1. CJM Online first
 Bijakowski, Stephane

Partial Hasse invariants, partial degrees, and the canonical subgroup
If the Hasse invariant of a $p$divisible group is small enough,
then one can construct a canonical subgroup inside its $p$torsion.
We prove that, assuming the existence of a subgroup of adequate
height in the $p$torsion with high degree, the expected properties
of the canonical subgroup can be easily proved, especially the
relation between its degree and the Hasse invariant. When one
considers a $p$divisible group with an action of the ring of
integers of a (possibly ramified) finite extension of $\mathbb{Q}_p$,
then much more can be said. We define partial Hasse invariants
(they are natural in the unramified case, and generalize a construction
of Reduzzi and Xiao in the general case), as well as partial
degrees. After studying these functions, we compute the partial
degrees of the canonical subgroup.
Keywords:canonical subgroup, Hasse invariant, $p$divisible group Categories:11F85, 11F46, 11S15 

2. CJM 2000 (vol 52 pp. 1269)
 Spriano, Luca

Well Ramified Extensions of Complete Discrete Valuation Fields with Applications to the Kato Conductor
We study extensions $L/K$ of complete discrete valuation fields $K$
with residue field $\oK$ of characteristic $p > 0$, which we do not
assume to be perfect. Our work concerns ramification theory for such
extensions, in particular we show that all classical properties which
are true under the hypothesis {\it ``the residue field extension
$\oL/\oK$ is separable''} are still valid under the more general
hypothesis that the valuation ring extension is monogenic. We also
show that conversely, if classical ramification properties hold true
for an extension $L/K$, then the extension of valuation rings is
monogenic. These are the ``{\it well ramified}'' extensions. We show
that there are only three possible types of well ramified extensions
and we give examples. In the last part of the paper we consider, for
the three types, Kato's generalization of the conductor, which we show
how to bound in certain cases.
Categories:11S, 11S15, 11S20 

3. CJM 1998 (vol 50 pp. 1007)
 Elder, G. Griffith

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 

4. CJM 1997 (vol 49 pp. 722)
 Elder, G. Griffith; Madan, Manohar L.

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 structureintegral representation. Categories:11S15, 20C32 
