Expand all Collapse all | Results 1 - 5 of 5 |
1. CJM 2006 (vol 58 pp. 580)
Annihilators for the Class Group of a Cyclic Field of Prime Power Degree, II We prove, for a field $K$ which is cyclic of odd prime power
degree over the rationals, that the annihilator of the
quotient of the units of $K$ by a suitable large subgroup (constructed
from circular units) annihilates what we call the
non-genus part of the class group.
This leads to stronger annihilation results for the whole
class group than a routine application of the Rubin--Thaine method
would produce, since the
part of the class group determined by genus theory has an obvious
large annihilator which is not detected by
that method; this is our reason for concentrating on
the non-genus part. The present work builds on and strengthens
previous work of the authors; the proofs are more conceptual now,
and we are also able to construct an example which demonstrates
that our results cannot be easily sharpened further.
Categories:11R33, 11R20, 11Y40 |
2. CJM 2001 (vol 53 pp. 449)
Descending Rational Points on Elliptic Curves to Smaller Fields In this paper, we study the Mordell-Weil group of an elliptic curve
as a Galois module. We consider an elliptic curve $E$ defined over a
number field $K$ whose Mordell-Weil rank over a Galois extension $F$ is
$1$, $2$ or $3$. We show that $E$ acquires a point (points) of
infinite order over a field whose Galois group is one of $C_n \times C_m$
($n= 1, 2, 3, 4, 6, m= 1, 2$), $D_n \times C_m$ ($n= 2, 3, 4, 6, m= 1, 2$),
$A_4 \times C_m$ ($m=1,2$), $S_4 \times C_m$ ($m=1,2$). Next, we consider
the case where $E$ has complex multiplication by the ring of integers $\o$
of an imaginary quadratic field $\k$ contained in $K$. Suppose that the
$\o$-rank over a Galois extension $F$ is $1$ or $2$. If $\k\neq\Q(\sqrt{-1})$
and $\Q(\sqrt{-3})$ and $h_{\k}$ (class number of $\k$) is odd, we show that
$E$ acquires positive $\o$-rank over a cyclic extension of $K$ or over a
field whose Galois group is one of $\SL_2(\Z/3\Z)$, an extension of
$\SL_2(\Z/3\Z)$ by $\Z/2\Z$, or a central extension by the dihedral group.
Finally, we discuss the relation of the above results to the vanishing of
$L$-functions.
Categories:11G05, 11G40, 11R32, 11R33 |
3. CJM 2001 (vol 53 pp. 310)
On a Product Related to the Cubic Gauss Sum, III We have seen, in the previous works [5], [6], that the argument of a
certain product is closely connected to that of the cubic Gauss sum.
Here the absolute value of the product will be investigated.
Keywords:Gauss sum, Lagrange resolvent Categories:11L05, 11R33 |
4. 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 |
5. 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 |