Vol. 9, No. 2, 2015

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
$p$-adic Hodge theory in rigid analytic families

Rebecca Bellovin

Vol. 9 (2015), No. 2, 371–433

We study the functors DB(V ), where B is one of Fontaine’s period rings and V is a family of Galois representations with coefficients in an affinoid algebra A. We first relate them to (φ,Γ)-modules, showing that DHT(V ) = iZ(DSen(V ) ti)ΓK, DdR(V ) = Ddif(V )ΓK, and Dcris(V ) = Drig(V )[1t]ΓK; this generalizes results of Sen, Fontaine, and Berger. We then deduce that the modules DHT(V ) and DdR(V ) are coherent sheaves on Sp(A), and Sp(A) is stratified by the ranks of submodules DHT[a,b](V ) and DdR[a,b](V ) of “periods with Hodge–Tate weights in the interval [a,b]@”. Finally, we construct functorial B-admissible loci in Sp(A), generalizing a result of Berger and Colmez to the case where A is not necessarily reduced.

$p$-adic Hodge theory, rigid analytic geometry
Mathematical Subject Classification 2010
Primary: 11S20
Secondary: 14G22
Received: 23 January 2014
Revised: 21 November 2014
Accepted: 25 December 2014
Published: 5 March 2015
Rebecca Bellovin
Department of Mathematics
University of California, Berkeley
Evans Hall
Berkeley, CA 94720
United States