Plectic structures in p-adic de Rham cohomology

TL;DR

This paper investigates plectic structures in p-adic de Rham cohomology of Hilbert modular varieties, proving a conjecture in quadratic cases and providing evidence for the general case through canonical splittings.

Contribution

It proves the plectic comparison conjecture for quadratic totally real fields and offers partial results and evidence for the conjecture in general fields.

Findings

Proved plectic comparison conjecture for degree 2 fields.

Established canonical splitting of plectic Hodge filtration.

Provided evidence supporting the conjecture in higher degree cases.

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)