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A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field

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A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field

Perez Buendia, Jesus Rogelio (2014) A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field. PhD thesis, Concordia University.

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Abstract

In this thesis we prove a $p$-adic analogous of the Kulikov-Persson-Pinkham classification theorem for the central fibre of a degeneration of $K3$-surfaces in terms of the nilpotency degree of the monodromy of the family \citep{Persson:1981wp}.

Namely, let $X_K$ be a be a smooth, projective $K3$-surface which has a minimal semistable model $X$ over $\mcal O_K$.
If we let $N_{st}$ be the monodromy operator on $D_{st}(H^2_{\et}(X_{\oln K},\Qp))$, then we prove that the degree of nilpotency of $N_{st}$ determines the type of the special fibre of $X$. As a consequence we give a criterion for the good reduction of the semi-stable $K3$-surface $X_K$ over the $p$-adic field $K$ in terms of its $p$-adic representation $H^2_{\et}(X_{\oln K},\Qp)$, which is similar to the criterion of good reduction for $p$-adic abelian varieties and curves given by \citep{Coleman:1997vg} and \citep{Iovita:2013ts}.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (PhD)
Authors:Perez Buendia, Jesus Rogelio
Institution:Concordia University
Degree Name:Ph. D.
Program:Mathematics
Date:10 January 2014
Thesis Supervisor(s):Iovita, Adrian
Keywords:Arithmetic, Geometry, Number Theory, p-adic Geoemtry, K3-Surfaces
ID Code:978195
Deposited By: JESUS-ROGELIO PEREZ-BUENDIA
Deposited On:16 Jun 2014 14:07
Last Modified:18 Jan 2018 17:46
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