Expand all Collapse all | Results 1 - 14 of 14 |
1. CMB 2011 (vol 56 pp. 148)
On the Gras Conjecture for Imaginary Quadratic Fields In this paper we extend K. Rubin's methods to prove the Gras conjecture
for abelian extensions of a given imaginary quadratic field $k$ and
prime numbers $p$ that divide the number of roots of unity in $k$.
Keywords:elliptic units, Stark units, Gras conjecture, Euler systems Categories:11R27, 11R29, 11G16 |
2. CMB 2011 (vol 55 pp. 774)
Pell Equations: Non-Principal Lagrange Criteria and Central Norms We provide a criterion for the central norm to be
any value in the simple continued fraction expansion of $\sqrt{D}$
for any non-square integer $D>1$. We also provide a simple criterion
for the solvability of the Pell equation $x^2-Dy^2=-1$ in terms of
congruence conditions modulo $D$.
Keywords:Pell's equation, continued fractions, central norms Categories:11D09, 11A55, 11R11, 11R29 |
3. CMB 2011 (vol 54 pp. 330)
Sur la borne infÃ©rieure du rang du 2-groupe de classes de certains corps multiquadratiques
Soient $p_1,p_2,p_3$ et $q$ des nombres premiers distincts tels que
$p_1\equiv p_2\equiv p_3\equiv -q\equiv 1 \pmod{4}$, $k = \mathbf{Q}
(\sqrt{p_1}, \sqrt{p_2}, \sqrt{p_3}, \sqrt{q})$ et $\operatorname{Cl}_2(k)$ le
$2$-groupe de classes de $k$. A. FrÃ¶hlich a
dÃ©montrÃ© que $\operatorname{Cl}_2(k)$ n'est jamais trivial. Dans cet article,
nous donnons une extension de ce rÃ©sultat, en dÃ©montrant que le
rang de $\operatorname{Cl}_2(k)$ est toujours supÃ©rieur ou Ã©gal Ã $2$. Nous
dÃ©montrons aussi, que la valeur $2$ est optimale pour une famille
infinie de corps $k$.
Keywords:class group, units, multiquadratic number fields Categories:11R29, 11R11 |
4. CMB 2009 (vol 52 pp. 583)
Computing Polynomials of the Ramanujan $t_n$ Class Invariants We compute the minimal polynomials of the Ramanujan values $t_n$,
where $n\equiv 11 \mod 24$, using the Shimura reciprocity law.
These polynomials can be used for defining the Hilbert class field
of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$ and have
much smaller coefficients than the Hilbert polynomials.
Categories:11R29, 33E05, 11R20 |
5. 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 |
6. 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 |
7. CMB 2002 (vol 45 pp. 138)
The Discriminant of a Dihedral Quintic Field Defined by a Trinomial $X^5 + aX + b$ Let $X^5 + aX + b \in Z[X]$ have Galois group $D_5$. Let $\theta$ be
a root of $X^5 + aX + b$. An explicit formula is given for the
discriminant of $Q(\theta)$.
Keywords:dihedral quintic field, trinomial, discriminant Categories:11R21, 11R29 |
8. CMB 2002 (vol 45 pp. 86)
On Cyclic Fields of Odd Prime Degree $p$ with Infinite Hilbert $p$-Class Field Towers Let $k$ be a cyclic extension of odd prime degree $p$ of the field of
rational numbers. If $t$ denotes the number of primes that ramify in $k$,
it is known that the Hilbert $p$-class field tower of $k$ is infinite if
$t>3+2\sqrt p$. For each $t>2+\sqrt p$, this paper shows that a positive
proportion of such fields $k$ have infinite Hilbert $p$-class field towers.
Categories:11R29, 11R37, 11R45 |
9. CMB 2001 (vol 44 pp. 398)
Exponents of Class Groups of Quadratic Function Fields over Finite Fields We find a lower bound on the number of imaginary quadratic extensions
of the function field $\F_q(T)$ whose class groups have an element of
a fixed order.
More precisely, let $q \geq 5$ be a power of an odd prime and let $g$
be a fixed positive integer $\geq 3$. There are $\gg q^{\ell
(\frac{1}{2}+\frac{1}{g})}$ polynomials $D \in \F_q[T]$ with $\deg(D)
\leq \ell$ such that the class groups of the quadratic extensions
$\F_q(T,\sqrt{D})$ have an element of order~$g$.
Keywords:class number, quadratic function field Categories:11R58, 11R29 |
10. CMB 2001 (vol 44 pp. 97)
On the Density of Cyclic Quartic Fields An asymptotic formula is obtained for the number of cyclic quartic fields
over $Q$ with discriminant $\leq x$.
Keywords:cyclic quartic fields, density, discriminant Categories:11R16, 11R29 |
11. CMB 2000 (vol 43 pp. 218)
Continued Fractions, Jacobi Symbols, and Quadratic Diophantine Equations The results herein continue observations on norm form equations and
continued fractions begun and continued in the works
\cite{chows}--\cite{mol}, and \cite{mvdpw}--\cite{schinz}.
Categories:11R11, 11D09, 11R29, 11R65 |
12. CMB 1999 (vol 42 pp. 427)
Ramanujan and the Modular $j$-Invariant A new infinite product $t_n$ was introduced by S.~Ramanujan on the
last page of his third notebook. In this paper, we prove
Ramanujan's assertions about $t_n$ by establishing new connections
between the modular $j$-invariant and Ramanujan's cubic theory of
elliptic functions to alternative bases. We also show that for
certain integers $n$, $t_n$ generates the Hilbert class field of
$\mathbb{Q} (\sqrt{-n})$. This shows that $t_n$ is a new class
invariant according to H.~Weber's definition of class invariants.
Keywords:modular functions, the Borweins' cubic theta-functions, Hilbert class fields Categories:33C05, 33E05, 11R20, 11R29 |
13. CMB 1998 (vol 41 pp. 328)
Class number one and prime-producing quadratic polynomials revisited Over a decade ago, this author produced class number one criteria for
real quadratic fields in terms of prime-producing quadratic
polynomials. The purpose of this article is to revisit the problem
from a new perspective with new criteria. We look at the more general
situation involving arbitrary real quadratic orders rather than the
more restrictive field case, and use the interplay between the various
orders to provide not only more general results, but also simpler proofs.
Categories:11R11, 11R09, 11R29 |
14. CMB 1997 (vol 40 pp. 214)
Polynomials of quadratic type producing strings of primes The primary purpose of this paper is to provide necessary and
sufficient conditions for certain quadratic polynomials of negative
discriminant (which we call Euler-Rabinowitsch type), to produce
consecutive prime values for an initial range of input values less than
a Minkowski bound. This not only generalizes the classical work of
Frobenius, the later developments by Hendy, and the generalizations by
others, but also concludes the line of reasoning by providing a
complete list of all such prime-producing polynomials, under the
assumption of the generalized Riemann hypothesis ($\GRH$). We demonstrate
how this prime-production phenomenon is related to the exponent of the
class group of the underlying complex quadratic field. Numerous
examples, and a remaining conjecture, are also given.
Categories:11R11, 11R09, 11R29 |