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1. CJM 2010 (vol 62 pp. 1011)
Functoriality of the Canonical Fractional Galois Ideal
The fractional Galois ideal
is a conjectural improvement on the higher Stickelberger
ideals defined at negative integers, and is expected to provide
non-trivial annihilators for higher $K$-groups of rings of integers of
number fields. In this article, we extend the definition of the
fractional Galois ideal to arbitrary (possibly infinite and
non-abelian) Galois extensions of number fields under the assumption
of Stark's conjectures and prove naturality properties under
canonical changes of extension. We discuss applications of this to the
construction of ideals in non-commutative Iwasawa algebras.
Categories:11R42, 11R23, 11R70 |
2. CJM 2010 (vol 62 pp. 1060)
Heegner Points over Towers of Kummer Extensions
Let $E$ be an elliptic curve, and let $L_n$ be the Kummer extension
generated by a primitive $p^n$-th root of unity and a $p^n$-th root of
$a$ for a fixed $a\in \mathbb{Q}^\times-\{\pm 1\}$. A detailed case study
by Coates, Fukaya, Kato and Sujatha and V. Dokchitser has led these
authors to predict unbounded and strikingly regular growth for the
rank of $E$ over $L_n$ in certain cases. The aim of this note is to
explain how some of these predictions might be accounted for by
Heegner points arising from a varying collection of Shimura curve
parametrisations.
Categories:11G05, 11R23, 11F46 |
3. CJM 2009 (vol 61 pp. 518)
Global Units Modulo Circular Units: Descent Without Iwasawa's Main Conjecture Iwasawa's classical asymptotical formula relates the orders of the $p$-parts $X_n$ of the ideal
class groups along a $\mathbb{Z}_p$-extension $F_\infty/F$ of a number
field $F$ to Iwasawa structural invariants $\la$ and $\mu$
attached to the inverse limit $X_\infty=\varprojlim X_n$.
It relies on ``good" descent properties satisfied by
$X_n$. If $F$ is abelian and $F_\infty$ is cyclotomic, it is known
that the $p$-parts of the orders of the global units modulo
circular units $U_n/C_n$ are asymptotically equivalent to the
$p$-parts of the ideal class numbers. This suggests that these
quotients $U_n/C_n$, so to speak unit class groups, also satisfy
good descent properties. We show this directly, \emph{i.e.,} without using Iwasawa's Main Conjecture.
Category:11R23 |
4. CJM 2005 (vol 57 pp. 812)
On the Vanishing of $\mu$-Invariants of Elliptic Curves over $\qq$ Let $E_{/\qq}$ be an elliptic curve with good ordinary reduction at a
prime $p>2$. It has a well-defined Iwasawa $\mu$-invariant $\mu(E)_p$
which encodes part of the information about the growth of the Selmer
group $\sel E{K_n}$ as $K_n$ ranges over the subfields of the
cyclotomic $\zzp$-extension $K_\infty/\qq$. Ralph Greenberg has
conjectured that any such $E$ is isogenous to a curve $E'$ with
$\mu(E')_p=0$. In this paper we prove Greenberg's conjecture for
infinitely many curves $E$ with a rational $p$-torsion point, $p=3$ or
$5$, no two of our examples having isomorphic $p$-torsion. The core
of our strategy is a partial explicit evaluation of the global duality
pairing for finite flat group schemes over rings of integers.
Category:11R23 |
5. CJM 2004 (vol 56 pp. 194)
Selmer Groups of Elliptic Curves with Complex Multiplication Suppose $K$ is an imaginary quadratic field and $E$ is an elliptic curve over a
number field $F$ with complex multiplication by the ring of integers in $K$.
Let $p$ be a rational prime that splits as $\mathfrak{p}_{1}\mathfrak{p}_{2}$
in $K$. Let $E_{p^{n}}$ denote the $p^{n}$-division points on $E$. Assume
that $F(E_{p^{n}})$ is abelian over $K$ for all $n\geq 0$. This paper proves
that the Pontrjagin dual of the $\mathfrak{p}_{1}^{\infty}$-Selmer group of
$E$ over $F(E_{p^{\infty}})$ is a finitely generated free $\Lambda$-module,
where $\Lambda$ is the Iwasawa algebra of $\Gal\bigl(F(E_{p^{\infty}})/
F(E_{\mathfrak{p}_{1}^{\infty}\mathfrak{p}_{2}})\bigr)$. It also gives a simple
formula for the rank of the Pontrjagin dual as a $\Lambda$-module.
Categories:11R23, 11G05 |
6. CJM 2003 (vol 55 pp. 673)
A Note on Cyclotomic Euler Systems and the Double Complex Method Let $\FF$ be a finite real abelian extension of $\QQ$. Let $M$ be an odd
positive integer. For every squarefree positive integer $r$ the prime
factors of which are congruent to $1$ modulo $M$ and split completely
in $\FF$, the corresponding Kolyvagin class $\kappa_r\in\FF^{\times}/
\FF^{\times M}$ satisfies a remarkable and crucial recursion which
for each prime number $\ell$ dividing $r$ determines the order of
vanishing of $\kappa_r$ at each place of $\FF$ above $\ell$ in terms
of $\kappa_{r/\ell}$. In this note we give the recursion a new and
universal interpretation with the help of the double complex method
introduced by Anderson and further developed by Das and Ouyang. Namely,
we show that the recursion satisfied by Kolyvagin classes is the
specialization of a universal recursion independent of $\FF$ satisfied
by universal Kolyvagin classes in the group cohomology of the universal
ordinary distribution {\it \`a la\/} Kubert tensored with $\ZZ/M\ZZ$.
Further, we show by a method involving a variant of the diagonal shift
operation introduced by Das that certain group cohomology classes belonging
(up to sign) to a basis previously constructed by Ouyang also satisfy the
universal recursion.
Categories:11R18, 11R23, 11R34 |
7. 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 |