Expand all Collapse all | Results 101 - 125 of 207 |
101. CMB 2006 (vol 49 pp. 560)
A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues In this article we will show that there are infinitely many
symmetric, integral $3 \times 3$ matrices, with zeros on the
diagonal, whose eigenvalues are all integral. We will do this by
proving that the rational points on a certain non-Kummer, singular
K3 surface
are dense. We will also compute the entire N\'eron--Severi group of
this surface and find all low degree curves on it.
Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory Categories:14G05, 14J28, 11D41 |
102. CMB 2006 (vol 49 pp. 481)
On Sequences of Squares with Constant Second Differences The aim of this paper is to study sequences of integers
for which the second differences between their squares are
constant. We show that there are infinitely many nontrivial
monotone sextuples having this property and discuss some related
problems.
Keywords:sequence of squares, second difference, elliptic curve Categories:11B83, 11Y85, 11D09 |
103. CMB 2006 (vol 49 pp. 526)
The Values of Modular Functions and Modular Forms Let $\Gamma_0$ be a Fuchsian group of the first kind of genus zero
and $\Gamma$ be a subgroup of $\Gamma_0$
of finite index of genus zero. We find universal recursive
relations giving the $q_{r}$-series coefficients of
$j_0$ by using those of the $q_{h_{s}}$-series of $j$, where $j$ is
the canonical Hauptmodul for $\Gamma$ and $j_0$ is a Hauptmodul
for $\Gamma_0$ without zeros on the complex upper half plane
$\mathfrak{H}$ (here $q_{\ell} := e^{2 \pi i z / \ell}$). We find universal recursive formulas for
$q$-series coefficients of any modular form on
$\Gamma_0^{+}(p)$ in terms of those of the canonical Hauptmodul $j_p^{+}$.
Categories:10D12, 11F11 |
104. CMB 2006 (vol 49 pp. 428)
Vector-Valued Modular Forms of Weight Two Associated With Jacobi-Like Forms We construct vector-valued modular forms of weight 2 associated to
Jacobi-like forms with respect to a symmetric tensor representation of
$\G$ by using the method of Kuga and Shimura as well as the
correspondence between Jacobi-like forms and sequences of modular forms.
As an application, we obtain vector-valued modular forms determined by
theta functions and by pseudodifferential operators.
Categories:11F11, 11F50 |
105. CMB 2006 (vol 49 pp. 472)
Cyclic Cubic Fields of Given Conductor and Given Index The number of cyclic cubic fields with a given conductor and a given index is determined.
Keywords:Discriminant, conductor, index, cyclic cubic field Categories:11R16, 11R29 |
106. CMB 2006 (vol 49 pp. 448)
A Lower Bound on the Number of Cyclic Function Fields With Class Number Divisible by $n$ In this paper, we find a lower bound on the number of cyclic function
fields of prime degree~$l$ whose class numbers are divisible by a
given
integer $n$. This generalizes a previous result of D. Cardon and R.
Murty
which gives a lower bound on the number of quadratic function fields
with
class numbers divisible by $n$.
Categories:11R29, 11R58 |
107. CMB 2006 (vol 49 pp. 296)
On the Modularity of Three Calabi--Yau Threefolds With Bad Reduction at 11 This paper investigates the modularity of three
non-rigid Calabi--Yau threefolds with bad reduction at 11. They are
constructed as fibre products of rational elliptic surfaces,
involving the modular elliptic surface of level 5. Their middle
$\ell$-adic cohomology groups are shown to split into
two-dimensional pieces, all but one of which can be interpreted in
terms of elliptic curves. The remaining pieces are associated to
newforms of weight 4 and level 22 or 55, respectively. For this
purpose, we develop a method by Serre to compare the corresponding
two-dimensional 2-adic Galois representations with uneven trace.
Eventually this method is also applied to a self fibre product of
the Hesse-pencil, relating it to a newform of weight 4 and level
27.
Categories:14J32, 11F11, 11F23, 20C12 |
108. CMB 2006 (vol 49 pp. 196)
Another Proof of Totaro's Theorem on $E_8$-Torsors We give a short proof of Totaro's theorem that every$E_8$-torsor over
a field $k$ becomes trivial over a finiteseparable extension of $k$of
degree dividing $d(E_8)=2^63^25$.
Categories:11E72, 14M17, 20G15 |
109. CMB 2006 (vol 49 pp. 247)
A Szpilrajn--Marczewski Type Theorem for Concentration Dimension on Polish Spaces Let $X$ be a Polish space.
We will prove that
$$
\dim_T X=\inf \{\dim_L X': X'\text{ is homeomorphic to
} X\},
$$
where $\dim_L X$ and $\dim_T X$ stand
for the concentration dimension and
the topological dimension of $X$, respectively.
Keywords:Hausdorff dimension, topological dimension, LÃ©vy concentration function, concentration dimension Categories:11K55, 28A78 |
110. CMB 2006 (vol 49 pp. 21)
Evaluation of the Dedekind Eta Function We extend the methods of Van der Poorten and Chapman
for
explicitly evaluating the Dede\-kind eta function at quadratic
irrationalities. Via evaluation of Hecke
$L$-series we obtain new evaluations at points in
imaginary quadratic number fields with
class numbers 3 and 4. Further, we overcome the limitations
of the earlier methods and via modular equations provide
explicit evaluations where the class number is 5 or 7.
Category:11G15 |
111. CMB 2006 (vol 49 pp. 108)
A Dynamical Proof of Pisot's Theorem We give a geometric proof of classical results that characterize
Pisot numbers as algebraic $\lambda>1$ for which
there is $x\neq0$ with $\lambda^nx \to 0 \mod$ and identify such
$x$ as members of $\Z[\lambda^{-1}] \cdot \Z[\lambda]^*$ where $\Z[\lambda]^*$ is the dual module of $\Z[\lambda]$.
Category:11R06 |
112. CMB 2005 (vol 48 pp. 535)
On the Error Term in Duke's Estimate for the Average Special Value of $L$-Functions Let $\FF$ be an orthonormal basis for weight $2$
cusp forms of level $N$. We show that various weighted averages of
special values $L(f \tensor \chi, 1)$ over $f \in \FF$ are equal to $4
\pi c + O(N^{-1 + \epsilon})$, where $c$ is an explicit nonzero constant. A previous result of Duke gives an error
term of $O(N^{-1/2}\log N)$.
Categories:11F67, 11F11 |
113. CMB 2005 (vol 48 pp. 636)
Correction to: On the Diophantine Equation $n(n+d)\cdots(n+(k-1)d)=by^l$ In the article under consideration
(Canad. Math. Bull. \textbf{47} (2004), pp.~373--388),
Lemma 6 is not true in the form presented there.
Lemma 6 is used only in the proof of part (i) of Theorem 9.
We note, however, that part (i) of Theorem 9 in question is a special
case of a theorem by Bennet, Bruin, Gy\H{o}ry and Hajdu.
Category:11D41 |
114. CMB 2005 (vol 48 pp. 576)
On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II Let $m=p^e$ be a power of a prime number $p$.
We say that a number field $F$ satisfies the property $(H_m')$
when for any $a \in F^{\times}$, the cyclic extension
$F(\z_m, a^{1/m})/F(\z_m)$ has a normal $p$-integral basis.
We prove that $F$ satisfies $(H_m')$
if and only if the natural homomorphism $Cl_F' \to Cl_K'$ is trivial.
Here $K=F(\zeta_m)$, and $Cl_F'$ denotes the ideal class group of $F$
with respect to the $p$-integer ring of $F$.
Category:11R33 |
115. CMB 2005 (vol 48 pp. 428)
Reduction of Elliptic Curves in Equal Characteristic~3 (and~2) and fibre type for elliptic curves
over discrete valued fields of equal characteristic~3.
Along the same lines, partial results are obtained
in equal characteristic~2.
Categories:14H52, 14K15, 11G07, 11G05, 12J10 |
116. CMB 2005 (vol 48 pp. 394)
Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices Let $n=2m$ be even and denote by $\Sp_n(F)$ the symplectic group
of rank $m$ over an infinite field $F$ of characteristic different
from $2$. We show that any $n\times n$ symmetric matrix $A$ is
equivalent under symplectic congruence transformations to the
direct sum of $m\times m$ matrices $B$ and $C$, with $B$ diagonal
and $C$ tridiagonal. Since the $\Sp_n(F)$-module of symmetric
$n\times n$ matrices over $F$ is isomorphic to the adjoint module
$\sp_n(F)$, we infer that any adjoint orbit of $\Sp_n(F)$ in
$\sp_n(F)$ has a representative in the sum of $3m-1$ root spaces,
which we explicitly determine.
Categories:11E39, 15A63, 17B20 |
117. CMB 2005 (vol 48 pp. 333)
Monotonicity Properties of the Hurwitz Zeta Function Let
$$
\zeta(s,x)=\sum_{n=0}^{\infty}\frac{1}{(n+x)^s} \quad{(s>1,\, x>0)}
$$
be the Hurwitz zeta function and let
$$
Q(x)=Q(x;\alpha,\beta;a,b)=\frac{(\zeta(\alpha,x))^a}{(\zeta(\beta,x))^b},
$$
where $\alpha, \beta>1$
and $a,b>0$ are real numbers. We prove:
(i) The function $Q$ is decreasing on $(0,\infty)$ if{}f $\alpha a-\beta b\geq \max(a-b,0)$.
(ii) $Q$ is increasing on $(0,\infty)$ if{}f $\alpha a-\beta b\leq
\min(a-b,0)$.
An application of part (i) reveals that for all $x>0$ the function $s\mapsto [(s-1)\zeta(s,x)]^{1/(s-1)}$ is decreasing on $(1,\infty)$. This settles
a conjecture of Bastien and Rogalski.
Categories:11M35, 26D15 |
118. CMB 2005 (vol 48 pp. 211)
The Distribution of Totatives The integers coprime to $n$ are called the {\it totatives} \rm of $n$.
D. H. Lehmer and Paul Erd\H{o}s were interested in understanding when
the number of totatives between $in/k$ and $(i+1)n/k$ are $1/k$th of
the total number of totatives up to $n$. They provided criteria in
various cases. Here we give an ``if and only if'' criterion which
allows us to recover most of the previous results in this literature
and to go beyond, as well to reformulate the problem in terms of
combinatorial group theory. Our criterion is that the above holds if
and only if for every odd character $\chi \pmod \kappa$ (where
$\kappa:=k/\gcd(k,n/\prod_{p|n} p)$) there exists a prime $p=p_\chi$
dividing $n$ for which $\chi(p)=1$.
Categories:11A05, 11A07, 11A25, 20C99 |
119. CMB 2005 (vol 48 pp. 147)
Baker-Type Estimates for Linear Forms in the Values of $q$-Series We obtain lower estimates for the absolute values
of linear forms of the values of generalized Heine
series at non-zero points of an imaginary quadratic field~$\II$,
in particular of the values of $q$-exponential function.
These estimates depend on the individual coefficients,
not only on the maximum of their absolute values.
The proof uses a variant of classical Siegel's method
applied to a system of functional Poincar\'e-type equations
and the connection between the solutions of these functional
equations and the generalized Heine series.
Keywords:measure of linear independence, $q$-series Categories:11J82, 33D15 |
120. CMB 2005 (vol 48 pp. 121)
Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ |
Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ We look at the simple continued fraction expansion of $\sqrt{D}$
for any $D=2^hc $ where $c>1$ is odd with a goal of
determining necessary and
sufficient conditions for the central norm (as determined by
the infrastructure of the underlying real quadratic order therein) to be
$2^h$. At the end of the paper, we also address the case where $D=c$
is odd and the central norm of $\sqrt{D}$ is equal to $2$.
Keywords:quadratic Diophantine equations, simple continued fractions,, norms of ideals, infrastructure of real quadratic fields Categories:11A55, 11D09, 11R11 |
121. CMB 2005 (vol 48 pp. 16)
On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves Let $ E $ be an elliptic curve defined over
$\Q,$ of conductor $N$ and without complex multiplication. For any
positive integer $l$, let $\phi_l$ be the Galois representation
associated to the $l$-division points of~$E$. From a celebrated
1972 result of Serre we know that $\phi_l$ is surjective for any
sufficiently large prime $l$. In this paper we find conditional
and unconditional upper bounds in terms of $N$ for the primes $l$
for which $\phi_l$ is {\emph{not}} surjective.
Categories:11G05, 11N36, 11R45 |
122. CMB 2004 (vol 47 pp. 589)
A Generalization of the ErdÃ¶s-Kac Theorem and its Applications We axiomatize the main properties of the classical Erd\"os-Kac Theorem
in order to apply it to a general context. We provide applications in the
cases of number fields, function fields, and geometrically irreducible
varieties over a finite field.
Categories:11N60, 11N80 |
123. CMB 2004 (vol 47 pp. 573)
A Generalization of the TurÃ¡n Theorem\\ and Its Applications We axiomatize the main properties of the classical Tur\'an Theorem
in order to apply it to a general context. We provide applications in the
cases of number fields, function fields, and geometrically irreducible
varieties over a finite field.
Categories:11N37, 11N80 |
124. CMB 2004 (vol 47 pp. 373)
On the Diophantine Equation $n(n+d)\cdots(n+(k-1)d)=by^l$ We show that the product of four or five consecutive positive
terms in arithmetic progression can never be a perfect power whenever the
initial term is coprime to the common difference of the arithmetic
progression. This is a generalization of the results of Euler and Obl\'ath
for the case of squares, and an extension of a theorem of Gy\H ory on three
terms in arithmetic progressions. Several other results concerning the
integral solutions of the equation of the title are also obtained. We extend
results of Sander on the rational solutions of the equation in $n,y$ when
$b=d=1$. We show that there are only finitely many solutions in $n,d,b,y$
when $k\geq 3$, $l\geq 2$ are fixed and $k+l>6$.
Category:11D41 |
125. CMB 2004 (vol 47 pp. 398)
A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces Let $V$ be a $K3$ surface defined over a number field $k$. The
Batyrev-Manin conjecture for $V$ states that for every nonempty open
subset $U$ of $V$, there exists a finite set $Z_U$ of accumulating
rational curves such that the density of rational points on $U-Z_U$ is
strictly less than the density of rational points on $Z_U$. Thus,
the set of rational points of $V$ conjecturally admits a stratification
corresponding to the sets $Z_U$ for successively smaller sets $U$.
In this paper, in the case that $V$ is a Kummer surface, we prove that
the Batyrev-Manin conjecture for $V$ can be reduced to the
Batyrev-Manin conjecture for $V$ modulo the endomorphisms of $V$
induced by multiplication by $m$ on the associated abelian surface
$A$. As an application, we use this to show that given some restrictions
on $A$, the set of rational points of $V$ which lie on rational curves
whose preimages have geometric genus 2 admits a stratification of
Keywords:rational points, Batyrev-Manin conjecture, Kummer, surface, rational curve, abelian surface, height Categories:11G35, 14G05 |