quation de Fermat gÃ©nÃ©ralisÃ©e Soient $A$, $B$ et $C$ trois entiers
non nuls premiers entre eux deux \`a deux, et $p$ un nombre premier.
Comme cons\'equence des travaux de A. Wiles et F. Diamond sur la
conjecture de Taniyama-Weil, on explicite une constante $f(A,B,C)$
telle que, sous certaines conditions portant sur $A$, $B$ et $C$,
l'\'equation $Ax^p+By^p+Cz^p=0$ n'a aucune solution non triviale
dans $\Z$, si $p$ est $>f(A,B,C)$. On d\'emontre par ailleurs,
sans condition suppl\'ementaire portant sur $A$, $B$ et $C$, que
cette \'equation n'a aucune solution non triviale dans $\Z$, si
$p$ divise $xyz$, et si $p$ est $>f(A,B,C)$.
Category:11G |
212. CJM 1997 (vol 49 pp. 1265)
Hecke algebras and class-group invariant Let $G$ be a finite group. To a set of subgroups of order two we associate
a $\mod 2$ Hecke algebra and construct a homomorphism, $\psi$, from its
units to the class-group of ${\bf Z}[G]$. We show that this homomorphism
takes values in the subgroup, $D({\bf Z}[G])$. Alternative constructions of
Chinburg invariants arising from the Galois module structure of
higher-dimensional algebraic $K$-groups of rings of algebraic integers
often differ by elements in the image of $\psi$. As an application we show
that two such constructions coincide.
Categories:16S34, 19A99, 11R65 |
213. CJM 1997 (vol 49 pp. 887)
Polynomials with $\{ 0, +1, -1\}$ coefficients and a root close to a given point For a fixed algebraic number $\alpha$ we
discuss how closely $\alpha$ can be approximated by
a root of a $\{0,+1,-1\}$ polynomial of given degree.
We show that the worst rate of approximation tends to
occur for roots of unity, particularly those of small degree.
For roots of unity these bounds depend on
the order of vanishing, $k$, of the polynomial at $\alpha$.
In particular we obtain the following. Let
${\cal B}_{N}$ denote the set of roots of all
$\{0,+1,-1\}$ polynomials of degree at most $N$ and
${\cal B}_{N}(\alpha,k)$ the roots of those
polynomials that have a root of order at most $k$
at $\alpha$. For a Pisot number $\alpha$ in $(1,2]$
we show that
\[
\min_{\beta \in {\cal B}_{N}\setminus \{ \alpha \}} |\alpha
-\beta| \asymp \frac{1}{\alpha^{N}},
\]
and for a root of unity $\alpha$ that
\[
\min_{\beta \in {\cal B}_{N}(\alpha,k)\setminus \{\alpha\}}
|\alpha -\beta|\asymp \frac{1}{N^{(k+1) \left\lceil
\frac{1}{2}\phi (d)\right\rceil +1}}.
\]
We study in detail the case of $\alpha=1$, where, by far, the
best approximations are real.
We give fairly precise bounds on the closest real root to 1.
When $k=0$ or 1 we
can describe the extremal polynomials explicitly.
Keywords:Mahler measure, zero one polynomials, Pisot numbers, root separation Categories:11J68, 30C10 |
214. CJM 1997 (vol 49 pp. 641)
Fine spectra and limit laws II First-order 0--1 laws. Using Feferman-Vaught techniques a condition on the fine
spectrum of an admissible class of structures is found
which leads to a first-order 0--1 law.
The condition presented is best possible in the
sense that if it is violated then one can find an admissible
class with the same fine spectrum which does not have
a first-order 0--1 law.
If the condition is satisfied (and hence we have a first-order %% 0--1 law)
Categories:03N45, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81 |
215. CJM 1997 (vol 49 pp. 749)
Twisted Hasse-Weil $L$-functions and the rank of Mordell-Weil groups Following a method outlined by Greenberg, root
number computations give a conjectural lower bound for the ranks of
certain Mordell-Weil groups of elliptic curves. More specifically,
for $\PQ_{n}$ a \pgl{{\bf Z}/p^{n}{\bf Z}}-extension of ${\bf Q}$ and
$E$ an elliptic curve over {\bf Q}, define the motive $E \otimes
\rho$, where $\rho$ is any irreducible representation of
$\Gal (\PQ_{n}/{\bf Q})$. Under some restrictions, the root number in
the conjectural functional equation for the $L$-function of $E
\otimes \rho$ is easily computible, and a `$-1$' implies, by the
Birch and Swinnerton-Dyer conjecture, that $\rho$ is found in
$E(\PQ_{n}) \otimes {\bf C}$. Summing the dimensions of such $\rho$
gives a conjectural lower bound of
$$
p^{2n} - p^{2n - 1} - p - 1
$$
for the rank of $E(\PQ_{n})$.
Categories:11G05, 14G10 |
216. CJM 1997 (vol 49 pp. 722)
Galois module structure of the integers in wildly ramified $C_p\times C_p$ extensions Let $L/K$ be a finite Galois extension of local fields which are finite
extensions of $\bQ_p$, the field of $p$-adic numbers. Let $\Gal (L/K)=G$,
and $\euO_L$ and $\bZ_p$ be the rings of integers in $L$ and $\bQ_p$,
respectively. And let $\euP_L$ denote the maximal ideal of $\euO_L$. We
determine, explicitly in terms of specific indecomposable $\bZ_p[G]$-modules,
the $\bZ_p[G]$-module structure of $\euO_L$ and $\euP_L$, for $L$, a
composite of two arithmetically disjoint, ramified cyclic extensions of
$K$, one of which is only weakly ramified in the sense of Erez \cite{erez}.
Keywords:Galois module structure---integral representation. Categories:11S15, 20C32 |
217. CJM 1997 (vol 49 pp. 468)
Fine spectra and limit laws I. First-order laws Using Feferman-Vaught techniques we show a certain property of the fine
spectrum of an admissible class of structures leads to a first-order law.
The condition presented is best possible in the sense that if it is
violated then one can find an admissible class with the same fine
spectrum which does not have a first-order law. We present three
conditions for verifying that the above property actually holds.
The first condition is that the count function of an admissible class
has regular variation with a certain uniformity of convergence. This
applies to a wide range of admissible classes, including those
satisfying Knopfmacher's Axiom A, and those satisfying Bateman
and Diamond's condition.
The second condition is similar to the first condition, but designed
to handle the discrete case, {\it i.e.}, when the sizes of the structures
in an admissible class $K$ are all powers of a single integer. It applies
when either the class of indecomposables or the whole class satisfies
Knopfmacher's Axiom A$^\#$.
The third condition is also for the discrete case, when there is a
uniform bound on the number of $K$-indecomposables of any given size.
Keywords:First order limit laws, generalized number theory Categories:O3C13, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81 |
218. CJM 1997 (vol 49 pp. 499)
Gorenstein Witt rings II The abstract Witt rings which are Gorenstein have been classified
when the dimension is one and the classification problem for those of
dimension zero has been reduced to the case of socle degree three. Here we
classifiy the Gorenstein Witt rings of fields with dimension zero and
socle degree three. They are of elementary type.
Categories:11E81, 13H10 |
219. CJM 1997 (vol 49 pp. 405)
On Hurwitz constants for Fuchsian groups Explicit bounds for the Hurwitz constants for general cofinite
Fuchsian groups have been found. It is shown that the bounds
obtained are exact for the Hecke groups and triangular groups with
signature $(0:2,p,q)$.
Categories:11J04, 20H10 |
220. CJM 1997 (vol 49 pp. 283)
The $2$-rank of the class group of imaginary bicyclic biquadratic fields A formula is obtained for the rank of the $2$-Sylow subgroup of the
ideal class group of imaginary bicyclic biquadratic fields. This
formula involves the number of primes that ramify in the field, the
ranks of the $2$-Sylow subgroups of the ideal class groups of the
quadratic subfields and the rank of a $Z_2$-matrix determined by
Legendre symbols involving pairs of ramified primes. As
applications, all subfields with both $2$-class and class group
$Z_2 \times Z_2$ are determined. The final results assume the
completeness of D.~A.~Buell's list of imaginary fields with small
class numbers.
Categories:11R16, 11R29, 11R20 |