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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 structure---integral representation.
Categories:11S15, 20C32

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