Plectic structures in p-adic de Rham cohomology

Abstract

Given a Hilbert modular form for a totally real field $F$, and a prime $p$ split completely in $F$, the $f$-eigenspace in $p$-adic de Rham cohomology of the Hilbert modular variety has a family of partial filtrations and partial Frobenius maps, indexed by the primes of $F$ above $p$. The general plectic conjectures of Nekovar and Scholl suggest a "plectic comparison isomorphism" comparing these structures to etale cohomology. We prove this conjecture in the case $[F:\mathbf{Q}]=2$ under some mild assumptions; and for general $F$ we prove a weaker statement which is strong evidence for the conjecture, showing that plectic Hodge filtration has a canonical splitting given by intersecting with simultaneous eigenspaces for the partial Frobenii. (In memory of Jan Nekovar)

Revised version with minor corrections. To appear in Journal of Number Theory

References in corpus (1)

• On Goren-Oort stratification for quaternionic Shimura varieties

Cited by in corpus (1)

• The plectic conjecture over local fields